Skip to main content Accessibility help
×
×
Home
Equivalents of the Riemann Hypothesis
  • Volume 2: Analytic Equivalents
  • Kevin Broughan, University of Waikato, New Zealand

  • Export citation
  • Recommend to librarian
  • Recommend this book

    Email your librarian or administrator to recommend adding this book to your organisation's collection.

    Equivalents of the Riemann Hypothesis
    • Online ISBN: 9781108178266
    • Book DOI: https://doi.org/10.1017/9781108178266
    Please enter your name
    Please enter a valid email address
    Who would you like to send this to *
    ×
  • Buy the print book

Book description

The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appendices containing fully developed proofs.

Reviews

'Throughout the book careful proofs are given for all the results discussed, introducing an impressive range of mathematical tools. Indeed, the main achievement of the work is the way in which it demonstrates how all these diverse subject areas can be brought to bear on the Riemann hypothesis. The exposition is accessible to strong undergraduates, but even specialists will find material here to interest them.'

D. R. Heath-Brown Source: Mathematical Reviews

'This two volume catalogue of many of the various equivalents of the Riemann Hypothesis by Kevin Broughan is a valuable addition to the literature … all in all these two volumes are a must have for anyone interested in the Riemann Hypothesis.'

Steven Decke Source: MAA Reviews

Refine List
Actions for selected content:
Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive
  • Send content to

    To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to .

    To send content items to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

    Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

    Find out more about the Kindle Personal Document Service.

    Please be advised that item(s) you selected are not available.
    You are about to send
    ×

