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This paper is available on arxiv under CC 4.0 license.
Authors:
(1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1
2
3 Proof of Theorem 1
References 3 Proof of Theorem 1 Remark. If N ≥ T then the simple Theorem 4.11 of [Tit1951] (provable by simple partial summation) yields Similarly to (4.2) of [Pin2023] we start with Summarizing the above we get This paper is available on arxiv under CC 4.0 license. Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1 2 3 Proof of Theorem 1 References 1 1 2 2 3 Proof of Theorem 1 3 Proof of Theorem 1 References References 3 Proof of Theorem 1 Remark. If N ≥ T then the simple Theorem 4.11 of [Tit1951] (provable by simple partial summation) yields Remark. Similarly to (4.2) of [Pin2023] we start with Summarizing the above we get

A Remark on Density Theorems for Riemann’s Zeta-function: Proof of Theorem 1

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1
2
3 Proof of Theorem 1
References 2 The goal of the present work is (i) to give a possibly simple proof for all values of σ > 3/4, i.e. η < 1/4, which (ii) breaks the DH for the non-negligible range σ ≥ 7/8, i.e. η ≤ 1/8, further gives (iii) explicit density theorems for different ranges of all η < 1/4 with B(η) → 0 as η → 0 and (iv) besides the (possibly indispensable) simple idea of Hal´asz uses only classical knowledge of the theory of Riemann’s zeta-function (classical in the sense that known since at least hundred years). This means that we avoid the use of Tur´an’s method, the large sieve, Vinogradov’s method, further mean and large value theorems. However, to break the DH we need the estimate reached by the Weyl–Hardy–Littlewood method [Lit1922] (see also (5.15.1) of the book of Titchmarsh [Tit1951], 2nd edition, or the paper of Landau [Lan1924]): which is proved by Hardy and Littlewood in [Lit1922] for ℓ ≥ 1, but it is also true for ℓ = 0 (µ(0) = 1/2) by the functional equation and µ(1) = 0. Alternatively, the method of Van der Corput would yield results of similar type [Cor1921], [Cor1922]. However, the use of (2.1) seems to be simpler, although Van der Corput’s result may yield even stronger estimates for N(1 − η, T) in many ranges of η. In order to formulate our results we need some more notation. For ℓ ≥ 1 let By the estimate (2.1) of Hardy and Littlewood we have for η < 1/2 ℓ Theorem 1. For ℓ ≥ 1, η < max(1/3, 2 −ℓ) we have This implies a series of explicit density theorems for small values of η and the conditional result of Hal´asz and Tur´an [HT1969] that assuming (LH) we have N(σ, T) ≪ Tε for σ > 3/4, which is for σ > 3/4 a strong improvement of the theorem of Ingham [Ing1937] stating A(σ) = 2 for σ > 1/2 on LH. Corollary 1. The Lindel¨of Hypothesis implies Proof. Choosing ℓ = 2, LH and the definition (2.4)–(2.5) yield uℓ(η) = vℓ(η) = 0 for η < 1/4. Corollary 2. For ℓ ≥ 2, η < 1/2ℓ we have Proof. 4/(ℓ + 2) ≥ 3/(ℓ + 1) for ℓ ≥ 2 with equality for ℓ = 2. Denoting by log2 x the logarithm of base 2 and choosing C large (C = C(η) → ∞ as η → 0, but with C(η) = o(log 1/η)), we obtain for η ∈ [1/2 ℓ(1+o(1)) , 1/2 ℓ+C] from (2.7). This is naturally weaker than a result obtainable by Vinogradov’s method (see [Pin2023] and [Hea2017]) but shows still B(η) → 0 as η → 0 beyond breaking DH in some interval for σ = 1 − η close to the point s = 1. However, Corollary 2 yields also interesting (although already known) consequences form small values of ℓ (ℓ = 2, 3, for example), i.e., for relatively large ranges of η. On the other hand, they are often stronger than earlier results proved in a more complicated way (as 11.30 of Ivic [Ivi1985], for example). This breaks the DH also for η ≤ 1/8, that is, for σ ≥ 7/8, but it is weaker than Corollary 4 for η < 1/8 (sharper for η > 1/8). Remark 1. Corollary 6 is trivial for η ≥ 2/7, i.e. σ ≤ 5/7. Remark 2. It is an interesting feature of Corollary 6 that it gives a non-trivial result even in the range η ∈ [1/4, 2/7] in contrary to most applications of Hal´asz’s method, although it is weaker there than the classical theorem of Carlson [Car1920] or the one of Ingham [Ing1940]. Remark 3. It is also interesting to note that the case ℓ = 1 would yield the inequality of Corollary 6 with any estimate µ(1/2) ≤ 3/16 and any estimate of type µ(1/2) ≤ 1/4 − c2 (c2 > 0) would break the DH for η < c3(c2) with c3(c2) > 0. Finally, similarly to Theorem 1 we can give an alternative proof of the case (12.10) of Theorem 12.1 of Montgomery [Mon1971] which states Remark 4. This represents a slight improvement of the Theorem of Tur´an (cf. (1.12)–(1.13)) since Remark 5. The proof of Theorem 2 does not need for any ℓ the estimate (2.1) of Hardy and Littlewood or the estimate of Van der Corput (apart from the “trivial” case µ(0) = 1/2 following from the functional equation). This paper is available on arxiv under CC 4.0 license. Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1 2 3 Proof of Theorem 1 References 1 1 2 2 3 Proof of Theorem 1 3 Proof of Theorem 1 References References 2 The goal of the present work is (i) to give a possibly simple proof for all values of σ > 3/4, i.e. η < 1/4, which (ii) breaks the DH for the non-negligible range σ ≥ 7/8, i.e. η ≤ 1/8, further gives (iii) explicit density theorems for different ranges of all η < 1/4 with B(η) → 0 as η → 0 and (iv) besides the (possibly indispensable) simple idea of Hal´asz uses only classical knowledge of the theory of Riemann’s zeta-function (classical in the sense that known since at least hundred years). This means that we avoid the use of Tur´an’s method, the large sieve, Vinogradov’s method, further mean and large value theorems. However, to break the DH we need the estimate reached by the Weyl–Hardy–Littlewood method [Lit1922] (see also (5.15.1) of the book of Titchmarsh [Tit1951], 2nd edition, or the paper of Landau [Lan1924]): which is proved by Hardy and Littlewood in [Lit1922] for ℓ ≥ 1, but it is also true for ℓ = 0 (µ(0) = 1/2) by the functional equation and µ(1) = 0. Alternatively, the method of Van der Corput would yield results of similar type [Cor1921], [Cor1922]. However, the use of (2.1) seems to be simpler, although Van der Corput’s result may yield even stronger estimates for N(1 − η, T) in many ranges of η. In order to formulate our results we need some more notation. For ℓ ≥ 1 let By the estimate (2.1) of Hardy and Littlewood we have for η < 1/2 ℓ Theorem 1. For ℓ ≥ 1, η < max(1/3, 2 −ℓ) we have Theorem 1. This implies a series of explicit density theorems for small values of η and the conditional result of Hal´asz and Tur´an [HT1969] that assuming (LH) we have N(σ, T) ≪ Tε for σ > 3/4, which is for σ > 3/4 a strong improvement of the theorem of Ingham [Ing1937] stating A(σ) = 2 for σ > 1/2 on LH. Corollary 1. The Lindel¨of Hypothesis implies Proof. Choosing ℓ = 2, LH and the definition (2.4)–(2.5) yield uℓ(η) = vℓ(η) = 0 for η < 1/4. Corollary 2. For ℓ ≥ 2, η < 1/2ℓ we have Proof. 4/(ℓ + 2) ≥ 3/(ℓ + 1) for ℓ ≥ 2 with equality for ℓ = 2. Denoting by log2 x the logarithm of base 2 and choosing C large (C = C(η) → ∞ as η → 0, but with C(η) = o(log 1/η)), we obtain for η ∈ [1/2 ℓ(1+o(1)) , 1/2 ℓ+C] from (2.7). This is naturally weaker than a result obtainable by Vinogradov’s method (see [Pin2023] and [Hea2017]) but shows still B(η) → 0 as η → 0 beyond breaking DH in some interval for σ = 1 − η close to the point s = 1. However, Corollary 2 yields also interesting (although already known) consequences form small values of ℓ (ℓ = 2, 3, for example), i.e., for relatively large ranges of η. On the other hand, they are often stronger than earlier results proved in a more complicated way (as 11.30 of Ivic [Ivi1985], for example). This breaks the DH also for η ≤ 1/8, that is, for σ ≥ 7/8, but it is weaker than Corollary 4 for η < 1/8 (sharper for η > 1/8). Remark 1. Corollary 6 is trivial for η ≥ 2/7, i.e. σ ≤ 5/7. Remark 2. It is an interesting feature of Corollary 6 that it gives a non-trivial result even in the range η ∈ [1/4, 2/7] in contrary to most applications of Hal´asz’s method, although it is weaker there than the classical theorem of Carlson [Car1920] or the one of Ingham [Ing1940]. Remark 3. It is also interesting to note that the case ℓ = 1 would yield the inequality of Corollary 6 with any estimate µ(1/2) ≤ 3/16 and any estimate of type µ(1/2) ≤ 1/4 − c2 (c2 > 0) would break the DH for η < c3(c2) with c3(c2) > 0. Finally, similarly to Theorem 1 we can give an alternative proof of the case (12.10) of Theorem 12.1 of Montgomery [Mon1971] which states Remark 4. This represents a slight improvement of the Theorem of Tur´an (cf. (1.12)–(1.13)) since Remark 5. The proof of Theorem 2 does not need for any ℓ the estimate (2.1) of Hardy and Littlewood or the estimate of Van der Corput (apart from the “trivial” case µ(0) = 1/2 following from the functional equation).

