Skip to main content Accessibility help
×
  • Cited by 2
Publisher:
Cambridge University Press
Online publication date:
November 2012
Print publication year:
2012
Online ISBN:
9781139236973

Book description

Hardy's Z-function, related to the Riemann zeta-function ζ(s), was originally utilised by G. H. Hardy to show that ζ(s) has infinitely many zeros of the form ½+it. It is now amongst the most important functions of analytic number theory, and the Riemann hypothesis, that all complex zeros lie on the line ½+it, is perhaps one of the best known and most important open problems in mathematics. Today Hardy's function has many applications; among others it is used for extensive calculations regarding the zeros of ζ(s). This comprehensive account covers many aspects of Z(t), including the distribution of its zeros, Gram points, moments and Mellin transforms. It features an extensive bibliography and end-of-chapter notes containing comments, remarks and references. The book also provides many open problems to stimulate readers interested in further research.

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

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.
×

Contents

References
References
[And] R. J., Anderson, On the function Z(t) associated with the Riemann zetafunction, J. Math. Anal. Appl. 118 (1986), 323–340.
[AGZ] G. W., Anderson, A., Guionnet and O., Zeitouni, An Introduction to Random Matrices, Cambridge University Press, Cambridge, 2010.
[Ape] R., Apéry, Interpolation de fractions continues et irrationalité de certaines constantes, Math. CTHS Bull. Sec. Sci. II (Bibl. Nat. Paris), 1981, 37–53.
[Atk1] F. V., Atkinson, The mean value of the zeta-function on the critical line, Quart. J. Math. Oxford 10 (1939), 122–128.
[Atk2] F. V., Atkinson, The mean value of the zeta-function on the critical line, Proc. London Math. Soc. 47 (1941), 174–200.
[Atk3] F. V., Atkinson, A mean value property of the Riemann zeta-function, J. London Math. Soc. 23 (1948), 128–135.
[Atk4] F. V., Atkinson, The mean value of the Riemann zeta-function, Acta Math. 81 (1949), 353–376.
[Bac] R. J., Backlund, Sur les zéros de la fonction ζ(s) de Riemann, Comptes Rendus Acad. Sci. 158 (1914), 1979–1982.
[Bet] S., Bettin, The second moment of the Riemann zeta-function with unbounded shifts, preprint available at arXiv:1111.0925.
[Boc] S., Bochner, On Riemann's functional equation with multiple gamma factors, Annals of Math. 67 (1958), 29–41.
[BoHe] E., Bombieri and D., Hejhal, Sur les zéros des fonctions zeta d'Epstein, Comptes Rendus Acad. Sci. Paris 304 (1987), 213–217.
[BoIw1] E., Bombieri and H., Iwaniec, On the order of ζ(½ + it), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)13 (1986), 449–472.
[BoIw2] E., Bombieri and H., Iwaniec, Some mean value theorems for exponential sums, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 473–486.
[BoGh] E., Bombieri and A., Ghosh, Around the Davenport-Heilbronn function, Russian Math. Surveys 66 (2011), 221–270,
Uspekhi Mat. Nauk 66 (2011), 15–66.
[Bom1] E., Bombieri, Riemann Hypothesis. The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 107–124.
[Bom2] E., Bombieri, The classical theory of zeta and L-functions, Milan J.Math. 78 (2010), 11–59.
[BCRW] P., Borwein, S., Choi, B., Rooney and A., Weirathmueller, The Riemann Hypothesis, a Resource for the Afficionado and the Virtuoso Alike, CMS Books in Mathematics, Canadian Math. Soc., 2008.
[BCY] H., Bui, B., Conrey and M. P., Young, More than 41% of the zeros of the zeta function are on the critical line, to appear in Acta Arith., see arXiv:1002.4127.
[BhS1] G., Bhowmik and J.-C., Schlage-Puchta, Natural boundaries of Dirichlet series, Func. Approx. Comment. Math. 37 (2007), 17–29.
[BhS2] G., Bhowmik and J.-C., Schlage-Puchta, Essential singularities of Euler products, to appear, preprint available at arXiv:1001.1891.
[Bre] J., Bredberg, Large gaps between consecutive zeros on the critical line of the Riemann zeta-function, to appear, preprint available at arXiv:1101.3197.
[Bui] H. M., Bui, Large gaps between consecutive zeros of the Riemann zeta-function, J. Number Theory 131 (2011), 67–95.
[Cha] K., Chandrasekharan, Arithmetical Functions, Springer Verlag, Berlin-Heidelberg-New York, 1970.
[ChNa] K., Chandrasekharan and R., Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetic functions, Annals of Math. 76 (1962), 93–136.
[ChSo] V., Chandee and K., Soundararajan, Bounding ∣ζ(½ + it)∣ on the Riemann Hypothesis, Bull. London Math. Soc. (2011), 243–250.
[CMoP] E., Carletti, G. Monti, Bragadin and A., Perelli, On general L-functions, Acta Arith. 66 (1994), 147–179.
[Con1] J. B., Conrey, The fourth moment of derivatives of the Riemann zeta-function, Quarterly J. Math., Oxf. II. Ser. 39 (1988), No. 153, 21–36.
[Con2] J. B., Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. Reine Angew. Math. 399 (1989), 1–26.
[Con3] J. B., Conrey, A note on the fourth power moment of the Riemann zeta-function, in: B. C., Berndt et al.(eds.) Analytic Number Theory. Vol. 1. Proc. of a Conf. in Honor of Heini Halberstam, Urbana, 1995, Birkhäuser, Prog. Math. 138 (1996), 225–230.
[Con4] J. B., Conrey, The Riemann hypothesis, Notices Amer. Math. Soc. 50 (2003), 341–353.
