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BACTERIA AND DISEASE

Bacteria: Chapter VI - BACTERIA IN MILK, MILK PRODUCTS, AND OTHER FOODS

Bacteria: Chapter V - BACTERIA IN THE SOIL

Bacteria: Chapter IV - BACTERIA AND FERMENTATION

Bacteria: Chapter III - BACTERIA IN THE AIR

Bacteria: Chapter II - BACTERIA IN WATER

Bacteria: Chapter 1 - THE BIOLOGY OF BACTERIA

Bacteria: Chapter IX - DISINFECTION

Bacteria by George Newman is part of the HackerNoon Books Series. You can jump to any chapter in this book here . APPENDIX APPENDIX It is proposed to add one or two notes on certain technical points in bacteriological work, with a view to assisting those medical men not able to obtain the advantages of a well-equipped laboratory, and yet desirous of occasionally attempting some practical bacteriology. 1. All fluids may be examined for bacteria in two chief ways: General Examination. ( ) A small quantity may be placed on a cover-glass or slide, dried over a lamp or bunsen flame, and stained with aniline dyes for a few minutes. It is then ready for microscopic examination. It is obvious that the result will generally be a of bacteria, for which differentiating stains may be used (Gram, Ziehl-Neelsen, etc.). a mixture ( ) A minute drop of the suspected fluid may be added to various fluid media (broth, liquefied gelatine, etc.) and then plated out upon small sterilised sheets of glass. In the course of two or three days the contained bacteria will reveal themselves in characteristic colonies, which may be examined, and if possible sub-cultured, and carefully studied. b These are various, and are used when it is desired to stain the bacteria themselves one colour, and the matrix or ground substance in which they are situated another colour. Three of the commoner methods are those of Ehrlich, Neelsen, and Gram. They are as follows: Double-Staining Methods. "Five parts of aniline oil are shaken up with 100 parts distilled water, and the emulsion filtered through moistened filter paper. A saturated alcoholic solution of fuchsine, methyl-violet, or gentian-violet is added to the filtrate in a watch-glass, drop by drop, until precipitation commences. Cover-glass preparations are floated in this mixture for fifteen to thirty minutes, then washed for a few seconds in dilute nitric acid (one part nitric acid to two of water), and then rinsed in distilled water. The stain is removed from everything except the bacilli; but the ground substance can be after-stained brown if the bacilli are violet, or blue if they have been stained red" (Crookshank, , p. 89). Ehrlich's Method. 338 Bacteriology and Infective Diseases The primary stain in this method is a solution of aniline gentian-violet (saturated alcoholic solution of gentian-violet 30 cc., aniline water 100 cc.), which stains both ground substance and bacteria in purple. The preparation is next immersed in the following solution for half a minute or a little more: Gram's Method. In this short space of time the iodine solution acts as a mordant of the purple colour in the bacteria, but not in the ground substance. Hence, if the preparation be now (when it has assumed a colour) washed in alcohol (methylated spirit), the ground substance slowly loses its colour and becomes clear. But the bacteria retain their colour, and thus stand out in a well-defined manner. Cover-glass preparations decolourise more quickly than sections of hardened tissue, and they should only be left in the methylated spirit until no more colour comes away. brown The preparation may now be washed in water, dried, and mounted for microscopic examination, or it may be double-stained, that is, immersed in some contrast colour which will lightly stain the ground substance. Eosin or Bismarck brown are commonly used for this purpose. The former is applied for a minute or two, the latter for five minutes, after which the specimen is passed through methylated spirit (and preferably xylol also) and mounted. The result is that the bacteria appear in a dark purple colour on a background of faint pink or brown. Carbol-thionine blue, picro-carmine, and other stains are occa sionally used in place of the aniline gentian-violet, and there are other slight modifications of the method. Here the primary stain is a solution of carbol-fuchsin: Ziehl-Neelsen Method. It is best to heat the dye in a sand-bath, in order to distribute the heat evenly. The various stages in the staining process are as follows: ( ) The cover-glass with the dried film upon it is immersed in the hot stain for one to three minutes. a ( ) Remove the cover-glass from the carbol-fuchsin, and place it in a capsule containing a 25 per cent. solution of sulphuric acid to decolourise it. Here its redness is changed into a slate-grey colour. b ( ) Wash in water, and alternately in the acid and water, until it is of a faint pink colour. c ( ) Now place the cover-glass for a minute or two in a saturated aqueous solution of methylene-blue, which will counter-stain the decolourised ground substance blue. d ( ) Wash in water. e ( ) Dehydrate by rinsing in methylated spirit, dry, and mount. A pure culture of bacteria will not necessarily require the counter-stain (methylene-blue). Sections of tissue may require twenty to thirty minutes in the primary stain (carbol-fuchsin). This stain is used for tubercle and leprosy. With a little practice