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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 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

The Last Man by Mary Wollstonecraft Shelley is part of the HackerNoon Books Series. Read this book online for free on HackerNoon! : The Last Man Title : Mary Wollstonecraft Shelley Author : April 24, 2006 [eBook #18247] Most recently updated: October 29, 2022 Release date : English Language TABLE OF LINKS VOL. I. INTRODUCTION. CHAPTER I. CHAPTER II. CHAPTER III. CHAPTER IV. CHAPTER IV. CHAPTER V. CHAPTER VI. CHAPTER VII. CHAPTER VIII. CHAPTER IX. CHAPTER X. VOL. II. CHAPTER I. CHAPTER II. CHAPTER III. CHAPTER IV. CHAPTER V. CHAPTER VI. CHAPTER VII. CHAPTER VIII. CHAPTER IX. VOL. III. CHAPTER I. CHAPTER II. CHAPTER III. CHAPTER IV. CHAPTER V. CHAPTER VI. CHAPTER VII. CHAPTER VIII. CHAPTER IX. CHAPTER X. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Mary Wollstonecraft Shelley (2006). The Last Man. Urbana, Illinois: Project Gutenberg. Retrieved https://www.gutenberg.org/cache/epub/18247/pg18247-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

The Last Man by Mary Wollstonecraft Shelley - Table of Links

Tales and Stories by Mary Wollstonecraft Shelley is part of the HackerNoon Books Series. Read this book online for free on HackerNoon! : Tales and Stories Title : Mary Wollstonecraft Shelley Author : Richard Garnett Author of introduction, etc. : March 1, 2018 [eBook #56665] Most recently updated: December 14, 2021 Release date : English Language TABLE OF LINKS INTRODUCTION I. The Sisters of Albano II. Ferdinando Eboli III. The Evil Eye IV. The Dream V. The Mourner VI. The False Rhyme VII. A Tale of the Passions; or, The Death of Despina VIII. The Mortal Immortal IX. Transformation X. The Swiss Peasant XI. The Invisible Girl XII. The Brother and Sister XIII. The Parvenue XIV. The Pole XV. Euphrasia XVI. The Elder Son XVII. The Pilgrims About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Mary Wollstonecraft Shelley (2018). Tales and Stories. Urbana, Illinois: Project Gutenberg. Retrieved https://www.gutenberg.org/cache/epub/56665/pg56665-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

Tales and Stories by Mary Wollstonecraft Shelley - Table of Links

Mathilda by Mary Wollstonecraft Shelley is part of the HackerNoon Books Series. Read this book online for free on HackerNoon! : Mathilda Title : Mary Wollstonecraft Shelley Author : March 2, 2005 [eBook #15238] Most recently updated: September 25, 2022 Release date : English Language TABLE OF LINKS PREFACE Introduction Mathilda I. II. II. IV. V. VI. VII. VIII. IX. X. XI. XII. The Fields Of Fancy About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Mary Wollstonecraft Shelley (2005). Mathilda. Urbana, Illinois: Project Gutenberg. Retrieved https://www.gutenberg.org/cache/epub/15238/pg15238-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

Mathilda by Mary Wollstonecraft Shelley - Table of Links

I awoke, and my first thought was, Adrian and Clara are dead.

Now—soft awhile—have I arrived so near the end?

We had now reached Switzerland

These events occupied so much time, that June had numbered more than half its days

Eventful winter passed; winter, the respite of our ills.

We held a council, to decide on our future movements

Our escort had been directed to prepare our abode for the night at the inn

Idris stirred and awoke; alas! she awoke to misery.

The spirit of emigration crept in among the few survivors

Hear you not the rushing sound of the coming tempest?

After a long interval, I am again impelled by the restless spirit within me to continue my narration

I did proceed to Windsor, but not with the intention of remaining there

I have lingered thus long on the extreme bank

I returned to my family estate in the autumn of the year 2092

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