[CoGh1] J. B., Conrey and A., Ghosh, A mean value theorem for the Riemann zeta-function at its relative extrema on the critical line, J. Lond. Math. Soc., II. Ser. 32 (1985), 193–202.
[CoGh2] J. B., Conrey and A., Ghosh, A simpler proof of Levinson's theorem, Math. Proc. Camb. Phil. Soc. 97 (1985), 385–395.
[CoGh3] J. B., Conrey and A., Ghosh, On the Selberg class of Dirichlet series: small degrees, Duke Math. J. 72 (1993), 673–693.
[CGG1] 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.
[CGG2] J. B., Conrey, A., Ghosh and S. M., Gonek, Large gaps between zeros of the zeta-function, Mathematika 33 (1986), 212–238.
[CGG3] J. B., Conrey, A., Ghosh and S. M., Gonek, Simple zeros of the Riemann zeta-function, Proc. Lond. Math. Soc., III. Ser. 76 (1998), No. 3, 497–522.
[CFKRS1] J. B., Conrey, D. W., Farmer, J. P., Keating, M. O., Rubinstein and N. C., Snaith, Integral moments of L-functions, Proc. London Math. Soc. (3) 91 (2005), 33–104.
[CFKRS2] J. B., Conrey, D. W., Farmer, J. P., Keating, M. O., Rubinstein and N. C., Snaith, Lower order terms in the full moment conjecture for the Riemann zeta function, J. Number Theory 128 (2008), 1516–1554.
[CoGo] J. B., Conrey and S. M., Gonek, High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), 577–604.
[COSV] G., Csordas, A. M., Odlyzko, W., Smith and R. S., Varga, A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant LAMBDA, Electr. Trans. Num. Anal. 1 (1993), 104–111.
[Dah] G., Dahlquist, On the analytic continuation of Eulerian products, Ark. Mat. 1 (1952), 533–554.
[DaHe] H., Davenport and H., Heilbronn, On the zeros of certain Dirichlet series I, II, J. London Math. Soc. 11 (1936), 181–185
H., Davenport and H., Heilbronn, On the zeros of certain Dirichlet series I, II, J. London Math. Soc. 11 (1936), 307–312.
[Del] H., Delange, Généralisation du théorème de Ikehara, Annales scien. E.N.S. 71 (1954), 213–242.
[Deli] P., Deligne, La conjecture de Weil. I., Publications Mathématiques de l'IHÉS 43 (1974), 273–307.
[DGG] A., Diaconu, P., Garrett and D., Goldfeld, Natural boundaries and a correct notion of integral moments of L-functions, preprint, 2009.
[DGH] A., Diaconu, D., Goldfeld and J., Hoffstein, Multiple Dirichlet series and moments of zeta and L-functions, Compositio Math. 139, No. 3 (2003), 297–360.
[Edw] H. M., Edwards, Riemann's Zeta-Function, Academic Press, New York- London, 1974 (Dover Publications Inc., Mineola, NY, 2001. Reprint of the 1974 original).
[EMOT] A., Erdélyi, W., Magnus, F., Oberhettinger and F. G., Tricomi, Higher Transcendental Functions, Volume I, McGraw-Hill, 1953.
[Est1] T., Estermann, On certain functions represented by Dirichlet series, Proc. London Math. Soc. 27 (1928), 435–448.
[Est2] T., Estermann, Introduction to Modern Prime Number Theory, Cambridge University Press, Cambridge, 1969.
[Eul] L., Euler, Remarques sur un beau rapport entre les séries des puissances tout directes que réciproques, Mém. Acad. Roy. Sci. Belles Lettres 17 (1768), 83–106.
[Fen] S., Feng, Zeros of the Riemann zeta-function on the critical line, to appear, preprint available at arXiv:1003.0059.
[FGH] D. W., Farmer, S. M., Gonek and C. P., Hughes, The maximum size of L-functions. J. Reine Angew. Math. 609 (2007), 215–236.
[Fla] C., Flammer, Spheroidal Wave Functions, Stanford University Press, Stanford, CA, 1957.
[For1] K., Ford, Vinogradov's integral and bounds for the Riemann zeta-function, Proc. London Math. Soc. III Ser. 85 (2002), 565–633.
[For2] K., Ford, Zero-free regions for the Riemann zeta-function, in: M. A., Bennett et al. (ed.) Number Theory for the Millennium II. Proc. of the Conf. on number theory, Urbana-Champaign, IL, USA, 2000. Natick, MA: A. K. Peters, 25–56 (2002).
[Fuj1] A., Fujii, On the distribution of zeros of the Riemann zeta function in short intervals, Bull. Amer. Math. Soc. 81 (1975), 139–142.
[Fuj2] A., Fujii, On the difference between r consecutive ordinates of the zeros of the Riemann zeta function, Proc. Japan Acad. 51 (1975), 741–743.
[Gab] W., Gabcke, Neue Herleitung und explizite Restabschätzung der Riemann-Siegel Formel, Univ. Göttingen, Ph.D. Dissertation, Göttingen, 1979, pp. 153.
[Gho] A., Ghosh, On the Riemann zeta function-mean value theorems and the distribution of |S(T)|, J. Number Theory 17 (1983), 93–102.
[Gol1] D. A., Goldston, Large differences between consecutive prime numbers, Ph.D. thesis, University of California, Berkeley, 1981.
[Gol2] D. A., Goldston, On the pair correlation conjecture for zeros of the Riemann zeta-function, J. Reine Angew. Math. 385 (1988), 24–40.