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
×

Page 1 of 2



Page 1 of 2


References
[1] A., Akbary and K., Hambrook, A variant of the Bombieri–Vinogradov theorem with explicit constants and applications, Math. Comp. 84 (2015), 1901–1932.
[2] N. I., Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, 1965.
[3] N. I., Akhiezer, Elements of the Theory of Elliptic Functions, American Mathematical Society, 1990.
[4] S., Alaca and K. S., Williams, Introductory Algebraic Number Theory, Cambridge University Press, 2004.
[5] L., Alfors, Complex Analysis, 2nd edn, McGraw-Hill, 1966.
[6] H., Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), 273–389.
[7] H., Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), 181–221.
[8] F., Amoroso, On the heights of the product of cyclotomic polynomials, Rend. Sem. Mat. Univ. Politec. Torino 53 (1995), 183–191.
[9] F., Amoroso, Algebraic numbers close to 1 and variants ofMahler's measure, J. Number Theory 60 (1996), 80–96.
[10] J., Anderson, Hyperbolic Geometry, 2nd edn, Springer, 2005.
[11] J., Andrade, A., Chang and S. J., Miller, Newman's conjecture in various settings, J. Number Theory 144 (2014), 70–91.
[12] G. E., Andrews, R., Askey and R., Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, 1999.,
[13] T. M., Apostol, Mathematical Analysis, 2nd edn, Addison-Wesley, 1974.
[14] T. M., Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer, 1976.
[15] T. M., Apostol, Introduction to Analytic Number Theory, 2nd edn, Springer, 1990.
[16] J., Arias de Reyna, Asymptotics of Keiper–Li coefficients, Funct. Approx. Comment. Math. 45 (2011), 7–21.
[17] L., Báez-Duarte, M., Balazard, B., Landreau and E., Saias, Notes on the Riemann zeta function III, Adv. Math. 149 (2000), 130–144.
[18] L., Báez-Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis, Atti Accad. Naz. Lincei Cl. Sci Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), 5–11.
[19] L., Báez-Duarte, A sequential Riesz-like criterion for the Riemann hypothesis, Int. J. Math. Math. Sci. 21 (2005), 3527–3537.
[20] L., Báez-Duarte, Möbius convolutions and the Riemann hypothesis, Int. J. Math.Math. Sci. 22 (2005), 3599–3608.
[21] B., Bagchi, On Nyman, Beurling and Báez-Duarte's Hilbert space reformulation of the Riemann hypothesis, Proc. Indian Acad. Sci. Math. Sci. 116 (2006), 137–146.
[22] R., Balasubramanian, A note on Dirichlet's L-functions, Acta. Arith. 38 (1980), 273–283.
[23] R., Balasubramanian and V., Kumar Murty, Zeros of Dirichlet L-functions, Ann. Sci. Ecole Norm. Sup. 25 (1992), 567–615.
[24] B., Beckman, Arne Beurling and the Swedish Crypto Program During World War II, American Mathematical Society, 2003.
[25] J., Bertrand, P., Bertrand and J.-P., Ovarlez, The Mellin transform, in The Transforms and Applications Handbook, Ed. A. D. Poularikas, chapter 11, CRC Press, 1996.
[26] A., Beurling, A closure problem related to the Riemann zeta-function, Proc. Natl. Acad. Sci. USA 41 (1955), 312–314.
[27] M., Balazard and E., Saias, Notes on the Riemann zeta function I, Adv.Math. 139 (1998), 310–321.
[28] M., Balazard, E., Saias and M., Yor, Notes sur la fonction ζ de Riemann, 2, Adv. Math. 143 (1999) 284–287.
[29] E., Bombieri, On the large sieve, Mathematika 12 (1965), 201–225.
[30] E., Bombieri, Le Grand Crible dans la Théorie Analytique des Nombres, Astérisque, no. 18, Société Mathématiques de France, 1974.
[31] E., Bombieri, J.B., Friedlander and H., Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), 203–251.
[32] E., Bombieri and J. C., Lagarias, Complements to Li's criterion for the Riemann hypothesis, J. Number Theory 77 (1999), 274–287.
[33] E., Bombieri, Remarks on Weil's quadratic functional in the theory of prime numbers I, Rend. Mat. Accad. Lincei 11 (2000), 183–233.
[34] E., Bombieri, The Riemann Hypothesis: Official Problem Description, Clay Mathematics Institute, 2000.
[35] E., Bombieri, A variational approach to the explicit formula, Comm. Pure Appl. Math. 56 (2003), 1151–1164.
[36] E., Bombieri, The classical theory of zeta and L-functions, Milan J. Math. 78 (2010), 11–59.
[37] P., Borwein, S., Choi, B., Rooney and A., Weirathmueller, The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, Springer, 2008.
[38] K. A., Broughan, Holomorphic flow of the Riemann xi function, Nonlinearity 18 (2005), 1269–1294.
