On the Riemann–Siegel formula for the derivatives of the Hardy function

Korolev, M.

doi:10.1090/spmj/1508

On the Riemann–Siegel formula for the derivatives of the Hardy function
HTML articles powered by AMS MathViewer

by M. A. Korolev
Translated by: A. Plotkin

St. Petersburg Math. J. 29 (2018), 581-601

DOI: https://doi.org/10.1090/spmj/1508

Published electronically: June 1, 2018

PDF | Request permission

Abstract:

Analogs are obtained of the asymptotic Riemann–Siegel formulas for the first and second order derivatives of the Hardy function $Z(t)$ and the Riemann zeta function on the critical line.

References

E. C. Titchmarsh, On van der Corput’s method and the zeta-function of Riemann (IV), Quart. J. Math. 5 (1934), 98–105.
Jan Mozer, A certain sum in the theory of the Riemann zeta-function, Acta Arith. 31 (1976), no. 1, 31–43. (errata insert) (Russian). MR 427244
Jan Mozer, A certain Hardy-Littlewood theorem in the theory of the Riemann zeta-function, Acta Arith. 31 (1976), no. 1, 45–51 (Russian). MR 427245
Ján Mozer, Gram’s law in the theory of the Riemann zeta-function, Acta Arith. 32 (1977), no. 2, 107–113 (Russian). MR 447144
Ján Mozer, An arithmetical analogue of a formula of Hardy-Littlewood in the theory of the Riemann zeta function, Acta Math. Univ. Comenian. 37 (1980), 109–120 (Russian, with English and Slovak summaries). MR 645778
Jan Mozer, On a self-correlated sum in the theory of the Riemann zeta function, Acta Math. Univ. Comenian. 37 (1980), 121–134 (Russian, with English and Slovak summaries). MR 645779
Justas Kalpokas and Jörn Steuding, On the value-distribution of the Riemann zeta-function on the critical line, Mosc. J. Comb. Number Theory 1 (2011), no. 1, 26–42. MR 2948324
Carl Ludwig Siegel, Gesammelte Abhandlungen. Bände I, II, III, Springer-Verlag, Berlin-New York, 1966 (German). Herausgegeben von K. Chandrasekharan und H. Maass. MR 0197270
M. V. Berry, The Riemann-Siegel expansion for the zeta function: high orders and remainders, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1939, 439–462. MR 1349513, DOI 10.1098/rspa.1995.0093
A. A. Karatsuba, On the distance between consecutive zeros of the Riemann zeta function that lie on the critical line, Trudy Mat. Inst. Steklov. 157 (1981), 49–63, 235 (Russian). Number theory, mathematical analysis and their applications. MR 651758
A. A. Lavrik, Uniform approximations and zeros in short intervals of the derivatives of the Hardy $Z$-function, Anal. Math. 17 (1991), no. 4, 257–279 (Russian, with English summary). MR 1171962, DOI 10.1007/BF01905933
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485