[Gol3] D. A., Goldston, Notes on pair correlation of zeros and prime numbers, in: F., Mezzadri et al.(eds.) Recent Perspectives in Random Matrix Theory and Number Theory, Proc. “Random Matrix Approaches in Number Theory”, London Mathematical Society Lecture Note Series, 322, Cambridge University Press, Cambridge (2005), 79–110.
[GoGo] D. A., Goldston and S. M., Gonek, A note on S(t) and the zeros of the Riemann zeta-function, Bull. London Math. Soc. 39 (2007), 482–486.
[GoMo] D. A., Goldston and H. L, Montgomery, Pair Correlation and Primes in Short Intervals, Analytic Number Theory and Diophantine Problems, Birkhäuser, Boston, Mass., 1987, pp. 187–203.
[Gon] S. M., Gonek, On negative moments of the Riemann zeta-function, Mathematika 36 (1989), 71–88.
[GrKo] S. W., Graham and G., Kolesnik, Van der Corput's method for exponential sums, London Mathematical Society Lecture Note Series, 126, Cambridge University Press, Cambridge, 1991.
[Gourd] X., Gourdon, The first 1013 zeros of the Riemann zeta-function and zeros computation at very large height, 2004, http://numbers.computation.free.fr/Constants/Miscellaneous.
[Gours]É., Goursat, Cours d'Analyse Mathématique, Dover, New York, 1959.
[Gra] J. P., Gram, Sur les zéros de la fonction de Riemann, Acta Mathematica 27 (1903), 289–304.
[Hal1] R. R., Hall, The behaviour of the Riemann zeta-function on the critical line, Mathematica 46 (1999), 281–313.
[Hal2] R. R., Hall, A Wirtinger type inequality and the spacing between the zeros of the Riemann zeta-function, J. Number Theory 93 (2002), 235–245.
[Hal3] R. R., Hall, On the extreme values of the Riemann zeta-function between its zeros on the critical line, J. Reine Angew. Math. 560 (2003), 29–41.
[Hal4] R. R., Hall, Generalized Wirtinger inequalities, random matrix theory, and the zeros of the Riemann zeta-function, J. Number Theory 97 (2002), 397–409.
[Hal5] R. R., Hall, On the stationary points of Hardy's function Z(t), Acta Arith. 111 (2004), 125–140.
[Hal6] R. R., Hall, A new unconditional result about large spaces between zeta zeros, Mathematika 53 (2005), 101–113.
[Hal7] R. R., Hall, Extreme values of the Riemann zeta-function on short zero intervals, Acta Arith. 121 (2006), 259–273.
[Ham] H., Hamburger, Über die Riemmansche Funkionalgleichung der ζ- Funktion I, II, III, Math. Zeit. 10 (1921), 240–254
H., Hamburger, Über die Riemmansche Funkionalgleichung der ζ- Funktion I, II, III, Math. Zeit. (1922), 224–245
H., Hamburger, Über die Riemmansche Funkionalgleichung der ζ- Funktion I, II, III, Math. Zeit. 13 (1922), 283–311.
[HaIv1] J. L., Hafner and A., Ivić, On some mean value results for the Riemann zeta-function, Proceedings International Number Theory Conference Québec 1987, Walter de Gruyter and Co., 1989, Berlin-New York, pp. 348–358.
[HaIv2] J. L., Hafner and A., Ivić, On the mean square of the Riemann zeta-function on the critical line, J. Number Theory 32 (1989), 151–191.
[Har] G., Harcos, Uniform approximate functional equation for principal L-functions, Inter. Math. Research Notices 18 (2002), 923–932.
[Har1] G. H., Hardy, On the zeros of Riemann's zeta-function, Proc. London Math. Soc. Ser. 2 13 (records of proceedings at meetings), March 1914.
[Har2] G. H., Hardy, Sur les zéros de la fonction ζ(s) de Riemann, Comptes Rendus Acad. Sci. (Paris) 158 (1914), 1012–1014.
[Har3] G. H., Hardy, A Mathematician's Apology, Cambridge University Press, Cambridge (1940), 2004 reissue.
[Har4] G. H., Hardy, Ramanujan, Cambridge University Press, London, 1940, reissue AMS Chelsea Pub., 1999.
[Har5] G. H., Hardy, Divergent Series, Clarendon Press, Oxford, 1949.
[Har6] G. H., Hardy, A Course of Pure Mathematics (10th edn). Cambridge University Press, Cambridge (1952) [1908], 2008 reissue.
[Har7] G. H., Hardy, Collected Papers of G. H. Hardy; Including Joint papers with J.E. Littlewood and Others, London Mathematical Society, 1966.
[HaLi1] G. H., Hardy and J. E., Littlewood, Contributions to the theory of the Riemann zeta-function and the distribution of primes, Acta Math. 41 (1918), 119–196.
[HaLi2] G. H., Hardy and J. E., Littlewood, The zeros of Riemann's zeta-function on the critical line, Math. Zeitschrift 10 (1921), no. 3–4, 283–317.
[HaLi3] G. H., Hardy and J. E., Littlewood, The approximate functional equation for ζ(s)and ζ2(s), Proc. London Math. Soc.(2) 29 (1929), 81–97.
[HaRi] G. H., Hardy and M., Riesz, The General Theory of Dirichlet Series, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1915.
[Has] C. B., Haselgrove, Tables of the Riemann Zeta Function, Cambridge University Press, Cambridge, 1960.
[HaWr] G. H., Hardy and E. M., Wright, An Introduction to the Theory of Numbers (6th edn), Oxford University Press, Oxford, 2008.
[Hea1] D. R., Heath-Brown, The twelfth power moment of the Riemann zeta-function, Quart. J. Math.(Oxford) 29 (1978), 443–462.