A Remark on Density Theorems for Riemann’s Zeta-function: 2

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1
2
3 Proof of Theorem 1
References References [BL1914] Bohr, H., Landau, E., Sur les zeros de la fonction ζ(s) de Riemann, Comptes Rendus de l’Acad. des Sciences (Paris) 158 (1914), 106–110. [Bom1971] Bombieri, Enrico, Density theorems for the zeta function, Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 352–358. Amer. Math. Soc., Providence, R.I., 1971. [Bou1] J. Bourgain, Remarks on Hal´asz–Montgomery type inequalities, in: Geometric aspects of functional analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77, Birkh¨auser, Basel, 1995, 25–39. [Bou2000] Bourgain, J., On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function, Internat. Math. Res. Notices (2000), no. 3, 133–146. [BGD2016] Bourgain, J., Demeter, C., Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633-–682. [Car1920] Carlson, F., Uber die Nullstellen der Dirichletschen Reihen und der ¨ Riemannschen ζ-Funktion, Arkiv f¨ur Math. Astron. Fys. 15 (20) (1920), 1–18. [Cor1921] van der Corput, J. G., Zahlentheoretische Absch¨atzungen, Math. Ann. 84 (1921), 53–79. [Cor1922] van der Corput, J. G., Versch¨arfung der Absch¨atzungen beim Teilerproblem, Math. Ann. 87 (1922), 39–65. [Cra1921] Cram´er, H., Some theorems concerning prime numbers, Ark. Mat. Astron. Fys. 15(5 (1921), 1–33. [Hal1968] Hal´asz, G., Uber die Mittelwerte multiplikativer zahlentheoretische ¨ r Funktionen, Acta Math. Hungar. 19 (1968), No. 3-4, 365–404. [HT1969] Hal´asz, G. and P. Tur´an, P., On the distribution of roots of Riemann zeta and allied functions, I, Journal of Number Theory 1 (1969), 121–137. [Hea2017] Heath-Brown, D. R., A new kth derivative estimate for a trigonometric sum via Vinogradov’s integral (in Russian). English version published in Proc. Steklov Inst. Math. 296 (2017), 88–103. [Hoh1930] Hoheisel, G., Primzahlprobleme in der Analysis, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. (1930), 580–588. [Hux1972] Huxley, M. N., On the difference between consecutive primes, Invent. Math. 15 (1972), 164–170. [Ing1937] Ingham, A. E., On the difference between consecutive primes, Quart. J. Math. Oxford Ser. 8 (1937), 255–266. [Ing1940] Ingham, A. E., On the estimation of N(σ, T), Quart. J. Math. Oxford Ser. 11 (1940), 291–292. [Ivi1985] Ivic, A., The Riemann Zeta-function, Wiley, 1985. [Jut1977] Jutila, Matti, Zero-density estimates for L-functions, Acta Arithmetica 32 (1977), no. 1, 55–62. [Lan1913] Landau, E., Gel¨oste und ungel¨oste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. In: Proc. 5th Internat. Congress of Math., Vol. 1, 93–108. Cambridge University Press, Cambridge 1913. [Lan1924] Landau, E., Uber die ¨ ζ-Funktion und die L-funktionen, Math. Z. 20 (1924), 98–104. [Lit1922] Littlewood, J. E., Researches in the theory of the Riemann ζ-function, Proc. London Math. Soc. (2) 20 (1922), Records xxii–xxviii. [Mon1969] Montgomery, H. L., Zeros of L-functions, Invent. Math. 8 (1969), 346– 354. [Mon1971] Montgomery, Hugh L., Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin–New York, 1971. ix+178 pp. [Per1908] Perron, O., Zur Theorie der Dirichletschen L-Reihen, J. Reine Angew. Math. 134 (1908), 95–143. [Pin2022] Pintz, J., On the density theorem of Hal´asz and Tur´an, Acta Math. Hungar. 166 (2022), no. 1, 48–56. [Pin2023] Pintz, J., Density theorems for Riemann’s zeta-function near the line Re s = 1, Acta Arithmetica, to appear (online first version DOI: 10.4064/aa210824-10-5). [Tit1951] Titchmarsh, E. C., The theory of the Riemann zeta-function, Oxford, Clarendon Press (2nd edition), 1951. [Tur1953] Tur´an P´al, Az anal´ızis egy ´uj m´odszer´er˝ol ´es annak egyes alkalmaz´asair´ol (in Hungarian) [On a new method in analysis and on some of its applications], Akad´emiai Kiad´o, Budapest, 1953. 197 pp. [Tur1954] Tur´an P´al, On the roots of the Riemann zeta function (in Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. K¨ozl. 4 (1954), 357–368. [Tur1984] P. Tur´an, On a new method of analysis and its applications, John Wiley & Sons, Inc., New York, 1984. xvi+584 pp. This paper is available on arxiv under CC 4.0 license. Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1 2 3 Proof of Theorem 1 References 1 1 2 2 3 Proof of Theorem 1 3 Proof of Theorem 1 References References References [BL1914] Bohr, H., Landau, E., Sur les zeros de la fonction ζ(s) de Riemann, Comptes Rendus de l’Acad. des Sciences (Paris) 158 (1914), 106–110. [Bom1971] Bombieri, Enrico, Density theorems for the zeta function, Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 352–358. Amer. Math. Soc., Providence, R.I., 1971. [Bou1] J. Bourgain, Remarks on Hal´asz–Montgomery type inequalities, in: Geometric aspects of functional analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77, Birkh¨auser, Basel, 1995, 25–39. [Bou2000] Bourgain, J., On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function, Internat. Math. Res. Notices (2000), no. 3, 133–146. [BGD2016] Bourgain, J., Demeter, C., Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633-–682. [Car1920] Carlson, F., Uber die Nullstellen der Dirichletschen Reihen und der ¨ Riemannschen ζ-Funktion, Arkiv f¨ur Math. Astron. Fys. 15 (20) (1920), 1–18. [Cor1921] van der Corput, J. G., Zahlentheoretische Absch¨atzungen, Math. Ann. 84 (1921), 53–79. [Cor1922] van der Corput, J. G., Versch¨arfung der Absch¨atzungen beim Teilerproblem, Math. Ann. 87 (1922), 39–65. [Cra1921] Cram´er, H., Some theorems concerning prime numbers, Ark. Mat. Astron. Fys. 15(5 (1921), 1–33. [Hal1968] Hal´asz, G., Uber die Mittelwerte multiplikativer zahlentheoretische ¨ r Funktionen, Acta Math. Hungar. 19 (1968), No. 3-4, 365–404. [HT1969] Hal´asz, G. and P. Tur´an, P., On the distribution of roots of Riemann zeta and allied functions, I, Journal of Number Theory 1 (1969), 121–137. [Hea2017] Heath-Brown, D. R., A new kth derivative estimate for a trigonometric sum via Vinogradov’s integral (in Russian). English version published in Proc. Steklov Inst. Math. 296 (2017), 88–103. [Hoh1930] Hoheisel, G., Primzahlprobleme in der Analysis, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. (1930), 580–588. [Hux1972] Huxley, M. N., On the difference between consecutive primes, Invent. Math. 15 (1972), 164–170. [Ing1937] Ingham, A. E., On the difference between consecutive primes, Quart. J. Math. Oxford Ser. 8 (1937), 255–266. [Ing1940] Ingham, A. E., On the estimation of N(σ, T), Quart. J. Math. Oxford Ser. 11 (1940), 291–292. [Ivi1985] Ivic, A., The Riemann Zeta-function, Wiley, 1985. [Jut1977] Jutila, Matti, Zero-density estimates for L-functions, Acta Arithmetica 32 (1977), no. 1, 55–62. [Lan1913] Landau, E., Gel¨oste und ungel¨oste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. In: Proc. 5th Internat. Congress of Math., Vol. 1, 93–108. Cambridge University Press, Cambridge 1913. [Lan1924] Landau, E., Uber die ¨ ζ-Funktion und die L-funktionen, Math. Z. 20 (1924), 98–104. [Lit1922] Littlewood, J. E., Researches in the theory of the Riemann ζ-function, Proc. London Math. Soc. (2) 20 (1922), Records xxii–xxviii. [Mon1969] Montgomery, H. L., Zeros of L-functions, Invent. Math. 8 (1969), 346– 354. [Mon1971] Montgomery, Hugh L., Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin–New York, 1971. ix+178 pp. [Per1908] Perron, O., Zur Theorie der Dirichletschen L-Reihen, J. Reine Angew. Math. 134 (1908), 95–143. [Pin2022] Pintz, J., On the density theorem of Hal´asz and Tur´an, Acta Math. Hungar. 166 (2022), no. 1, 48–56. [Pin2023] Pintz, J., Density theorems for Riemann’s zeta-function near the line Re s = 1, Acta Arithmetica, to appear (online first version DOI: 10.4064/aa210824-10-5). [Tit1951] Titchmarsh, E. C., The theory of the Riemann zeta-function, Oxford, Clarendon Press (2nd edition), 1951. [Tur1953] Tur´an P´al, Az anal´ızis egy ´uj m´odszer´er˝ol ´es annak egyes alkalmaz´asair´ol (in Hungarian) [On a new method in analysis and on some of its applications], Akad´emiai Kiad´o, Budapest, 1953. 197 pp. [Tur1954] Tur´an P´al, On the roots of the Riemann zeta function (in Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. K¨ozl. 4 (1954), 357–368. [Tur1984] P. Tur´an, On a new method of analysis and its applications, John Wiley & Sons, Inc., New York, 1984. xvi+584 pp.