[39] K. A., Broughan, Equivalents of the Riemann Hypothesis I: Arithmetic Equivalents, Cambridge University Press, 2017.
[40] F. C. S., Brown, Li's criterion and the zero-free regions of L-functions, J. Number Theory 111 (2005), 1–32.
[41] A. M., Bruckner, J.B., Bruckner and B. S., Thomson, Real Analysis, 2nd edn, ClassicalRealAnalysis.com, 2008.
[42] N. G., de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197–226.
[43] J.-F., Burnol, A lower bound in an approximation problem involving the zeros of the Riemann zeta function, Adv. Math. 170 (2002), 56–70.
[44] D. A., Cardon and S. A., Roberts, An equivalence for the Riemann hypothesis in terms of orthogonal polynomials, J. Approx. Theory 138 (2006), 54–64.
[45] T., Carleman, Uber die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Arkiv Mat. Astron. Fys. 17 (1922), no. 9.
[46] T., Carleman, Fonctions Quasi Analytiques, Gauthier-Villars, 1926.
[47] J. W. S., Cassels, Lectures on Elliptic Curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, 1991.
[48] A., Chang, D., Mehrle, S.J., Miller, T., Reiter, J., Stahl and D., Yott, Newman's conjecture in function fields, J. Number Theory 157 (2015), 154–169.
[49] J. R., Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176.
[50] E. W., Cheney, Analysis for Applied Mathematics, Springer, 2001.
[51] T. S., Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978.
[52] J., Cislo and M., Wolf, Equivalence of Riesz and Báez-Duarte criterion for the Riemann hypothesis, Preprint, 2006.
[53] M. W., Coffey, Relations and positivity results for the derivatives of the Riemann ξ-function, J. Comput. Appl. Math. 166 (2004), 525–534.
[54] H., Cohen and H. W., Lenstra, Jr., Heuristics on class groups, in Number Theory (New York, 1982), Lecture Notes in Mathematics, vol. 1052, pp. 26–36. Springer, 1984.
[55] A., Connes, An essay on the Riemann hypothesis, Preprint, arXiv:1509.05576v1, 2015.
[56] J. B., Conrey and K., Soundararajan, Real zeros of quadratic Dirichlet L-functions, Invent. Math. 150 (2002), 1–44.
[57] J. B., Conrey, The Riemann hypothesis, Notices Amer. Math. Soc. 50 (2003), 341–353.
[58] J. B., Conrey and N. C., Snaith, Applications of the L-functions ratios conjecture, Proc. London Math. Soc. (3) 94 (2007), 497–522.
[59] J. B., Conrey, D.W., Farmer and M. R., Zirnbauer, Autocorrelation of ratios of L-functions, Commun. Number Theory Phys. 2 (2008), 593–636.
[60] G., Csordas, T.S., Norfolk and R. S., Varga, The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc. 296 (1986), 521–541.
[61] G., Csordas, T.S., Norfolk and R. S., Varga, A lower bound for the de Bruijn–Newman constant Λ, Numer. Math. 52 (1988), 483–497.
[62] G., Csordas, A., Ruttan and R. S., Varga, The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis, Numer. Algorithms 1 (1991), 305–329.
[63] G., Csordas, A., Odlyzko, W., Smith and R. S., Varga, A new Lehmer pair of zeros and a new lower bound for the de Bruijn–Newman constant Λ, Electron. Trans. Numer. Anal. 1 (1993), 104–111.
[64] G., Csordas, W., Smith and R. S., Varga, Lehmer pairs of zeros, the de Bruijn–Newman constant, and the Riemann hypothesis, Constr. Approx. 10 (1994), 107–129.
[65] G., Csordas, W., Smith and R. S., Varga, Lehmer pairs of zeros and the Riemann ξ-function, Proc. Symp. Appl. Math. 48 (1994), 553–556.
[66] B., Dacorogna, Direct Methods in the Calculus of Variations, Springer, 1989.
[67] H., Davenport, Multiplicative Number Theory, 3rd edn, Springer, 2000.
[68] B., Davies, Integral Transforms and Their Applications, 3rd edn, Springer, 2002.
[69] P., Deligne, La Conjecture de Weil I, Publ. Math. IHES 43 (1974), 273–308.
[70] P., Deligne, La Conjecture de Weil II, Publ. Math. IHES 52 (1980), 137–252.
[71] J.-M., Deshouillers, G., Effinger, H., te Riele and D., Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 99–104.
[72] M., Deuring, Zetafunktionen quadratischer formen, J. reine angew. Math. 1972 (1935), 226–252.
[73] K., Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys. 22 (1930), 1–14.
[74] J. A., Dieudonné, On the history of the Weil conjectures, Math. Intelligencer 10 (1975), 7–21.
[75] W. F., Donoghue, Jr., Distributions and Fourier Transforms, Academic Press, 1969.
[76] A. D., Droll, Variations of Li's criterion for an extension of the Selberg class, Thesis, Queen's University, Kingston, ON, 2012.