[Hea2] D. R., Heath-Brown, The mean value theorem for the Riemann zeta-function, Mathematika 25 (1978), 177–184.
[Hea3] D. R., Heath-Brown, The fourth power moment of the Riemann zeta function, J. London Math. Soc. (3)38 (1979), 385–422.
[Hea4] D. R., Heath-Brown, Fractional moments of the Riemann zeta-function, J. London Math. Soc. 24 (1981), 65–78.
[Hej1] D. A., Hejhal, On the distribution of ∣log ζ′ (½ + it)∣, in: K. E., Aubert et al. (eds.) Number Theory, Trace Formulas and Discrete Groups, Proceedings Selberg 1987 Symposium, Academic Press, 1989, 343–370.
[Hej2] D. A., Hejhal, On a result of Selberg concerning zeros of linear combinations of L-functions, Int. Math. Res. Not. 11 (2000), 551–577.
[HiOd1] G. A., Hiary and A. M., Odlyzko, The zeta function on the critical line: numerical evidence for moments and random matrix theory models, to appear, preprint available at arXiv:1008.2173.
[HiOd2] G. A., Hiary and A. M., Odlyzko, Numerical study of the derivative of the Riemann zeta-function at zeros, to appear, preprint available at arXiv:1105.4312.
[HKN] C. P., Hughes, J. P., Keating and O., Neil, Random matrix theory and the derivative of the Riemann zeta function, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 456 (2000), 2611–2627.
[Hut] J. I., Hutchinson, On the roots of the Riemann zeta-function, V. Trans. Amer. Math. Soc. 27 (1925), 27–49.
[Hux1] M. N., Huxley, Area, Lattice Points and Exponential Sums, Oxford Science Publications, Clarendon Press, Oxford, 1996.
[Hux2] M. N., Huxley, Exponential sums and the Riemann zeta function V, Proc. London Math. Soc.(3) 90 (2005), 1–41.
[HuIv] M. N., Huxley and A., Ivić, Subconvexity for the Riemann zeta-function and the divisor problem, Bulletin CXXXIV de l'Académie Serbe des Sciences et des Arts - 2007, Classe des Sciences Mathématiques et Naturelles, Sciences Mathématiques 32, 13–32.
[Ing] A. E., Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc.(2) 27 (1926), 273–300.
[Isr1] M. I., Israilov, Coefficients of the Laurent expansion of the Riemann zeta-function (Russian), Dokl. Akad. Nauk SSSR 12 (1979), 9–10.
[Isr2] M. I., Israilov, The Laurent expansion of the Riemann zeta-function (Russian), Trudy Mat. Inst. Steklova 158 (1981), 98–104.
[Iv1] A., Ivić, The Riemann Zeta-Function, John Wiley & Sons, New York, 1985 (reissue, Dover, Mineola, New York, 2003).
[Iv2] A., Ivić, On consecutive zeros of the Riemann zeta-function on the critical line, in: Séminaire de Théorie des Nombres, Université de Bordeaux 1986/87, Exposé no. 29, 14 pp.
[Iv3] A., Ivić, On a problem connected with zeros of ζ(s) on the critical line, Monatshefte Math. 104 (1987), 17–27.
[Iv4] A., Ivić, Mean Values of the Riemann Zeta-function, LN's 82, Tata Inst. of Fundamental Research, Bombay, 1991 (Springer Verlag, Berlin).
[Iv5] A., Ivić, On the fourth moment of the Riemann zeta-function, Publs. Inst. Math. (Belgrade) 57(71) (1995), 101–110.
[Iv6] A., Ivić, On sums of gaps between the zeros of ζ(s) on the critical line, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 6 (1995), 55–62.
[Iv7] A., Ivić, An approximate functional equation for a class of Dirichlet series, Journal of Analysis 3 (1995), 241–252.
[Iv8] A., Ivić, The Mellin transform and the Riemann zeta-function, in: W. G., Nowak and J. Schoißengeier, Vienna (eds.) Proceedings of the Conference on Elementary and Analytic Number Theory (Vienna, July 18–20, 1996), Universität Wien & Universität für Bodenkultur, 1996, 112–127.
[Iv9] A., Ivić, On the error term for the fourth moment of the Riemann zeta-function, J. London Math. Soc. 60 (2)(1999), 21–32.
[Iv10] A., Ivić, The Laplace transform of the fourth moment of the zeta-function, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11 (2000), 41–48.
[Iv11] A., Ivić, On some conjectures and results for the Riemann zeta-function, Acta. Arith. 99 (2001), 115–145.
[Iv12] A., Ivić, The Laplace transform of the fourth moment of of the zeta-function, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11 (2000), 41–48.
[Iv13] A., Ivić, On small values of the Riemann zeta-function on the critical line and gaps between zeros, Lietuvos Mat. Rinkinys 42 (2002), 31–45.
[Iv14] A., Ivić, On the estimation of Ƶ2(s), in: Anal. Probab. Methods Number Theory, A., Dubickas et al. (eds.) TEV, Vilnius, 2002, 83–98.
[Iv15] A., Ivić, On the integral of Hardy's function, Arch. Mathematik 83 (2004), 41–47.
[Iv16] A., Ivić, The Mellin transform of the square of Riemann's zeta-function, International J. of Number Theory 1 (2005), 65–73.
[Iv17] A., Ivić, On the estimation of some Mellin transforms connected with the fourth moment of ∣ζ(½ + it)∣, in: W., Schwarz and J., Steuding (eds.) Elementare und Analytische Zahlentheorie (Tagungsband), Proceedings ELAZ-Conference May 24–28, 2004, Franz Steiner Verlag, 2006, 77–88.