A Remark on Density Theorems for Riemann’s Zeta-function: Reference

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1
2
3 Proof of Theorem 1
References Keywords and phrases: Riemann’s zeta function, density hypothesis, density theorems. 2020 Mathematics Subject Classification: Primary 11M26, Secondary 11M06. 1 The plausibility of the Riemann Hypothesis (RH) is supported (among others) by results of type (1.1) valid for all σ > 1/2 as T → ∞. (1.1) was first shown by Bohr and Landau in 1914 ([BL1914]). The first estimate of type was shown few years later by Carlson [Car1920] with It was Hoheisel [Hoh1930] who first observed that such estimates lead to arithmetic consequences about the difference of consecutive primes. He proved the first approximation towards the famous conjecture that there exists always a prime between two consecutive squares. This conjecture was characterised by Landau [Lan1913] in his plenary talk at ICM1912 in Cambridge as one of the four main problems of the distribution of primes, besides the Riemann Hypothesis. Hoheisel [Hoh1930] could show (pn denotes the nth prime) In the proof important role was played by Carlson’s density theorem (1.2)– (1.3). Later it was realized that a uniform estimate of the form yields (combined with a slightly better zero-free region than the classical one of de la Vall´ee Poussin) In particular, the best possible estimate A = 2 would almost imply Landau’s conjecture. This is especially remarkable in light of the fact that even assuming the Riemann Hypothesis the best estimate we know is (with C = 2 easily by the Riemann–Von Mangoldt explicit formula, with C = 1 by a deeper argument of Cram´er [Cra1921]). This explains the significance of the Density Hypothesis (DH) which states or, in a slightly weaker form, using the notation (1.2), The Riemann Hypothesis clearly implies the Density Hypothesis. However, Ingham [Ing1937] showed that also the Lindel¨of Hypothesis (LH) (1.10) implies DH. In 1954 Tur´an [Tur1954] used his celebrated power-sum method [Tur1953], [Tur1984] to give a different proof of Ingham’s result that LH implies DH. In the same work he came very close to breaking the DH in the vicinity of the boundary line σ = 1. In the following we will use the notation: His result was with a small constant c1 [Tur1954] or with our new notation It was 16 years later when G. Hal´asz and Tur´an succeeded to break the DH [HT1969], that is, to prove (in the refined form appearing as Theorem 38.2 of Tur´an’s book [Tur1984]) with the notation (1.1)–(1.2), (1.11) The proof was based on (i) Vinogradov’s estimate µ(1 − η) ≪ η 3/2, (ii) Tur´an’s power-sum method; (iii) a simple but ingenious idea of Hal´asz [Hal1968]. Soon after this Bombieri [Bom1971] gave a different proof using ideas from the large sieve coupled with (i) and (iii). Further, Montgomery has shown the DH for σ ≥ 9/10, i.e. η ≤ 1/10 ([Mon1969], [Mon1971]). Using another simple but ingenious idea Huxley [Hux1972] (with a refinement of Montgomery) reached the DH for η ≤ 1/6 and proved that The DH was shown for larger and larger ranges, until improving the result σ > 11/14 of Jutila [Jut1977] Bourgain proved its validity [Bou2000] for Another direction of the research was to give strong density estimates for small values of η (often especially for η → 0). In this direction the most important results were reached (in alphabetical order) by Bourgain, Ford, Heath-Brown, Huxley, Ivic, Jutila and Montgomery. All these results use the large sieve and in some form the mentioned idea of Hal´asz, which needs η ≤ 1/4. This is also the reason that they cannot give improvements of the classical zero-density theorem of Ingham [Ing1940], A(σ) ≤ 3/(2 − σ), which is still the best today in the whole range Recently I gave two alternative variants for proving the DH for small values of η. In [Pin2022] the goal was to reach a possibly simple proof while in [Pin2023] to show the strongest possible result both in specific ranges of η < 1/12 and for η → 0. Both proofs were based on Vinogradov’s method and Hal´asz’s idea. However, in the second [Pin2023] it was important to use recent deep results of Heath-Brown [Hea2017], further of Bourgain, Demeter and Guth [BGD2016]. This paper is available on arxiv under CC 4.0 license. Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1 2 3 Proof of Theorem 1 References 1 1 2 2 3 Proof of Theorem 1 3 Proof of Theorem 1 References References Keywords and phrases: Riemann’s zeta function, density hypothesis, density theorems. Keywords and phrases: 2020 Mathematics Subject Classification: Primary 11M26, Secondary 11M06. 1 The plausibility of the Riemann Hypothesis (RH) is supported (among others) by results of type (1.1) valid for all σ > 1/2 as T → ∞. (1.1) was first shown by Bohr and Landau in 1914 ([BL1914]). The first estimate of type was shown few years later by Carlson [Car1920] with It was Hoheisel [Hoh1930] who first observed that such estimates lead to arithmetic consequences about the difference of consecutive primes. He proved the first approximation towards the famous conjecture that there exists always a prime between two consecutive squares. This conjecture was characterised by Landau [Lan1913] in his plenary talk at ICM1912 in Cambridge as one of the four main problems of the distribution of primes, besides the Riemann Hypothesis. Hoheisel [Hoh1930] could show (pn denotes the nth prime) In the proof important role was played by Carlson’s density theorem (1.2)– (1.3). Later it was realized that a uniform estimate of the form yields (combined with a slightly better zero-free region than the classical one of de la Vall´ee Poussin) In particular, the best possible estimate A = 2 would almost imply Landau’s conjecture. This is especially remarkable in light of the fact that even assuming the Riemann Hypothesis the best estimate we know is (with C = 2 easily by the Riemann–Von Mangoldt explicit formula, with C = 1 by a deeper argument of Cram´er [Cra1921]). This explains the significance of the Density Hypothesis (DH) which states or, in a slightly weaker form, using the notation (1.2), The Riemann Hypothesis clearly implies the Density Hypothesis. However, Ingham [Ing1937] showed that also the Lindel¨of Hypothesis (LH) (1.10) implies DH. In 1954 Tur´an [Tur1954] used his celebrated power-sum method [Tur1953], [Tur1984] to give a different proof of Ingham’s result that LH implies DH. In the same work he came very close to breaking the DH in the vicinity of the boundary line σ = 1. In the following we will use the notation: His result was with a small constant c1 [Tur1954] or with our new notation It was 16 years later when G. Hal´asz and Tur´an succeeded to break the DH [HT1969], that is, to prove (in the refined form appearing as Theorem 38.2 of Tur´an’s book [Tur1984]) with the notation (1.1)–(1.2), (1.11) The proof was based on (i) Vinogradov’s estimate µ(1 − η) ≪ η 3/2, (ii) Tur´an’s power-sum method; (iii) a simple but ingenious idea of Hal´asz [Hal1968]. Soon after this Bombieri [Bom1971] gave a different proof using ideas from the large sieve coupled with (i) and (iii). Further, Montgomery has shown the DH for σ ≥ 9/10, i.e. η ≤ 1/10 ([Mon1969], [Mon1971]). Using another simple but ingenious idea Huxley [Hux1972] (with a refinement of Montgomery) reached the DH for η ≤ 1/6 and proved that The DH was shown for larger and larger ranges, until improving the result σ > 11/14 of Jutila [Jut1977] Bourgain proved its validity [Bou2000] for Another direction of the research was to give strong density estimates for small values of η (often especially for η → 0). In this direction the most important results were reached (in alphabetical order) by Bourgain, Ford, Heath-Brown, Huxley, Ivic, Jutila and Montgomery. All these results use the large sieve and in some form the mentioned idea of Hal´asz, which needs η ≤ 1/4. This is also the reason that they cannot give improvements of the classical zero-density theorem of Ingham [Ing1940], A(σ) ≤ 3/(2 − σ), which is still the best today in the whole range Recently I gave two alternative variants for proving the DH for small values of η. In [Pin2022] the goal was to reach a possibly simple proof while in [Pin2023] to show the strongest possible result both in specific ranges of η < 1/12 and for η → 0. Both proofs were based on Vinogradov’s method and Hal´asz’s idea. However, in the second [Pin2023] it was important to use recent deep results of Heath-Brown [Hea2017], further of Bourgain, Demeter and Guth [BGD2016].