the staining of the bacillus of tubercle when present in pus or sputum becomes a very simple and accurate method of diagnosis. A small particle of sputum or pus is placed between two clean cover-glasses and thus pressed between the thumb and finger into a thin film. This is readily dried and stained as above, the bacillus of tubercle appearing as a delicately-beaded red rod with a background of blue. f The following points must be ascertained in order to identify any particular micro-organism: Bacteriological Diagnosis. (1) Its morphology, bacillus, coccus, spirillum, etc.; the presence or absence of involution forms. (2) Motility by the unstained cover-glass preparation ("hanging drop"); note presence of flagella. (3) Presence of spores, their appearance and position. (4) Whether or not the organism stains with Gram's method. (5) The character of the growth upon various media (gelatine, agar, milk, potato, broth); the presence or absence of liquefaction in the gelatine culture; its power of producing acid, gas, or indol. (6) Whether it is aërobic or anaërobic. (7) Its colour in cultivation. (8) If it is a disease-producing organism under examination, its effect upon the animal tissues and the course of the disease should be observed. There are other points of importance, but the above are essential to a right conclusion. Diagnosis in Special Diseases: (1) This disease may be bacteriologically diagnosed with a minimum of apparatus and equipment. By means of a swab a rubbing from a suspected throat is readily obtained. This may be examined by the microscope, or sub-cultured on favourable medium. Blood serum is perhaps the best, but, as Hewlett remarks, "If no serum tubes can be had, an egg may be used. It is boiled hard, the shell chipped away from one end with a knife sterilised by heating, and the inoculation made on the exposed white surface; the egg is then placed, inoculated end down, in a wine-glass of such a size that it rests on the rim and does not touch the bottom. Diphtheria. A few drops of water may with advantage be put at the bottom of the glass to keep the egg-white moist. The preparation is kept in a warm place for twenty-four to forty-eight hours and then examined." The examination, of course, consists in staining and preparing for the microscope and observing the form, arrangement, and characters of the organism or organisms present. A small piece of the membrane may be detached, washed in water, and stained for the bacilli. (2) (Ziehl-Neelsen's stain, ). Tubercle vide supra (3) ( ). Typhoid Enteric Fever This diagnostic test depends upon the effect which the blood of a person suffering from typhoid fever has upon the . The effect is twofold. In the first place, the actively motile becomes immotile; and secondly, there is an agglutination, or grouping together in colonies, of the . Neither of these features occurs if healthy human blood is brought into contact with a culture of the typhoid bacillus. There are various ways in which this "serum diagnosis" can be carried out. Widal's Reaction. Bacillus typhosus B. typhosus B. typhosus The simplest and quickest method is as follows: To ten drops of a twenty-four or forty-eight-hours-old neutral broth culture of the typhoid bacillus one drop of the blood serum to be tested is added. The serum and culture are rapidly mixed in the trough of a hollow ground slide (such as is used for the "hanging drop"), and a single drop is taken, placed upon an ordinary clean slide, and a cover-glass superimposed. The positive reaction of agglutination and immotility, if the blood comes from a case of typhoid fever, will probably appear within fifteen or twenty minutes. The fluid culture of typhoid may be taken from an agar culture as well as from broth. In both cases it may be desirable to filter through ordinary filter paper to remove any normally agglutinated masses of bacilli before commencing the test. In his first experiments Widal used a test-tube in the following manner: The blood to be tested is diluted by one part of it being added to fifteen parts of broth in a test-tube. The mixture is inoculated with a drop of a typical culture. The tube is then incubated at 37° C. for twenty-four hours, after which it is examined. If the reaction be positive, the broth appears comparatively clear, but at the bottom of the test-tube a more or less abundant sediment will be found. This is due to the clumps of bacilli having fallen owing to gravity. If, on the other hand, the reaction is negative, the broth will appear more or less uniformly turbid. Bacillus typhosus For the required to carry out the simpler methods of bacteriological work reference should be made to the standard laboratory text-books, which furnish all necessary details. A good microscope, with a 1/12 oil immersion lens, is, of course, essential. This can now be obtained for about £16 (Beck, Swift, Baker, Watson, etc.), and the other necessary apparatus is readily obtainable of Baird and Tatlock, Hatton Garden, E. C., and other makers. apparatus 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. Newman, George. 2015.Bacteria. Urbana, Illinois: Project Gutenberg. Retrieved May 2022 from https://www.gutenberg.org/files/48793/48793-h/48793-h.htm 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 www.gutenberg.org , located at https://www.gutenberg.org/policy/license.html .