[77] H. M., Edwards, Riemann's Zeta Function, Academic Press, 1974; reprinted by Dover, 2001.
[78] P. D. T. A., Elliott and H., Halberstam, A conjecture in prime number theory, Symposia Mathematica, Vol. IV, INDAM, Rome, 1968/69, pp. 59–72, Academic Press, 1970.
[79] W., Ellison and F., Ellison, Prime Numbers, John Wiley, 1985.
[80] B., Epstein, Linear Functional Analysis, W. B. Saunders, 1970.
[81] A., Erdélyi, W., Magnus, F., Oberhettinger and F. G., Tricomi, Tables of Integral Transforms, vol. I, Bateman Manuscript Project, McGraw-Hill, 1954.
[82] S., Estala-Arias, Distribution of cusp sections in the Hilbert modular orbifold, J. Number Theory 155 (2015), 202–225.
[83] K. J., Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985.
[84] D., Fiorilli, On the non-vanishing of Dirichlet L-functions at the central point, Quart. J. Math. 66 (2015), 517–528.
[85] J., Franel, Les suites de Farey et le probléme des nombres premiers, Göttinger Nachrichten (1924), 198–201.
[86] P., Freitas, A Li-type criterion for zero-free half planes of Riemann's zeta function, J. London Math. Soc. (2) 73 (2006), 399–414.
[87] J., Friedlander and A., Granville, Limitations to the equi-distribution of primes I, Ann. Math. 129 (1989), 363–382.
[88] J. B., Friedlander and H., Iwaniec, What is the parity phenomenon?, Notices Amer.Math. Soc. 56 (2009), 817–818.
[89] J. B., Friedlander and H., Iwaniec, Opera de Cribro, American Mathematical Society, 2010.
[90] W., Fulton, Algebraic Curves: An Introduction to Algebraic Geometry,W. A. Benjamin, 1969.
[91] S. D., Galbraith, The Mathematics of Public Key Cryptography, Cambridge University Press, 2012.
[92] P. X., Gallagher, The large sieve, Mathematika 14 (1967), 14–20.
[93] P. X., Gallagher, Bombieri's mean value theorem, Mathematika 15 (1968), 1–6.
[94] R., Garunkstis, On a positivity property of the Riemann ξ-function, Lithuanian Math. J. 43 (2002), 140–145.
[95] C. F., Gauss, Disquisitiones Arithmeticae, 1801; English transl., Yale University Press, 1966.
[96] I. M., Gelfand and S. V., Fomin, Calculus of Variations, Prentice-Hall, 1965.
[97] D., Goldfeld, An asymptotic formula relating the Siegel zero and the class number of quadratic fields, Ann. Scu. Norm. Sup. Pisa (4) 2 (1975), 611–615.
[98] D., Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scu. Norm. Sup. Pisa (4) 3 (1976), 623–663.
[99] D., Goldfeld, Gauss’ class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. 13 (1985), 23–37. References 477
[100] D., Goldfeld, Automorphic Forms and L-Functions for the Group GL(n,R), Cambridge University Press, 2006.
[101] D. A., Goldston, J., Pintz and C. Y., Yildirim, Primes in tuples I, Ann. Math. (2) 170 (2009), 819–862.
[102] T., Gowers (Ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008.
[103] I. S., Gradshteyn and I. M., Ryzhik, Tables of Integrals, Series and Products, 6th edn, Academic Press, 2000.
[104] A., Granville, Least Primes in Arithmetic Progressions, Théorie des Nombres, Quebec, 1987, pp. 306–321, Walter de Gruyter, 1989.
[105] 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.
[106] A., Granville, Smooth numbers: computational number theory and beyond, Algorithmic Number Theory 42 (2008), 267–323.
[107] A., Granville, Primes in intervals of bounded length, Bull. Amer. Math. Soc. 52 (2015), 171–222.
[108] R., Gupta and M., Ram Murty, A remark on Artin's conjecture, Invent. Math. 78 (1984), 127–130.
[109] L., Habsieger, On the Nyman–Beurling criterion for the Riemann hypothesis, Funct. Approx. Comment. Math. 37 (2007), 187–201.
[110] G. H., Hardy and J. E., Littlewood, Contributions to the theory of the Riemann zetafunction and the theory of the distribution of primes, Acta Math. 41 (1918), 119–196.
[111] G. H., Hardy, Remarks in addition to Dr. Widder's note on inequalities, J. London Math. Soc. 4 (1929), 199–202.
[112] G. H., Hardy, A Mathematician's Apology, Cambridge University Press, 1940.
[113] G. H., Hardy, J.E., Littlewood and G., Pólya, Inequalities, Cambridge University Press, 1964.
[114] M., Harris, Mathematics Without Apologies, Princeton University Press, 2015.
[115] D. R., Heath-Brown, Simple zeros of the Riemann zeta function on the critical line, Bull. London Math. Soc 11 (1979), 17–18.
[116] D. R., Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc 64 (1992), 265–338.
