Mathematics Without Boundaries pp 245-258 | Cite as
Geometric Patterns in Uniform Distribution of Zeros of the Riemann Zeta Function
Abstract
We introduce a geometrical investigation of the distribution of several sequences involving the imaginary parts of the nontrivial zeros of the Riemann zeta function.
Keywords
Distribution modulo 1 Nonreal zeros of the Riemann zeta functionNotes
Acknowledgement
I am greatly indebted to Prof. A. Akbary for pointing out a big number of grammatical mistakes, and for clearing mathematical content of the paper by asking some important questions. Also, I gratefully acknowledge the many helpful suggestions by Prof. J.-M. Deshouillers during the preparation of the chapter, and specially his suggestion for adding Fig. 10, and introducing me the paper [12]. Finally, I deem my duty to thank Prof. R. Heath-Brown for giving very valuable comments on the mathematical justification of the geometric patterns described in this chapter.
References
- 1.Akbary, A., Murty, M.R.: Uniform distribution of zeros of Dirichlet series. In: de Koninck J.M., Granville A., Luca F. (eds.) Anatomy of integers. CRM proceedings & lecture notes, vol. 46, pp. 143–158. American Mathematical Society, Providence (2008)Google Scholar
- 2.Dekking, F., Mendès France, M.: Uniform distribution modulo one: a geometrical viewpoint. J. Reine Angew. Math. 329, 143–153 (1981)MathSciNetMATHGoogle Scholar
- 3.Deshouillers, J.-M.: Geometric aspect of Weyl sums. In: Iwaniec H. (ed.) Elementary and analytic theory of numbers, Banach Center Publications. vol. 17, pp. 75–82. PWN-Polish Scientific Publishers, Warsaw (1985)Google Scholar
- 4.Ford, K., Zaharescu, A.: On the distribution of imaginary parts of zeros of the Riemann zeta function. J. Reine Angew. Math. 579, 145–158 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 5.Fujii, A.: On the uniformity of the distribution of the zeros of the Riemann zeta function. J. Reine Angew. Math. 302, 167–205 (1978)MathSciNetMATHGoogle Scholar
- 6.Fujii, A.: Some problems of diophantine approximation in the theory of the Riemann zeta function III. Comment. Math. Univ. St. Paul. 42, 161–187 (1993)MathSciNetMATHGoogle Scholar
- 7.Hlawka, E.: Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafuncktionen zusammenhängen. Österr. Akad. Wiss., Math.-NaturwKl. Abt. II. 184, 459–471 (1975)MathSciNetGoogle Scholar
- 8.Ivić, A.: The Riemann zeta function: theory and applications. John Wiley & Sons, New York (1985)MATHGoogle Scholar
- 9.Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. Dover Publications, Mineola (2006)Google Scholar
- 10.Landau, E.: Über die Nullstellen der ζ-Funktion. Math. Ann. 71, 548–568 (1911)CrossRefGoogle Scholar
- 11.Odlyzko, A.M.: Home Page. Tables of zeros of the Riemann zeta function. Available at http://www.dtc.umn.edu/~odlyzko/ (2011)
- 12.Odlyzko, A.M.: On the distribution of spacings between zeros of the zeta function. Math. Comp. 48, 273–308 (1987)MathSciNetCrossRefMATHGoogle Scholar
- 13.Rademacher, H.: Fourier analysis in number theory, Symposium on harmonic analysis and related integral transforms: Final technical report. Cornell University, Ithica, N.Y., 25 p (1956) (Also in: H. Rademacher, Collected Works, pp. 434–458)Google Scholar
- 14.Tenenbaum, G., Mendès France, M.: The prime numbers and their distribution (Student Mathematical Library), vol. 6. American Mathematical Society, Providence (2001)Google Scholar
- 15.Titchmarsh, E.C.: The theory of the Riemann zeta function, 2nd ed. (Revised by D.R. Heath-Brown). Oxford University Press, Oxford (1986)Google Scholar
- 16.Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)MathSciNetCrossRefMATHGoogle Scholar