Bacteria: APPENDIX

Bacteria: Chapter VII - THE QUESTION OF IMMUNITY AND ANTITOXINS

Why Is the AI Surge Boosting HDD Stocks? Finis Conner Says: Drives Are to AI as Batteries Are to EVs

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References References [1] R. Balasubramanian, J. B. Conrey and D. R. Heath-Brown, Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. reine angew. Math. 357(1985), 161-181. [2] J. B. Conrey, A. Ghosh and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull. London Math. Soc. 16(1984), 421-424. [3] J. B. Conrey, A. Ghosh and S. M. Gonek, Mean values of the Riemann zeta-function with application to the distribution of zeros, Number theory, trace formulas and discrete groups (Academic Press, Boston, 1989), 185-199. [4] J. B. Conrey, A. Ghosh and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3)76(1998), 497-522. 5] J. B. Conrey, H. Iwaniec and K. Soundararajan, Asymptotic large sieve, Preprint 2011, arxiv 1105.1176 [6] J. B. Conrey, H. Iwaniec and K. Soundararajan, Critical zeros of Dirichlet Lfunctions, Preprint 2011, arxiv 1105.1177 [7 ] H. Davenport, Multiplicative Number Theory, 3rd. ed. (revised by H. L. Montgomery), Springer-Verlag, New York, 2000 [8] D. Goldfeld, A simple proof of Siegel’s theorem, Proc. Nat. Acad. Sci. USA 71(1974), 1055. [9] D. Goldfeld, An asymptotic formula relating the Siegel zero and the class number of quadratic fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(1975), 611-615. [10] D. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(1976), 624-663. [11] A. Granville and H. M. Stark, ABC implies no “Siegel zeros” for L-functions of characters with negative discriminant, Invent. Math. 139(2000), 509-523. [12] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math., 84(1986), 225-320. [13] D. R. Heath-Brown, Small class numbers and the pair correlation of zeros, Conference in Celebration of the Centenary of the the Proof of the Prime Number Theorem (A Symposium on the Riemann Hypothesis), Seattle, 1996. [14] H. Iwaniec and E. Kowalski, Analytic number theory, Coll. Pulb. vol. 53, Amer. Math. Soc. 2004 [15] H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic Lfunctions and Landau-Siegel zeros, Israel J. Math., 120(2000), 155-177. [16] A. Karatsuba, Basic Analytic Number Theory, (translated from Russian by M. Nathanson), Springer-Verlag, New York, 1993. [17] H. L. Montgomery, The pair correlation of zeros of the zeta-function, Analytic number theory, Proc. Symp. Pure Math. Vol 24, 181-193, (Amer. Math. Soc., Providence, RI. 1973) [18] C. L. Siegel, Uber die Classenzahl quadratischer Zuhlk¨orper, ¨ Acta Arith. 1(1936), 83-86. [19] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd. ed. (revised by D. R. Heath-Brown), Oxford Univ. Press, Oxford, 1986 This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References References [1] R. Balasubramanian, J. B. Conrey and D. R. Heath-Brown, Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. reine angew. Math. 357(1985), 161-181. [2] J. B. Conrey, A. Ghosh and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull. London Math. Soc. 16(1984), 421-424. [3] J. B. Conrey, A. Ghosh and S. M. Gonek, Mean values of the Riemann zeta-function with application to the distribution of zeros, Number theory, trace formulas and discrete groups (Academic Press, Boston, 1989), 185-199. [4] J. B. Conrey, A. Ghosh and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3)76(1998), 497-522. 5] J. B. Conrey, H. Iwaniec and K. Soundararajan, Asymptotic large sieve, Preprint 2011, arxiv 1105.1176 [6] J. B. Conrey, H. Iwaniec and K. Soundararajan, Critical zeros of Dirichlet Lfunctions, Preprint 2011, arxiv 1105.1177 [7 ] H. Davenport, Multiplicative Number Theory, 3rd. ed. (revised by H. L. Montgomery), Springer-Verlag, New York, 2000 [8] D. Goldfeld, A simple proof of Siegel’s theorem, Proc. Nat. Acad. Sci. USA 71(1974), 1055. [9] D. Goldfeld, An asymptotic formula relating the Siegel zero and the class number of quadratic fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(1975), 611-615. [10] D. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(1976), 624-663. [11] A. Granville and H. M. Stark, ABC implies no “Siegel zeros” for L-functions of characters with negative discriminant, Invent. Math. 139(2000), 509-523. [12] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math., 84(1986), 225-320. [13] D. R. Heath-Brown, Small class numbers and the pair correlation of zeros, Conference in Celebration of the Centenary of the the Proof of the Prime Number Theorem (A Symposium on the Riemann Hypothesis), Seattle, 1996. [14] H. Iwaniec and E. Kowalski, Analytic number theory, Coll. Pulb. vol. 53, Amer. Math. Soc. 2004 [15] H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic Lfunctions and Landau-Siegel zeros, Israel J. Math., 120(2000), 155-177. [16] A. Karatsuba, Basic Analytic Number Theory, (translated from Russian by M. Nathanson), Springer-Verlag, New York, 1993. [17] H. L. Montgomery, The pair correlation of zeros of the zeta-function, Analytic number theory, Proc. Symp. Pure Math. Vol 24, 181-193, (Amer. Math. Soc., Providence, RI. 1973) [18] C. L. Siegel, Uber die Classenzahl quadratischer Zuhlk¨orper, ¨ Acta Arith. 1(1936), 83-86. [19] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd. ed. (revised by D. R. Heath-Brown), Oxford Univ. Press, Oxford, 1986 This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Discrete Mean Estimates and the Landau-Siegel Zero: References