[117] H., Heilbronn, On the class number of imaginary quadratic fields, Quart. J. Math. 5 (1934), 150–160.
[118] H. A., Helfgott, Minor arcs for Goldbach's problem, Preprint, arXiv:1205.5252, 2012.
[119] H. A., Helfgott, Major arcs for Goldbach's problem, Preprint, arXiv:1305.2897, 2013.
[120] H. A., Helfgott, The ternary Goldbach conjecture (transl. by M. Bilu, rev. by author), Gaz. Math. 140 (2014), 5–18.
[121] A., Hildebrand, Integers free of large prime factors and the Riemann hypothesis, Mathematika 31 (1984), 258–271.
[122] A., Hildebrand and G., Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), 411–484.
[123] J., Hoffstein, J., Pipher and J. H., Silverman, An Introduction to Mathematical Cryptography, Springer, 2008.
[124] C., Hooley, On Artin's conjecture, J. reine angew. Math. 225 (1967), 209–220.
[125] L., Hörmander, Linear Partial Differential Operators, Academic Press, 1964.
[126] L., Hörmander, The Analysis of Linear Partial Differential Operators, Springer, 1983.
[127] J., Horváth, Topological Vector Spaces and Distributions, Addison-Wesley, 1966.
[128] K., Ireland and M., Rosen, A Classical Introduction toModern Number Theory, Springer, 1993.
[129] A., Ivić, Riemann Zeta Function: Theory and Applications, Dover, 2003.
[130] H., Iwaniec, Spectral Methods of Automorphic Forms, 2nd edn, American Mathematical Society, 2002.
[131] H., Iwaniec and E., Kowalski, Analytic Number Theory, AmericanMathematical Society, 2004.
[132] H., Iwaniec, Conversations on the exceptional character, in Analytic Number Theory, Lecture Notes in Mathematics, vol. 1891, pp. 97–132, Springer, 2006.
[133] H., Iwaniec, Prime numbers and L-functions, in Proceedings of the International Congress of Mathematicians, Madrid, 2006, vol. 1, pp. 280–306, European Mathematical Society, 2007.
[134] N., Jacobson, Basic Algebra I, 2nd edn, Dover, 2009.
[135] G. J. O., Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
[136] J., Jost and X., Li-Jost, Calculus of Variations, Cambridge University Press, 1998.
[137] M., Jutila, On character sums and class numbers, J. Number Theory 5 (1973), 203–214.
[138] A. A., Karatsuba and S. M., Voronin, Riemann Zeta Function (transl. from Russian by N. Koblitz), Walter de Gruyter, 1994.
[139] N., Katz, L-functions and monodromy: four lectures on Weil II, Adv. Math. 160 (2001), 81–132.
[140] H., Ki and Y.-O., Kim, On the de Bruijn–Newman constant, Adv. Math. 222 (2009), 281–306.
[141] N., Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd edn, Springer, 1993.
[142] J.-M., De Koninck and F., Luca, Analytic Number Theory: Exploring the Anatomy of Integers, American Mathematical Society, 2012.
[143] E., Kowalski, A survey of algebraic exponential sums and some applications, in Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry, vol. II, London Mathematical Society Lecture Note Series 384, pp. 178–201, Cambridge University Press, 2011.
[144] T., Kubota, Elementary Theory of Eisenstein Series, Halsted, 1973.
[145] J. C., Lagarias, On a positivity property of the Riemann ξ-function, Acta Arith. 99 (1999) 217–213.
[146] J. C., Lagarias, The Riemann hypothesis: arithmetic and geometry, in Surveys in Noncommutative Geometry, Clay Mathematics Proceedings, vol. 6, pp. 127–141, American Mathematical Society and Clay Mathematics Institute, 2006.
[147] J. C., Lagarias and K., Soundararajan, Smooth solutions to the abc equation: the xyz conjecture, J. Théor. Nombres Bordeaux 23 (2011) 209–234.
[148] J. C., Lagarias and K., Soundararajan, Counting smooth solutions to the equation A+B = C, Proc. London Math. Soc. (3) 104 (2012), 770–798.
[149] E., Landau, Handbuch der lehre von der Verteilung der Primzahlen, 2nd edn, vols 1 and 2, Chelsea, 1953.
[150] E., Landau, Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel, Göttinger Nachrichten (1924), 202–206.
[151] B., Landreau and F., Richard, The Beurling–Nyman criterion for the Riemann hypothesis: numerical aspects, Exp. Math. 11 (2002), 349–360.
[152] S., Lang, Analysis I, Addison-Wesley, 1968.
[153] N., Levinson, On closure problems and zeros of the Riemann zeta function, Proc. Amer. Math. Soc. 7 (1956), 838–845.
[154] X.-J., Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory 65 (1997), 325–333.
[155] U. V., Linnik, The large sieve, C. R. (Dokl.) Acad. Sci. URSS, N.S. 30 (1941), 292–294.