[Iv18] A., Ivić, On some reasons for doubting the Riemann Hypothesis, in: P., Borwein et al.(eds.) The Riemann Hypothesis, CMS Books in Mathematics, Springer, 2008.
[Iv19] A., Ivić, On the moments of the Riemann zeta-function in short intervals, Hardy-Ramanujan Journal 32 (2009), 4–23.
[Iv20] A., Ivić, On the Mellin transforms of powers of Hardy's function, Hardy- Ramanujan Journal 33 (2010), 32–58.
[Iv21] A., Ivić, On some problems involving Hardy's function, Central European J. Math. 8(6) (2010), 1029–1040.
[IvJu] A., Ivić and M., Jutila, Gaps between consecutive zeros of the Riemann zeta-function, Monatshefte Math. 105 (1988), 59–73.
[IJM] A., Ivić, M., Jutila and Y., Motohashi, The Mellin transform of powers of the Riemann zeta-function, Acta Arith. 95 (2000), 305–342.
[IvMo1] A., Ivić and Y., Motohashi, A note on the mean value of the zeta and L-functions VII, Proc. Japan Acad. Ser. A 66 (1990), 150–152.
[IvMo2] A., Ivić and Y., Motohashi, The mean square of the error term for the fourth moment of the zeta-function, Proc. London Math. Soc. (3)66 (1994), 309–329.
[IvMo3] A., Ivić and Y., Motohashi, The fourth moment of the Riemann zeta-function, J. Number Theory 51 (1995), 16–45.
[IvPe] A., Ivić and A., Perelli, Mean values of certain zeta-functions on the critical line, Litovskij Mat. Sbornik 29 (1989), 701–714.
[Iwa1] H., Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Bibl. de la Revista Iberoamericana, Madrid, 1995.
[Iwa2] H., Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics 17, American Mathematical Society, Providence, RI, 1997.
[Iwa3] H., Iwaniec, Spectral Methods of Automorphic Forms, 2nd edn, Graduate Studies in Mathematics, 53, American Mathematical Society, Providence, RI, 1997.
[Joy] D., Joyner, Distribution Theorems of L-functions, Pitman Research Notes in Mathematics Series 142, Longman Scientific & Technical, Harlow, John Wiley & Sons, New York, 1986.
[Jut1] M., Jutila, On the value distribution of the zeta-function on the critical line, Bull. London Math. Soc. 15 (1983), 513–518.
[Jut2] M., Jutila, Mean values of Dirichlet series via Laplace transforms, in: Y., Motohashi (ed.) Analytic Number Theory, London Math. Soc. LNS 247, Cambridge University Press, Cambridge, 1997, 169–207.
[Jut3] M., Jutila, Atkinson's formula revisited, in: Voronoï's Impact on Modern Science, Book 1, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 1998, 137–154.
[Jut4] M., Jutila, The Mellin transform of the square of Riemann's zeta-function, Periodica Math. Hung. 42 (2001), 179–190.
[Jut5] M., Jutila, The Mellin transform of the fourth power of the Riemann zeta-function, in: S. D, Adhikari, et al.(eds.) Number Theory. Proc. Inter. Conf. on Analytic Number Theory with special emphasis on L-functions, held at the Inst. Math. Sc., Chennai, India, January 2002. Ramanujan Math. Soc. LNS 1 (2005), 15–29.
[Jut6] M., Jutila, Atkinson's formula for Hardy's function, J. Number Theory 129 (2009), 2853–2878.
[Jut7] M., Jutila, An estimate for the Mellin transform of Hardy's function, Hardy- Ramanujan J. 33 (2010), 23–31.
[Jut8] M., Jutila, The Mellin transform of Hardy's function is entire (in Russian), Mat. Zametki 88 (4) (2010), 635–639.
[Jut9] M., Jutila, An asymptotic formula for the primitive of Hardy's function, Arkiv Mat. 49, No. 1, (2011), 97–107.
[Kac] J., Kaczorowski, Axiomatic theory of L-functions: the Selberg class, in: A., Perelli and C., Viola (eds.) Analytic Number Theory, Springer Verlag, Berlin-Heidelberg, 2006, 133–209.
[KaKo] A. A., Karatsuba and M. A., Korolev, The argument of the Riemann zeta function (English translation of Russian original)Russ. Math. Surv. 60 (2005), No. 3, 433–488;
translation from Usp. Mat. Nauk 60 (2005), No. 3, 41–96.
[KaPe1] J., Kaczorowski and A., Perelli, The Selberg class: a survey, in: Number Theory in Progress, Vol. 2 (Zakopane-Koscielisko, 1997), de Gruyter, Berlin, 1999, 953–992.
[KaPe2] J., Kaczorowski and A., Perelli, On the structure of the Selberg class, I: 0 ≤ d ≤ 1, Acta Math. 182 (1999), 207–241.
[KaPe3] J., Kaczorowski and A., Perelli, On the structure of the Selberg class, V: 1 < d < 5/3, Invent. Math. 150 (2002), 485–516.
[KaPe4] J., Kaczorowski and A., Perelli, On the structure of the Selberg class, VI: non-linear twists, Acta Arith. 116 (2005), 315–341.
[KaPe5] J., Kaczorowski and A., Perelli, On the structure of the Selberg class, VII: 1 < d < 2, Annals of Math. 173 (2011), 1397–1441.
[Kar1] A. A., Karatsuba, On the distance between adjacent zeros of the Riemann zeta-function lying on the critical line (Russian), Trudy Mat. Inst. Steklova 157 (1981), 49–63.
[Kar2] A. A., Karatsuba, On the zeros of the Davenport-Heilbronn function lying on the critical line (Russian), Izv. Akad. Nauk SSSR ser. mat. 54 no. 2 (1990), 303–315.