A Remark on Density Theorems for Riemann’s Zeta-function: 1

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References Conclusion In this mathematical study, we delve into the realm of statistical inference and introduce a novel approach to variational non-Bayesian inference. Most significantly, we propose a new method for uniquely determining the hidden PDF solely from a random sample while leveraging theory. Beyond merely approximating the shape of the unrevealed PDF, we yield results that provide practical assistance in inferring important moments, such as the mean and variance, from the estimated PDF. These days, methods utilizing artificial intelligence are widely employed, but they generally exhibit a drawback of depending on initial conditions, i.e., a prior distribution, and iterative computations, i.e., a backpropagation. The limitations of such approaches lie in their reliance on these conditions to achieve global optimization. We emphasize the significant contribution of the proposed method to predictive and classification models, even without any information on populations, highlighting its potential applications in various domains such as finance, economics, weather forecasting, and machine learning, all of which present unique challenges. The most well-known method for approximating hidden PDFs is the variational approach in a Bayesian context. In contrast, our study extends this problem by seeking to precisely determine unknown PDFs through a system of equations. Our approach includes proving the Fréchet differentiability of entropy to establish the uniqueness of the energy function space in the Wiener algebra. We then derive the unique determination of the energy function through the minimization of KL-divergence. Leveraging the Ergodic theorem, we elucidate that solutions to equations comprising polynomial function series are the coefficients of the energy function and numerically substantiate the convergence of partial sums of the energy function obtained from a finite number of equations. In summary, our mathematical exploration has unveiled the potential of variational Non-Bayesian inference in Wiener space. We anticipate that these mathematical ideas can offer an innovative framework for probability density estimation and predictive modeling. Ultimately, our research emphasizes its potential contribution to reshaping mathematical horizons and expanding the boundaries of knowledge in the field of statistical methodology. REFERENCES [1] K. Bharath A. Saha and S. Kurtek, A geometric variational approach to bayesian inference, Journal of the American Statistical Association 115 (2020), no. 530, 822–835. [2] P. Alquier and J. Ridgway, Concentration of tempered posteriors and of their variational approximations, The Annals of Statistics 48 (2020), no. 3, 1475–1497. [3] A. R. Barron and C. Sheu, Approximation of density functions by sequences of exponential families, The Annals of Statistics 19 (1991), no. 3, 1347–1369. [4] C. M. Bishop and M. E. Tipping, Probabilistic principal component analysis, J. R. Statist. Soc. B 61 (1999), no. 3, 611–622. [5] F. W. Olver, D. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. [6] C Brouder and F Patras, One-particle irreducibility with initial correlations, (2011). [7] I. Csiszar, I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975), no. 1, 146–158. [8] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (1989), 303–314. [9] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2003. [10] D. E. Rumelhart, G. E. Hinton and R. Williams, Learning representations by back-propagating errors, Nature 323 (1986), 533–536. [11] G. E. Hinton, A Practical Guide to Training Restricted Boltzmann Machines, University of Toronto, 2010. [12] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (1991), no. 2, 251–257. [13] Y. Katznelson, An introduction to harmonic analysis (third ed.), New York: Cambridge Mathematical Library, 2004. [14] D. M. Blei, A. Kucukelbir and J. D. McAuliffe, Variational inference: A review for statisticians, Jour. of the American Statistical Association 112 (2017), no. 518, 859–877. [15] S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), no. 1, 79–86. [16] H. M. Möller and R. Tenberg, Multivariate polynomial system solving using intersections of eigenspaces, Journal of Symbolic Computation 32 (2001), no. 5, 513–531. [17] B. Mourrain, V. Y. Pan and O. Ruatta, Accelerated solution of multivariate polynomial systems of equations, SIAM Journal on Computing 32 (2003), no. 2, 435–454. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] D. W. Stroock, Probability theory, an analytic view, Cambridge University Press, 1993. [20] B. Sturmfels, Solving systems of polynomial equations (cbms regional conference series in mathematics), Amer. Math. Soc., 2002. [21] N. Wiener, Tauberian theorems, Annals of Mathematics 33 (1932), no. 1, 1–100. [22] F. Zhang and C. Gao, Convergence rates of variational posterior distributions, The Annals of Statistics 48 (2020), no. 4, 2180–2207. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References Conclusion In this mathematical study, we delve into the realm of statistical inference and introduce a novel approach to variational non-Bayesian inference. Most significantly, we propose a new method for uniquely determining the hidden PDF solely from a random sample while leveraging theory. Beyond merely approximating the shape of the unrevealed PDF, we yield results that provide practical assistance in inferring important moments, such as the mean and variance, from the estimated PDF. These days, methods utilizing artificial intelligence are widely employed, but they generally exhibit a drawback of depending on initial conditions, i.e., a prior distribution, and iterative computations, i.e., a backpropagation. The limitations of such approaches lie in their reliance on these conditions to achieve global optimization. We emphasize the significant contribution of the proposed method to predictive and classification models, even without any information on populations, highlighting its potential applications in various domains such as finance, economics, weather forecasting, and machine learning, all of which present unique challenges. The most well-known method for approximating hidden PDFs is the variational approach in a Bayesian context. In contrast, our study extends this problem by seeking to precisely determine unknown PDFs through a system of equations. Our approach includes proving the Fréchet differentiability of entropy to establish the uniqueness of the energy function space in the Wiener algebra. We then derive the unique determination of the energy function through the minimization of KL-divergence. Leveraging the Ergodic theorem, we elucidate that solutions to equations comprising polynomial function series are the coefficients of the energy function and numerically substantiate the convergence of partial sums of the energy function obtained from a finite number of equations. In summary, our mathematical exploration has unveiled the potential of variational Non-Bayesian inference in Wiener space. We anticipate that these mathematical ideas can offer an innovative framework for probability density estimation and predictive modeling. Ultimately, our research emphasizes its potential contribution to reshaping mathematical horizons and expanding the boundaries of knowledge in the field of statistical methodology. REFERENCES [1] K. Bharath A. Saha and S. Kurtek, A geometric variational approach to bayesian inference, Journal of the American Statistical Association 115 (2020), no. 530, 822–835. [2] P. Alquier and J. Ridgway, Concentration of tempered posteriors and of their variational approximations, The Annals of Statistics 48 (2020), no. 3, 1475–1497. [3] A. R. Barron and C. Sheu, Approximation of density functions by sequences of exponential families, The Annals of Statistics 19 (1991), no. 3, 1347–1369. [4] C. M. Bishop and M. E. Tipping, Probabilistic principal component analysis, J. R. Statist. Soc. B 61 (1999), no. 3, 611–622. [5] F. W. Olver, D. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. [6] C Brouder and F Patras, One-particle irreducibility with initial correlations, (2011). [7] I. Csiszar, I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975), no. 1, 146–158. [8] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (1989), 303–314. [9] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2003. [10] D. E. Rumelhart, G. E. Hinton and R. Williams, Learning representations by back-propagating errors, Nature 323 (1986), 533–536. [11] G. E. Hinton, A Practical Guide to Training Restricted Boltzmann Machines, University of Toronto, 2010. [12] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (1991), no. 2, 251–257. [13] Y. Katznelson, An introduction to harmonic analysis (third ed.), New York: Cambridge Mathematical Library, 2004. [14] D. M. Blei, A. Kucukelbir and J. D. McAuliffe, Variational inference: A review for statisticians, Jour. of the American Statistical Association 112 (2017), no. 518, 859–877. [15] S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), no. 1, 79–86. [16] H. M. Möller and R. Tenberg, Multivariate polynomial system solving using intersections of eigenspaces, Journal of Symbolic Computation 32 (2001), no. 5, 513–531. [17] B. Mourrain, V. Y. Pan and O. Ruatta, Accelerated solution of multivariate polynomial systems of equations, SIAM Journal on Computing 32 (2003), no. 2, 435–454. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] D. W. Stroock, Probability theory, an analytic view, Cambridge University Press, 1993. [20] B. Sturmfels, Solving systems of polynomial equations (cbms regional conference series in mathematics), Amer. Math. Soc., 2002. [21] N. Wiener, Tauberian theorems, Annals of Mathematics 33 (1932), no. 1, 1–100. [22] F. Zhang and C. Gao, Convergence rates of variational posterior distributions, The Annals of Statistics 48 (2020), no. 4, 2180–2207.