[156] U. V., Linnik, A remark on the least quadratic non-residue, C. R. (Dokl.) Acad. Sci. URSS, N.S. 36 (1942), 119–120.
[157] U. V., Linnik, On the least prime in an arithmetic progression. I. The basic theorem; II. The Deuring–Heilbronn phenomenon, Mat. Sbornik 15 (1947), 139–178. 347–368.
[158] E. R., Lorch, Spectral Theory, Oxford University Press, 1962.
[159] M., Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field, Acta Arith. 14 (1868), 117–140.
[160] D. A., Marcus, Number Fields, Springer, 1977.
[161] D. W., Masser, On abc and discriminants, Proc. Amer. Math. Soc. 130 (2002) 3141–3150.
[162] Mathematics Genealogy Project, http://genealogy.math.ndsu.nodak.edu/index.php.
[163] Y., Matiyasevich, F., Saidak and P., Zvengrowski, Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions, Acta Arith. 166 (2014), 189–200.
[164] J., Maynard, Small gaps between primes, Ann. Math. (2) 181 (2015), 383–413.
[165] K., Mazhouda and S., Omar, The Cardon and Roberts’ criterion for the Riemann hypothesis, Analysis (Berlin) 33 (2013), 309–318.
[166] K., Mazhouda, Reformulation of the Li criterion for the Selberg class, Preprint, arXiv:1405.7354v3, 2015.
[167] B., Mazur and W. A., Stein, The Riemann Hypothesis, Cambridge University Press, 2016.
[168] H., Mellin, U berden Zusammenhang zwischen den linearen Differential- und Differenzengleichungen, Acta Math. 25 (1902), 139–164.
[169] F., Mertens, Uber einize asymptotische Gesetse der Zahlentheorie, J. reine angew. Math. 77 (1874), 46–62.
[170] S. J., Miller, A symplectic test of the L-functions ratios conjecture, Int. Math. Res. Notices 2008 (2008), art. 146 (36pp.).
[171] J., Milne, Elliptic Curves, BookSurge, 2006.
[172] H. L., Montgomery and R. C., Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge University Press, 2007.
[173] P., Moree, Artin's primitive root conjecture, a survey, Integers 10 (2012), 1305–1416.
[174] P., Moree, Nicolaas Govert de Bruijn, the enchanter of friable integers, Indag. Math. (N.S.) 24 (2013), 224–801.
[175] G. L., Mullen and C., Mummert, Finite Fields and Applications I, Student Mathematical Library, American Mathematical Society, 2007.
[176] M. R., Murty and V. K., Murty, A variant of the Bombieri–Vinogradov theorem, in Number Theory, Montreal, 1985, Canadian Mathematical Society Conference Proceedings, vol. 7, pp. 243–272, American Mathematical Society, 1987.
[177] M. R., Murty and V. K., Murty, Non-Vanishing of L-Functions and Applications, Birkhauser, 1997.
[178] M. R., Murty and K. L., Petersen, A Bombieri–Vinogradov theorem for all number fields, Trans. Amer. Math. Soc. 365 (2013), 4987–5032.
[179] D., Naccache and I. E., Shparlinski, Divisibility, smoothness and cryptographic applications, in Algebraic Aspects of Digital Communications, NATO Science for Peace and Security Series – D: Information and Communication Security, vol. 24, pp. 115–173, IOS Press, 2009.
[180] W., Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer, 1989.
[181] M. B., Nathanson, Elementary Methods in Number Theory, Springer, 2000.
[182] C. M., Newman, Fourier transforms with only real zeros, Proc. Amer. Math. Soc. 61 (1976), 245–251.
[183] T. S., Norfolk, A., Ruttan and R. S., Varga, A lower bound for the de Bruijn–Newman constant Λ II, in Progress in Approximation Theory, eds A. A. Gonchar and E. B. Saff, pp. 403–418, Springer, 1992.
[184] K. K., Norton, Numbers with Small Prime Factors, and the Least kth Power Non-Residue, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, 1971.
[185] B., Nyman, On the one dimensional translation group and semi-group in certain function spaces, Thesis, University of Uppsala, 1950.
[186] F., Oberhettinger, Tables of Mellin Transforms, Springer, 1974.
[187] A., Odlyzko, An improved lower bound for the de Bruijn–Newman constant, Numer. Algorithms 25 (2000), 293–303.
[188] F. W. J., Olver, D.W., Lozier, R.F., Boisvert and C. W., Clark (eds), NIST Handbook of Mathematical Functions, U.S. Department of Commerce, National Institute of Standards and Technology, Cambridge University Press, 2010.
[189] J., Oesterlé, Nouvelles Approches du “Théor`eme” de Fermat, Sém. Bourbake, Exp. No. 694, Astérisque, no. 161–162, pp. 165–186, Société Mathématiques de France, 1988.
[190] S., Omar and K., Mazhouda, Le crit`ere de Li et l'hypoth`ese de Riemann pour la classe de Selberg, J. Number Theory 125 (2007), 50–58.