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References Appendix B. Some arithmetic sums Proof of Lemma 15.1. Put First we claim that Since χ = µ ∗ ν, it follows that Hence This together with Lemma 3.2 yields (B.1). Next we claim that This yields (B.2). By (B.1) and (B.2), for µ = 2, 3, We proceed to prove theassertion with µ = 2. Since for σ > 1 and it follows that For µ = 1 the proof is therefore reduced to showing that By (4.2) and (4.3), the left side of (B.3) is equal to By a change of variable, for 0.5 ≤ z ≤ 0.504, Hence, in a way similar to the proof of, we find that the left side of (B.3) i Proof of Lemma 17.1. By Lemma 3.1, The sum on the right side is equal to Assume σ > 1. We have If χ(p) = 1, then (see [19, (1.2.10)]) if χ(p) = −1, then if χ(p) = 0, then Hence In a way similar to the proof of, by (A) and simple estimate, we find that the integral (14) is equal to the residue of the function at s = 0, plus an acceptable error O, which is equal to This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References Appendix B. Some arithmetic sums Proof of Lemma 15.1. Put Proof of Lemma 15.1. First we claim that Since χ = µ ∗ ν, it follows that Hence This together with Lemma 3.2 yields (B.1). Next we claim that This yields (B.2). By (B.1) and (B.2), for µ = 2, 3, We proceed to prove theassertion with µ = 2. Since for σ > 1 and it follows that For µ = 1 the proof is therefore reduced to showing that By (4.2) and (4.3), the left side of (B.3) is equal to By a change of variable, for 0.5 ≤ z ≤ 0.504, Hence, in a way similar to the proof of, we find that the left side of (B.3) i Proof of Lemma 17.1. By Lemma 3.1, Proof of Lemma 17.1. The sum on the right side is equal to Assume σ > 1. We have If χ(p) = 1, then (see [19, (1.2.10)]) if χ(p) = −1, then if χ(p) = 0, then Hence In a way similar to the proof of, by (A) and simple estimate, we find that the integral (14) is equal to the residue of the function at s = 0, plus an acceptable error O , which is equal to s O This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Discrete Mean Estimates and the Landau-Siegel Zero: Appendix B. Some Arithmetic Sums

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References Appendix A. Some Euler Products This appendix is devoted to proving Lemma 8.3, 15.2, 15.3, 16.1 and 16.2. For notational simplicity we shall write Proof of Lemma 8.3. Note that which are henceforth assumed. We discuss in three cases. Case 1. (q, dh) = 1. We have It follows that This together with the relations yields (A.1). Case 2. q|h. We have so that This yields (A.3). This completes the proof. Proof of Lemma 16.1. For any q, r, d and l we have Hence and On the other hand we have It follows that It is direct to verify that in either case the assertion holds. Proof of Lemma 16.2. We give a sketch only. If dl < D, (dl, D) = 1 and |s − 1| ≤ 5α, then with The assertion follows by discussing the cases χ(2) 6= 1 and χ(2) = 1 respectively This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References Appendix A. Some Euler Products This appendix is devoted to proving Lemma 8.3, 15.2, 15.3, 16.1 and 16.2. For notational simplicity we shall write Proof of Lemma 8.3. Note that Proof of Lemma 8.3. Note that which are henceforth assumed. We discuss in three cases. Case 1. (q, dh) = 1. Case 1. (q, dh) = 1. We have It follows that This together with the relations yields (A.1). Case 2. q|h. Case 2. q|h. We have so that This yields (A.3). This completes the proof. Proof of Lemma 16.1. For any q, r, d and l we have Proof of Lemma 16.1. For any q, r, d and l we have Hence and On the other hand we have It follows that It is direct to verify that in either case the assertion holds. Proof of Lemma 16.2. We give a sketch only. If dl < D, (dl, D) = 1 and |s − 1| ≤ 5α, then Proof of Lemma 16.2. dl < D, (dl, D) with The assertion follows by discussing the cases χ(2) 6= 1 and χ(2) = 1 respectively This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Discrete Mean Estimates and the Landau-Siegel Zero: Appendix A. Some Euler Products

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References 18. Proof of Proposition 2.5 By the discussion at the end of Section 2, it suffices to prove (2.32) and (2.33). Proof of (2.32). By (12.3), (12,17), (13.7), (15.24), (16.17) and (17.10), In view of (15.), we can write By calculation (there is a theoretical interpretation), Hence Direct calculation shows that It follows from (8.24), (9.8) and (18.2) that This with together (8.23), (9.7) and (18.1) yields (2.32). Proof of (2.33). By Lemma 8.1, We have The right side is split into three sums according to Thus we have the crude bound so that This yields (2.33) by Lemma 8.1 and Proposition 7.1 This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References 18. Proof of Proposition 2.5 By the discussion at the end of Section 2, it suffices to prove (2.32) and (2.33). Proof of (2.32). Proof of (2.32). By (12.3), (12,17), (13.7), (15.24), (16.17) and (17.10), In view of (15.), we can write By calculation (there is a theoretical interpretation), Hence Direct calculation shows that It follows from (8.24), (9.8) and (18.2) that This with together (8.23), (9.7) and (18.1) yields (2.32). Proof of (2.33). Proof of (2.33). By Lemma 8.1, We have The right side is split into three sums according to Thus we have the crude bound so that This yields (2.33) by Lemma 8.1 and Proposition 7.1 This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Discrete Mean Estimates and the Landau-Siegel Zero: Proof of Proposition 2.5

Closing the Skills Gap: Addressing the Surge in Demand for Digital Marketing Experts

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References 11. Proof of Proposition 2.6 By the result of Section 9, Hence, by Cauchy’s inequality, the proof of Proposition 2.6 is reduced to showing that This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References 11. Proof of Proposition 2.6 By the result of Section 9, Hence, by Cauchy’s inequality, the proof of Proposition 2.6 is reduced to showing that This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Discrete Mean Estimates and the Landau-Siegel Zero: Proof of Proposition 2.6

Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References 13. Approximation to Ξ14 In this section we establish an approximation to Ξ14. Assume that ψ ∈ Ψ1 and ρ ∈ Z(ψ). By Lemma 5.2 and (2.2), By Lemma 6.1, and, by Lemma 5.1, Hence By Lemma 6.1 and 5.1, We insert this into (13.2) and then insert the result into (13.1). Thus we obtain where where Inserting this into (12.4) we obtain Combining (2.34), Cauchy’s inequality, Proposition 7.1, Lemma 5.9, 6.1 and 3.3, we can verify that For example, by (2.34) the right side being estimated via Lemma 5.9, 6.1 and 3.3 This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References 13. Approximation to Ξ14 In this section we establish an approximation to Ξ14. Assume that ψ ∈ Ψ1 and ρ ∈ Z(ψ). By Lemma 5.2 and (2.2), By Lemma 6.1, and, by Lemma 5.1, Hence By Lemma 6.1 and 5.1, We insert this into (13.2) and then insert the result into (13.1). Thus we obtain where where Inserting this into (12.4) we obtain Combining (2.34), Cauchy’s inequality, Proposition 7.1, Lemma 5.9, 6.1 and 3.3, we can verify that For example, by (2.34) the right side being estimated via Lemma 5.9, 6.1 and 3.3 This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Stories, News & Mentions about SIRGE

Sirge’s Stories on HackerNoon