Variational Non-Bayesian Inference of the Probability Density Function: Conclusion & References

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References PROOF. By Taylor’s series expansion of the exponential function, the triangle inequality, and the uniform convergence, we have This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References PROOF. By Taylor’s series expansion of the exponential function, the triangle inequality, and the uniform convergence, we have

Variational Non-Bayesian Inference of the Probability Density: Characterization of Coefficients

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References This paper presents a research study focused on uncovering the hidden population distribution from the viewpoint of a variational non-Bayesian approach. It asserts that if the hidden probability density function (PDF) has continuous partial derivatives of at least half the dimension’s order, it can be perfectly reconstructed from a stationary ergodic process: First, we establish that if the PDF belongs to the Wiener algebra, its canonical ensemble form is uniquely determined through the Fréchet differentiation of the KullbackLeibler divergence, aiming to minimize their cross-entropy. Second, we utilize the result that the differentiability of the PDF implies its membership in the Wiener algebra. Third, as the energy function of the canonical ensemble is defined as a series, the problem transforms into finding solutions to the equations of analytic series for the coefficients in the energy function. Naturally, through the use of truncated polynomial series and by demonstrating the convergence of partial sums of the energy function, we ensure the efficiency of approximation with a finite number of data points. Finally, through numerical experiments, we approximate the PDF from a random sample obtained from a bivariate normal distribution and also provide approximations for the mean and covariance from the PDF. This study substantiates the excellence of its results and their practical applicability. 1. Introduction. Statistical inference enables us to draw meaningful conclusions about a population’s distribution by studying a representative sample. Numerous facets of scientific research implicitly or explicitly require the estimation of population’s distribution from a sample. By analyzing a subset of individuals or observations, we can gather information that applies to the larger group, even when studying the entire population is impractical or impossible. Drawing from the works of two authors ([8, 12]), machine learning methods have gained popularity for predicting hidden information from data. Today, most emerging artificial neural networks incorporate statistical methods. Nevertheless, these computations do not achieve global optimization of learning results because of the inherent challenges related to setting initial conditions and handling the small magnitudes of high-dimensional differential values. To address this issue, many scholars have applied statistical inferences. Among these methods, variational Bayesian method is a commonly used technique for estimating posterior information. It’s worth noting that there are numerous excellent studies on the method. One can refer to the citations [1, 2, 14, 22] as examples of the most recent works. Our study aims to identify the hidden probability density function (PDF) without the need for a prior distribution in a non-Bayesian context. This is achieved solely from a stationary ergodic process using entropy minimization. Additionally, we aim to estimate moments of the PDF, such as the mean and variance. We provide a proof of the norm convergence of sequences of approximated PDFs from a finite sample size, which is useful in practical applications. In most probability models arising from natural phenomena, assuming the boundedness of a random variable within a specific window becomes reasonable when considering a sufficiently wide range of variability. Moreover, we assume ergodicity in the context of a stationary process, as is common in many scientific communities. The basic framework of the idea to find the hidden PDF is as follows: First, we express the PDF in the form of a canonical ensemble, resembling a mechanical heat system. Second, by embedding the problem in an infinite-dimensional function space, we establish the existence of the Fréchet derivative of an entropy induced from the energy function of the PDF (for more information on the Fréchet derivative, refer to [18]). Third, we uniquely determine the energy function by minimizing the Kullback-Leibler divergence (KL-divergence) ([7, 15]). The concept of KL-divergence, valuable in information theory, finds widespread use, even in artificial intelligence (for example, see [10], [4], [11]). Finally, we derive a system of polynomial series that corresponds to the sample means of complex exponentials and present their numerical solutions from random samples in a bivariate normal distribution. Using the approximated PDFs that we obtained, we compare approximations of the mean and variance from the bivariate normal distribution. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References This paper presents a research study focused on uncovering the hidden population distribution from the viewpoint of a variational non-Bayesian approach. It asserts that if the hidden probability density function (PDF) has continuous partial derivatives of at least half the dimension’s order, it can be perfectly reconstructed from a stationary ergodic process: First, we establish that if the PDF belongs to the Wiener algebra, its canonical ensemble form is uniquely determined through the Fréchet differentiation of the KullbackLeibler divergence, aiming to minimize their cross-entropy. Second, we utilize the result that the differentiability of the PDF implies its membership in the Wiener algebra. Third, as the energy function of the canonical ensemble is defined as a series, the problem transforms into finding solutions to the equations of analytic series for the coefficients in the energy function. Naturally, through the use of truncated polynomial series and by demonstrating the convergence of partial sums of the energy function, we ensure the efficiency of approximation with a finite number of data points. Finally, through numerical experiments, we approximate the PDF from a random sample obtained from a bivariate normal distribution and also provide approximations for the mean and covariance from the PDF. This study substantiates the excellence of its results and their practical applicability. 1. Introduction. Statistical inference enables us to draw meaningful conclusions about a population’s distribution by studying a representative sample. Numerous facets of scientific research implicitly or explicitly require the estimation of population’s distribution from a sample. By analyzing a subset of individuals or observations, we can gather information that applies to the larger group, even when studying the entire population is impractical or impossible. Drawing from the works of two authors ([8, 12]), machine learning methods have gained popularity for predicting hidden information from data. Today, most emerging artificial neural networks incorporate statistical methods. Nevertheless, these computations do not achieve global optimization of learning results because of the inherent challenges related to setting initial conditions and handling the small magnitudes of high-dimensional differential values. To address this issue, many scholars have applied statistical inferences. Among these methods, variational Bayesian method is a commonly used technique for estimating posterior information. It’s worth noting that there are numerous excellent studies on the method. One can refer to the citations [1, 2, 14, 22] as examples of the most recent works. Our study aims to identify the hidden probability density function (PDF) without the need for a prior distribution in a non-Bayesian context. This is achieved solely from a stationary ergodic process using entropy minimization. Additionally, we aim to estimate moments of the PDF, such as the mean and variance. We provide a proof of the norm convergence of sequences of approximated PDFs from a finite sample size, which is useful in practical applications. In most probability models arising from natural phenomena, assuming the boundedness of a random variable within a specific window becomes reasonable when considering a sufficiently wide range of variability. Moreover, we assume ergodicity in the context of a stationary process, as is common in many scientific communities. The basic framework of the idea to find the hidden PDF is as follows: First, we express the PDF in the form of a canonical ensemble, resembling a mechanical heat system. Second, by embedding the problem in an infinite-dimensional function space, we establish the existence of the Fréchet derivative of an entropy induced from the energy function of the PDF (for more information on the Fréchet derivative, refer to [18]). Third, we uniquely determine the energy function by minimizing the Kullback-Leibler divergence (KL-divergence) ([7, 15]). The concept of KL-divergence, valuable in information theory, finds widespread use, even in artificial intelligence (for example, see [10], [4], [11]). Finally, we derive a system of polynomial series that corresponds to the sample means of complex exponentials and present their numerical solutions from random samples in a bivariate normal distribution. Using the approximated PDFs that we obtained, we compare approximations of the mean and variance from the bivariate normal distribution.