[KaS] J., Kalpokas and J., Steuding, On the value-distribution of the Riemann zeta-function on the critical line, Moscow Journal of Combinatorics and Number Theory 1 (2011), 26–42.
[KaVo] A. A., Karatsuba and S. M., Voronin, The Riemann Zeta-Function, Walter de Gruyter, Berlin-New York, 1992.
[Kea] J., Keating, The Riemann zeta-function and quantum chaology, Proc. Internat. School of Phys. Enrico Fermi CXIX (1993), 145–185.
[KeSn] J. P., Keating and N.C., Snaith, Random matrix theory and L-functions at s = 1/2, Comm. Math. Phys. 214 (2000), 57–89.
[Kob] H., Kober, Eine Mittelwertformel der Riemannschen Zetafunktion, Compositio Math. 3 (1936), 174–189.
[Kol] G., Kolesnik, On the estimation of multiple exponential sums, in: Recent Progress in Analytic Number Theory, Symp. Durham 1979 (Vol. 1), Academic Press, London, 1981, 231–246.
[Kore] J., Korevaar, Tauberian Theory, Grund. der math. Wissenschaften Vol. 329, Springer, Berlin, 2004.
[Kor1] M. A., Korolev, On the argument of the Riemann zeta function on the critical line. (English translation of Russian original)Izv. Math. 67 (2003), No. 2, 225–264;
translation from Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), No. 2, 21–60.
[Kor2] M. A., Korolev, Sign change of the function S(t) on short intervals (English translation of Russian original), Izv. Math. 69 (2005), No. 4, 719–731;
translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69 (2005), No. 4, 75–88.
[Kor3] M. A., Korolev, On the primitive of the Hardy function Z(t), Dokl. Math. 75, No. 2, 295–298 (2007);
translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 413, No. 5, 599–602 (2007).
[Kor4] M. A., Korolev, On the integral of Hardy's function Z(t), Izv. Math. 72, No. 3, 429–478 (2008);
translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 3, 19–68 (2008).
[Kor5] M. A., Korolev, Gram's law and the argument of the Riemann zeta-function, to appear, preprint available at arXiv:1106.0516.
[Kos] H., Kösters, On the occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function, J. Number Theory 130 (2005), 2596–2609.
[Kra] E., Krätzel, Lattice Points, Mathematics and its Applications: East European Series, 33. Dordrecht, Kluwer Academic Publishers; Berlin: VEB Deutscher Verlag der Wissenschaften, 1988.
[Lan] E., Landau, Euler und Functionalgleichung der Riemannschen Zeta-Funktion, Biblio. Math.(3) Bd. 7, Leipzig, 1906, pp. 69–79.
[Lau1] A., Laurinčikas, On the zeta-function of Riemann on the critical line (Russian), Litovskij Mat. Sbornik 25 (1985), 114–118.
[Lau2] A., Laurinčikas, On the moments of the zeta-function of Riemann on the critical line (Russian), Mat. Zametki 39 (1986), 483–493.
[Lau3] A., Laurinčikas, The limit theorem for the Riemann zeta-function on the critical line I (Russian), Litovskij Mat. Sbornik 27 (1987), 113–132;
A., Laurinčikas, The limit theorem for the Riemann zeta-function on the critical line II (Russian), Litovskij Mat. Sbornik 27 (1987), 489–500.
[Lau4] A., Laurinčikas, A limit theorem for Dirichlet L-functions on the critical line (Russian), Litovskij Mat. Sbornik 27 (1987), 699–710.
[Lau5] A., Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, 1996.
[Lau6] A., Laurinčikas, Limit theorems for the Mellin transforms of the Riemann zeta-function, Fiz. Mat. Fak. Moksl. Semin. Darb. 8 (2005), 63–75.
[Lav1] A. A., Lavrik, Uniform approximations and zeros of derivatives of Hardy's Z-function in short intervals (in Russian), Analysis Mathem. 17 (1991), 257–259.
[Lav2] A. A., Lavrik, Titchmarsh's problem in the discrete theory of the Riemann zeta-function. (English translation of Russian original)Proc. Steklov Inst. Math. 207 (1995), 179–209;
translation from Tr. Mat. Inst. Steklova 207 (1994), 197–230.
[Lavr] A. F., Lavrik, On the principal term in the divisor problem and power series of the Riemann zeta-function in a neighborhood of its pole (Russian), Trudy Mat. Inst. Steklova 142 (1976), 165–173, reprinted in Proc. Steklov Mat. Inst. 3 (1979), 175–183.
[LaIE] A. F., Lavrik, M. I., Israilov and Ž., Edgorov, On integrals containing the error term in the divisor problem (Russian), Acta Arith. 37 (1980), 381–389.
[LaTs] Y.-K., Lau and K.-M., Tsang, Omega result for the mean square of the Riemann zeta function, Manuscr. Math. 117 (2005), 373–381.
[Leh1] D. H., Lehmer, On the roots of the Riemann zeta function, Acta Math. 95 (1956), 291–298.
[Leh2] D. H., Lehmer, Extended computation of the Riemann zeta-function, Mathematika 3 (1956), 102–108.
[Lev] N., Levinson, More than one third of the zeros of Riemann's zeta-function are on σ = 1/2, Adv. Math. 18 (1975), 383–346.
[Lin] E., Lindelöf, Quelques remarques sur la croissance de la fonction ζ(s), Bull. Sci. Math. 32 (1908), 341–356.
[Lit] J. E., Littlewood, On the zeros of the Riemann zeta-function, Proc. Camb. Phil. Soc. 22 (1924), 295–318.