[191] S., Omar and K., Mazhouda, The Li criterion and the Riemann hypothesis for the Selberg class II, J. Number Theory 130 (2010), 1098–1108.
[192] K., Ono and K., Soundararajan, Ramanujan's ternary quadratic form, Invent. Math. 130 (1997), 415–454.
[193] M. L., Patrick, Extensions of inequalities of the Laguerre and Turán type, Pacific J. Math. 44 (1973), 675–682.
[194] S. J., Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Studies in Advanced Mathematics, vol. 14, Cambridge University Press, 1988.
[195] A., Perelli, J., Pintz and S., Salerno, Bombieri's theorem in short intervals, Ann. Scu. Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 529–539.
[196] A., Perelli, J., Pintz and S., Salerno, Bombieri's theorem in short intervals. II, Invent. Math. 79 (1985), 1–9.
[197] G., Pólya, George Pólya: Collected Papers, Vol. II, ed. R. P. Boas, MIT Press, 1974.
[198] G., Purdy, The real zeros of the Epstein zeta function, Ph.D. Thesis, University of Illinois, 1972.
[199] L. D., Pustyl'nikov, On a property of the classical zeta-function associated with the Riemann hypothesis, Russian Math. Surveys 55 (1999), 262–263.
[200] O., Ramaré, On Snirel'man's constant, Ann. Scu. Norm. Sup. Pisa Cl. Sci. 22 (1995), 645–706.
[201] A., Rényi, On the representation of an even number as the sum of a prime and of an almost prime, Amer. Math. Soc. Transl. (2) 19 (1962), 299–321.
[202] P., Ribenboim, My Numbers, My Friends, Springer, 2000.
[203] B., Riemann, Gesammelte Werke, Teubner, 1893; reprinted by Dover, 1953.
[204] M., Riesz, Sur l'hypoth`ese de Riemann, Acta Math. 40 (1916), 185–190.
[205] G., Robin, Grandes valeurs de la fonction sommes des diviseurs et hypoth`ese de Riemann, J. Math. Pures Appl. 63 (1984), 187–213.
[206] G., Robin, Sur la différence Li(θ(x)) − π(x), Ann. Fac. Sci. Toulouse Math. 6 (1984), 257–268.
[207] H. L., Royden, Real Analysis, 2nd edn, Macmillan, 1968.
[208] W., Rudin, Real and Complex Analysis, 2nd edn, McGraw-Hill, 1974.
[209] W., Rudin, Functional Analysis, 2nd edn, McGraw-Hill, 1991.
[210] K., Sabbagh, The Riemann Hypothesis, Farrar, Straus and Giroux, 2003.
[211] R., Salem, Sur une proposition équivalente `a l'hypoth`ese de Riemann. C. R. Acad. Sci. Paris 236 (1953), 127–128.
[212] Y., Saouter, X., Gourdon and P., Demichel, An improved lower bound for the de Bruijn– Newman constant, Math. Comp. 80 (2011), 2259–2279.
[213] P., Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math. 34 (1981), 719–739.
[214] P., Sarnak, Problems of the Millennium: the Riemann Hypothesis (2004), Annual Report of the Clay Mathematics Institute, 2004.
[215] M., du Sautoy, The Music of the Primes, HarperCollins, 2003.
[216] M., Schechter, An Introduction to Nonlinear Analysis, Cambridge University Press, 2004.
[217] L., Schwartz, Théorie des Distributions, vols 1 and 2, Hermann, 1951.
[218] S. K., Sekatskii, S., Beltraminelli and D., Merlini, On equalities involving integrals of the logarithm of the Riemann-function and equivalent to the Riemann hypothesis, Ukrain. Math. J. 64 (2012), 247–261.
[219] S. K., Sekatskii, Generalized Bombieri–Lagarias’ theorem and generalized Li's criterion with its arithmetic interpretation, Ukrain. Mat. Zh. 66 (2014), 371–383.
[220] J. A., Shohat and J. D., Tamarkin, The Problem of Moments, Mathematical Surveys, no. 1, American Mathematical Society, 1943.
[221] C. L., Siegel, On the zeros of the Dirichlet L-functions, Ann. Math. 46 (1945), 409–422.
[222] J. H., Silverman and J., Tate, Rational points on elliptic curves, Undergraduate Texts, Springer, 1968.
[223] J. H., Silverman, The Arithmetic of Elliptic Curves, 1st edn, Graduate Texts in Mathematics, vol. 106, Springer, 1986; 2nd edn, 2009.
[224] H., Skovgaard, On inequalities of the Turán type, Math. Scand. 2 (1954), 65–73.
[225] L., Smajlović, On Li's criterion for the Riemann hypothesis for the Selberg class, J. Number Theory 130 (2010), 828–851.
[226] J., Sondow and C., Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann hypothesis, Period. Math. Hungar. 60 (2010), 37–40.
[227] K., Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s= 12, Ann. Math. (2) 152 (2000), 447–488.
[228] S., Stahl, The Poincaré Half-Plane: A Gateway to Modern Geometry, Jones and Bartlett, 1993.
[229] A., Steiger, Course Notes, 2006.