Variational Non-Bayesian Inference of the Probability Density Function: Introduction

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References 2. Organization and notation. This article is organized as follows: In the next section, we formulate the problem by defining the energy and partition function for the hidden PDF in a generalized Wiener algebra. We observe that the nonlinearity of energy functions poses challenges in maximizing likelihood functions. To overcome this issue, we adopt the KLdivergence between two distribution functions. In Section 4, we embed energy functions into a generalized Wiener algebra, thereby transforming the nonlinearity of the energy function into a system of equations in the function space and simultaneously converting the min-max problem into a problem of solving equations. We showthe Fréchet differentiability of the KL-divergence with respect to coefficients in the generalized Wiener algebra. From this property, we then uniquely obtain the existence such that its energy function yields a KL-divergence of 0. We establish a characterization of the coefficients of the energy function by expressing them in relation to the Fourier coefficients found within the (classical) Wiener algebra, as demonstrated in Section 5. This serves as an example of the generalized Wiener algebra. In the concluding Section 6, when dealing with a random sample that consists of realizations of a stationary ergodic process, we formulate a system of polynomial series with the coefficients of the energy function as variables, which consists of infinitely many equations in infinitely many variables. From the convergence property of the partial sums of an energy function in Section 4, the truncated system, consisting of a finite number of equations with partial sums, offers an avenue for coefficient approximations. The estimated coefficients effectively serve as estimators for the coefficients. More precisely, the hidden PDF can be approximated using these estimators. Furthermore, we provide a numerical example that emphasizes the approximation of the PDF, as well as the calculation of the mean and variance from the PDF using a random sample. This sample is generated from a bivariate normal distribution. In this paper, we consistently employ the following general notations (Table 1). Some notations will be elaborated upon in more detail when they are introduced. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References 2. Organization and notation. This article is organized as follows: In the next section, we formulate the problem by defining the energy and partition function for the hidden PDF in a generalized Wiener algebra. We observe that the nonlinearity of energy functions poses challenges in maximizing likelihood functions. To overcome this issue, we adopt the KLdivergence between two distribution functions. In Section 4, we embed energy functions into a generalized Wiener algebra, thereby transforming the nonlinearity of the energy function into a system of equations in the function space and simultaneously converting the min-max problem into a problem of solving equations. We showthe Fréchet differentiability of the KL-divergence with respect to coefficients in the generalized Wiener algebra. From this property, we then uniquely obtain the existence such that its energy function yields a KL-divergence of 0. We establish a characterization of the coefficients of the energy function by expressing them in relation to the Fourier coefficients found within the (classical) Wiener algebra, as demonstrated in Section 5. This serves as an example of the generalized Wiener algebra. In the concluding Section 6, when dealing with a random sample that consists of realizations of a stationary ergodic process, we formulate a system of polynomial series with the coefficients of the energy function as variables, which consists of infinitely many equations in infinitely many variables. From the convergence property of the partial sums of an energy function in Section 4, the truncated system, consisting of a finite number of equations with partial sums, offers an avenue for coefficient approximations. The estimated coefficients effectively serve as estimators for the coefficients. More precisely, the hidden PDF can be approximated using these estimators. Furthermore, we provide a numerical example that emphasizes the approximation of the PDF, as well as the calculation of the mean and variance from the PDF using a random sample. This sample is generated from a bivariate normal distribution. In this paper, we consistently employ the following general notations (Table 1). Some notations will be elaborated upon in more detail when they are introduced.

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