[Luk] M., Lukkarinen, The Mellin transform of the square of Riemann's zeta-function and Atkinson's formula, Doctoral Dissertation, Annales Acad. Sci. Fennicae, No. 140, Helsinki, 2005, 74 pp.
[LRW] J., van de Lune, H. J. J., te Riele and D. T., Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), 667–681.
[Man] H., von Mangoldt, Zu Riemann's Abhandlung “Über die Anzahl …,”, Crelle's J. 114 (1895), 255–305.
[MaTa] K., Matsumoto and Y., Tanigawa, On the zeros of higher derivatives of Hardy's Z-function, J. Number Theory 75 (1999), 262–278.
[Meh] M. L., Mehta, Random Matrices (3rd edn), Pure and Applied Mathematics, 142. Elsevier/Academic Press, Amsterdam, 2004.
[Mil] M. B., Milinovich, Moments of the Riemann zeta-function at its relative extrema on the critical line, Bull. London Math. Soc. 43 (2011), 1119–1129.
[Mol] G., Molteni, A note on a result of Bochner and Conrey-Ghosh about the Selberg class, Arch. Math. 72 (1999), 219–222.
[Mon1] H. L., Montgomery, The pair correlation of zeros of the zeta-function, in: Proc. Symp. Pure Math. 24, AMS, Providence, RI, 1973, 181–193.
[Mon2] H. L., Montgomery, Extreme values of the Riemann zeta-function, Comment. Math. Helv. 52 (1977), 511–518.
[MOd] H. L., Montgomery and A. M., Odlyzko, Gaps between zeros of the zeta function, in: Coll. Math. Soc. János Bolyai 34, Topics in Classical Number Theory (Budapest, 1981), 1079–1106.
[Mos1] J., Moser, The proof of the Titchmarsh hypothesis in the theory of the Riemann zeta-function (Russian), Acta Arith. 36 (1980), 147–156.
[Mos2] J., Moser, On the order of a sum of E. C. Titchmarsh in the theory of the Riemann zeta-function (Russian), Czech. Math. J. 41(116) (1991), 663–684.
[Mot1] Y., Motohashi, A note on the approximate functional equation for ζ2(s), Proc. Japan Acad. 59A (1983), 392–396,
Y., Motohashi, A note on the approximate functional equation for ζ2(s), II Proc. Japan Acad., 59A (1983), 469–472.
[Mot2] Y., Motohashi, Riemann-Siegel Formula, Lecture Notes, University of Colorado, Boulder, 1987.
[Mot3] Y., Motohashi, An explicit formula for the fourth power mean of the Riemann zeta-function, Acta Math. 170 (1993), 181–220.
[Mot4] Y., Motohashi, A relation between the Riemann zeta-function and the hyperbolic Laplacian, Annali Scuola Norm. Sup. Pisa, Cl. Sci. IV ser. 22 (1995), 299–313.
[Mot5] Y., Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge, 1997.
[Mot6] Y., Motohashi, The Riemann zeta-function and Hecke congruence subgroups II, Journal of Research Institute of Science and Technology, Tokyo, 2009, to appear.
[MoVa] H. L., Montgomery and R. C., Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 2007.
[Mue] J., Mueller, On the difference between consecutive zeros of the Riemann zeta-function, J. Number Theory 14 (1982), 327–331.
[NaSt] H., Nagoshi and J., Steuding, Universality for L-functions in the Selberg class, Lithuanian Math. J. 50 (2010), 293–311.
[Odl1] A. M., Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), 273–308.
[Odl2] A. M., Odlyzko, The 1022-nd zero of the Riemann zeta function, in: M., van Frankenhuysen and M. L., Lapidus (eds.) Dynamical, Spectral, and Arithmetic Zeta Functions, Amer. Math. Soc., Contemporary Math. 290, 2001, 139–144.
[Odl3] A. M., Odlyzko, The 1020-th zero of the Riemann zeta-function and 175 million of its neighbors, unpublished, see www.dtc.umn.edu/~odlyzko/zeta_tables/index.html.
[Per] A., Perelli, A survey of the Selberg class of L-functions, Part I, Milan J. Math. 73 (2005), 19–52, and Part II, Riv. Mat. Univ. Parma 7 (2004), 83–118.
[Pia] I., Piatetski-Shapiro, Multiplicity one theorems, in: A., Borel and W., Casselman (eds.), Automorphic Forms, Representations and L-functions, Proc. Symp. Pure Math. 33, AMS Publications, 1979, 209–212.
[Pre] E., Preissmann, Sur la moyenne de la fonction zêta, in: K., Nagasaka (ed.) Analytic Number Theory and Related Topics, Proceedings of the Symposium, Tokyo, Japan, November 11–13, 1991, Singapore: World Scientific (1993), 119–125.
[Rad] M., Radziwill, The 4.36-th moment of the Riemann zeta-function, IMRN, to appear, preprint available at arXiv:1106.4806.
[RaSo] M., Radziwill and K., Soundararajan, Continuous lower bounds for moments of zeta and L-functions, preprint available at arXiv:1202.1351.
[Ram] K., Ramachandra, On the mean-value and omega-theorems for the Riemann zeta-function, LN's 85, Tata Inst. of Fundamental Research, Bombay, 1995 (Springer Verlag, Berlin).
[Ran] R. A., Rankin, Van der Corput's method and the theory of exponent pairs, Quart. J. Math. Oxford 6 (1955), 147–153.
[RaSa1] K., Ramachandra and A., Sankaranarayanan, Note on a paper by H.L. Montgomery-I, Publ. Inst. Math. 50 (64) (1991), 51–59.
[RaSa2] K., Ramachandra and A., Sankaranarayanan, On some theorems of Littlewood and Selberg I, J. Number Theory 44 (1993), 281–291.
[Ric] H.-E., Richert, Über Dirichlet Reihen mit Funktionalgleichung, Publs. Inst. Math. Serbe Sci. 1 (1957), 73–124.
[Rie] B., Riemann, Über die Anzahl der Primzahlen unter einer gegebener Grösse, Monatshefte Preuss. Akad. Wiss. (1859–1860), 671–680.
[RiLu] H. J. J., te Riele and J., van de Lune, Computational Number Theory at CWI in 1970–1994, CWI Quarterly 7(4) (1994), 285–335.
[Riv1] T., Rivoal, La fonction zêta de Riemann prend une infinitédevaleurs irrationnelles aux entiers impairs, C. R. Acad. Sci., Paris, Sér. I, Math. 331, No. 4 (2000), 267–270.
[Riv2] T., Rivoal, Irrationalité d'au moins un des neuf nombres ζ(5), ζ(7),…, ζ(21), Acta Arith. 103, No. 2 (2002), 157–167.
[RuYa] M. O., Rubinstein and S., Yamagishi, Computing the moment polynomials of the zeta function, to appear, preprint available at arXiv:1112.2201.
[Sar1] P., Sarnak, L-functions, in: Proc. International Congress of Mathematicians, Vol. I (Berlin, 1998), Doc. Math. 1998, Extra Vol. I, 453–465.
[Sar2] P., Sarnak, The grand Riemann hypothesis, Milan J. Math. 78 (2010), 61–63.
[Sel] A., Selberg, Selected Papers, Vol. I, Springer Verlag, Berlin 1989, and Vol. II, Springer Verlag, Berlin 1991.
[Sie] C. L., Siegel, Über Riemanns Nachlaß zur analytischen Zahlentheorie, Quell. Stud. Gesch. Mat. Astr. Physik 2 (1932), 45–80 (also in Gesammelte Abhandlungen, Band I, Springer Verlag, Berlin, 1966, 275–310).
[Sou1] K., Soundarajan, Omega results for the divisor and circle problems, Int. Math. Res. Not. 36 (2003), 1987–1998.
[Sou2] K., Soundarajan, Degree 1 elements of Selberg class, Expo. Math. 23 (2005), 65–70.
[Sou3] K., Soundarajan, Extreme values of zeta and L-functions, Math. Annalen 342 (2008), 467–486.
[Sou4] K., Soundarajan, Moments of the Riemann zeta function, Ann. Math. 170 (2010), 981–993.
[Spi] R., Spira, Some zeros of the Titchmarsh counterexample, Math. Comp. 63 (1994), 747–748.
[Sri] B. R., Srinivasan, Lattice point problems of many-dimensional hyperboloids II, Acta Arith. 8 (1963), 173–204, and II, Math. Annalen 160 (1965), 280–310.
[Ste] J., Steuding, Value-distribution of L-functions, Lecture Notes Math. 1877, Springer-Verlag, Berlin, 2007.
[Sti] T. J., Stieltjes, Correspondance d'Hermite et de Stieltjes, Tome 1, Gauthier-Villars, Paris, 1905.
[Sub] M. A., Subhankulov, Tauberian Theorems with Remainder Terms (Russian), Nauka, Moscow, 1976.
[Tit1] E. C., Titchmarsh, On van der Corput's method and the zeta-function of Riemann. IV, Quart. J. Math. 5 (1934), 98–105.
[Tit2] E. C., Titchmarsh, Introduction to the Theory of Fourier Integrals (2nd edn), Oxford University Press, Oxford, 1948.
[Tit3] E. C., Titchmarsh, The Theory of the Riemann Zeta-Function (2nd edn), Oxford University Press, Oxford, 1986.
[Tru1] T. S., Trudgian, Further results on Gram's law, DPhil Thesis, University of Oxford, Oxford, 2009, 95pp.
[Tru2] T. S., Trudgian, On the success and failure of Gram's law and the Rosser rule, Acta. Arith. 143 (2011), 225–256.
[Tru3] T. S., Trudgian, A modest improvement on the function S(T), to appear in Mathematics of Computation.
[Tsa1] K.-M., Tsang, Some Ω-theorems for the Riemann zeta-function, Acta Arith. 46 (1985), 369–395.
[Tsa2] K.-M., Tsang, The large values of the Riemann zeta-function, Mathematika 40 (1993), 203–214.
[Vin1] I. M., Vinogradov, Selected Works, Springer-Verlag, Berlin-New York, 1985.
[Vin2] I. M., Vinogradov, Method of Trigonometrical Sums in the Theory of Numbers, Dover Publications, Mineola, NY, 2004.
[Vor] S. M., Voronin, Theorem on the “universality” of the Riemann zeta-function (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), no. 3, 475–486.
[Wats] G. N., Watson, A Treatise on the Theory of Bessel Function (2nd edn), Cambridge University Press, Cambridge, 1952.
[Watt] N., Watt, A note on the mean square of ∣ζ(½ + it)∣, J. London Math. Soc. 82(2) (2010), 279–294.
[Wie] R., Wiebelitz, Über approximative Funktionalgleichungen der Potenzen der Riemannschen Zeta-funktion, Math. Nachr. 6 (19511952), 263–270.
[Wil] J. R., Wilton, An approximate functional equation for the product of two ζ-functions, Proc. London Math. Soc. 31 (1930), 11–17.
[Zud] V., Zudilin, Irrationality of values of the Riemann zeta function, Izv. Math. 66, No. 3 (2002), 489–542.
[You] M. P., Young, A short proof of Levinson's theorem, Arch. Math. 95 (2010), 539–548.

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.