[230] T., Tao, Every odd number greater than 1 is the sum of at most five primes,Math. Comp. 83 (2012), 997–1038.
[231] T., Tao, Web based lecture notes on the Bombieri–Vinogradov theorem, 2016.
[232] G., Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995.
[233] J., Thorner, A variant of the Bombieri–Vinogradov theorem in short intervals and some questions of Serre, Math. Proc. Cambridge Philos. Soc. 161 (2016), 53–63.
[234] E. C., Titchmarsh, A divisor problem, Rend. Circ. Mat. Palermo 54 (1930), 414–429.
[235] E. C., Titchmarsh, The Theory of Functions, 2nd edn, Oxford University Press, 1939.
[236] E. C., Titchmarsh and D. R., Heath-Brown, The Theory of the Riemann Zeta-Function, 2nd edn, Oxford University Press, 1986.
[237] F., Tréves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967.
[238] O. N., Vasilenko, Number-Theoretic Algorithms in Cryptography, American Mathematical Society, 2007.
[239] R. C., Vaughan, Mean value theorems in prime number theory, J. London Math. Soc. (2) 10 (1975), 153–162.
[240] R. C., Vaughan, An elementary method in prime number theory, Acta Arith. 37 (1980), 111–115.
[241] J.-L., Verger-Gaugry, Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and the dichotomy of Perron numbers, Unif. Distrib. Theory 3 (2008), 157–190.
[242] A., Verjovsky, Arithmetic geometry and dynamics in the unit tangent bundle of the modular orbifold, in Dynamical Systems (Santiago, 1990), Pitman Research Notes in Mathematics Series, 285, pp. 263–298, Longman, 1993.
[243] A., Verjovsky, Discrete measures and the Riemann hypothesis, Kodai Math. J. 17 (1994), 596–608.
[244] A. I., Vinogradov, On the density hypothesis for Dirichlet L-series, Izv. Akad. Nauk SSSR, Ser. Mat. 29 (1965), 903–934.
[245] A. I., Vinogradov, Corrections to the work of A. I. Vinogradov “On the density hypothesis for Dirichlet L-series”, Izv. Akad. Nauk SSSR, Ser. Mat. 30 (1965), 719–729.
[246] V. V., Volchkov, On an equality equivalent to the Riemann hypothesis, Ukrain. Math. J. 47 (1995), 422–423.
[247] A., Voros, Sharpenings of Li's criterion for the Riemann hypothesis, Math. Phys. Anal. Geom. 9 (2006), 53–63.
[248] F. T., Wang, A note on the Riemann zeta-function, Bull. Amer. Math. Soc. 52 (1946), 319–321.
[249] M., Watkins, Real zeros of real odd Dirichlet L-functions, Math. Comp. 73 (2003), 415–423.
[250] M., Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), 907–938.
[251] A., Weil, Sur les “formules explicites” de la théorie des nombres premiers, Meddel. Fran Lunds Univ. Mat. Sem. (1952), 252–265; see also Oeuvres Scientifiques – Collected Papers, Vol. II, corrected 2nd printing, pp. 48–61, Springer, 1980.
[252] A., Weil, Sur les formules explicites de la théorie des nombres premiers, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3–18.
[253] A., Weil, Number Theory: An Approach Through History. From Hammurapi to Legendre, Birkhauser, 1984.
[254] E. T., Whittaker and G. N., Watson, A Course of Modern Analysis, Cambridge University Press, 1962.
[255] D. V., Widder, The Laplace Transform, Princeton University Press, 1946.
[256] N., Wiener, Tauberian theorems, Ann Math. 33 (1932), 1–100.
[257] H. S., Wilf, On the zeros of Riesz’ function in the analytic theory of numbers, Illinois J. Math. 8 (1964), 639–641.
[258] M., Wolf, Evidence in favour of the Báez-Duarte criterion for the Riemann hypothesis, Comp. Meth. Sci. Technol. 14 (2008), 47–54.
[259] M., Wolf, Some remarks on the Báez-Duarte criterion for the Riemann hypothesis, Comp. Meth. Sci. Technol. 20 (2014), 39–47.
[260] S., Yakubovich, Integral and series transformations via Ramanujan's identities and Slaem's type equivalences to the Riemann hypothesis, Integral Transforms Spec. Funct. 25 (2014), 255–271.
[261] H., Yoshida, On Hermitian forms attached to zeta functions, in Zeta Functions in Geometry (Tokyo, 1990), Advanced Studies in Pure Mathematics, vol. 21, pp. 281–325, Kinokuniya, 1992.
[262] K., Yosida, Functional Analysis, 4th edn, Springer, 1974.
[263] D., Zagier, Eisenstein series and the Riemann zeta-function, in Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Tata Institute Studies in Mathematics, vol. 10, pp. 275–301, Springer, 1981.
[264] Y., Zhang, Bounded gaps between primes, Ann. Math. (2) 1979 (2014), 1121–1174.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed