Applications of uniform distribution theory to the Riemann zeta-function

Özbek, Selin Selen; Steuding, Jörn

doi:10.1515/tmmp-2015-0042

Abstract

We give two applications of uniform distribution theory to the Riemann zeta-function. We show that the values of the argument of are uniformly distributed modulo , where P(n) denotes the values of a polynomial with real coefficients evaluated at the positive integers. Moreover, we study the distribution of arg modulo π, where γ_n is the nth ordinate of a zeta zero in the upper half-plane (in ascending order).

References

[1] ARIAS DE REYNA, J.: On the distribution (mod 1) of the normalized zeros of the Riemann zeta-function, J. Number Theory 153 (2015), 37-53.10.1016/j.jnt.2015.01.006
Search in Google Scholar
[2] BOHR, H.-COURANT, R.: Neue Anwendungen der Theorie der diophantischen Approximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math. 144 (1914), 249-274.10.1515/crll.1914.144.249
Search in Google Scholar
[3] EDWARDS, H. M.: Riemann’s Zeta Function, in: Pure Appl. Math., Vol. 58, Academic Press, New York, 1974.
Search in Google Scholar
[4] ELLIOTT, P. D. T. A.: The Riemann zeta function and coin tossing, J. Reine Angew. Math. 254 (1972), 100-109.
Search in Google Scholar
[5] FORD, K.-SOUNDARARAJAN, K.-ZAHARESCU, A.: On the distribution of imaginary parts of zeros of the Riemann zeta function, J. Reine Angew. Math. 579 (2005), 145-158.10.1515/crll.2005.2005.579.145
Search in Google Scholar
[6] FUJII, A.: On the uniformity of the distribution of the zeros of the Riemann zeta function, J. Reine Angew. Math. 302 (1978), 167-205.
Search in Google Scholar
[7] FUJII, A.: On a conjecture of Shanks, Proc. Japan Acad. 70 (1994), 109-114.
Search in Google Scholar
[8] FUJII, A.: Some observations concerning the distribution of the zeros of the zeta functions, III, Proc. Japan Acad. Ser. A 68 (1992), 105-110.
Search in Google Scholar
[9] GARUNKŠTIS, R.-STEUDING, J.: On the roots of the equation ζ(s) = a, Abh. Math. Semin. Univ. Hamb. 84 (2014), 1-15.10.1007/s12188-014-0093-7
Search in Google Scholar
[10] HARDY, G. H.-LITTLEWOOD, J. E.: Contributions to the theory of the Riemann zetafunction and the theory of the distribution of primes, Acta Math. 41 (1916), 119-196.10.1007/BF02422942
Search in Google Scholar
[11] HLAWKA, E.: Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktion zusammenhängen, Österr. Akad. Wiss., Math.-Naturw. Kl. Abt. II 184 (1975), 459-471.
Search in Google Scholar
[12] KUIPERS, L.-NIEDERREITER, H.: Uniform Distribution of Sequences. John Wiley & Sons, New York, 1974.
Search in Google Scholar
[13] LANDAU, E.: Über die Nullstellen der Zetafunktion, Math. Ann. 71 (1912), 548-564.10.1007/BF01456808
Search in Google Scholar
[14] ÖZBEK, S. S.-STEUDING, J.: On the distribution of the argument of the Riemann zeta-function on the critical line, Unif. Distrib. Theory 10 (2015), (to appear).
Search in Google Scholar
[15] RADEMACHER, H. A.: Fourier analysis in number theory, in: Symposium on Harmonic Analysis and Related Integral Transforms, Cornell Univ., Ithaca, N.Y., 1956; Collected Papers of Hans Rademacher, Vol. II, Massachusetts Inst. Tech., Cambridge, Mass., 1974, pp. 434-458.
Search in Google Scholar
[16] SHANKS, D.: Review of ‘Tables of the Riemann Zeta Function’ by C.B. Haselgrove in collaboration with J.C.P. Miller, Math. Comp. 15 (1961), 84-86.10.2307/2003098
Search in Google Scholar
[17] TITCHMARSH, E. C.: The Theory of the Riemann Zeta-Function (2nd ed.). Oxford University Press, Oxford, 1986.
Search in Google Scholar
[18] TRUDGIAN, T. S.: On a conjecture of Shanks, J. Number Theory 130 (2010), 2635-2638.10.1016/j.jnt.2010.04.010
Search in Google Scholar
[19] VAN DER CORPUT, J. G.: Diophantische Ungleichungen. I: Zur Gleichverteilung modulo Eins. Acta Math. 56 (1931), 373-456.
Search in Google Scholar
[20] WEYL, H.: Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352. 10.1007/BF01475864
Search in Google Scholar

Applications of uniform distribution theory to the Riemann zeta-function

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