Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

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


Author: M. A. Korolev
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 29 (2017), nomer 4.
Journal: St. Petersburg Math. J. 29 (2018), 581-601
MSC (2010): Primary 33B15
DOI: https://doi.org/10.1090/spmj/1508
Published electronically: June 1, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] E. C. Titchmarsh, On van der Corput's method and the zeta-function of Riemann (IV), Quart. J. Math. 5 (1934), 98-105.
  • [2] 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 0427244
  • [3] 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 0427245
  • [4] Ján Mozer, Gram’s law in the theory of the Riemann zeta-function, Acta Arith. 32 (1977), no. 2, 107–113 (Russian). MR 0447144
  • [5] 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
  • [6] 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
  • [7] 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
  • [8] Carl Ludwig Siegel, Gesammelte Abhandlungen. Bände I, II, III, Herausgegeben von K. Chandrasekharan und H. Maass, Springer-Verlag, Berlin-New York, 1966 (German). MR 0197270
  • [9] 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, https://doi.org/10.1098/rspa.1995.0093
  • [10] A. A. Karatsuba, On the distance between consecutive zeros of the Riemann zeta function that lie on the critical line, Tr. Mat. Inst. Steklov. 157 (1981), 49-63; English transl., Proc. Steklov Inst. Math.  157 (1983), 51-66. MR 0651758 (83k:10071)
  • [11] 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, https://doi.org/10.1007/BF01905933
  • [12] 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
  • [13] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 33B15

Retrieve articles in all journals with MSC (2010): 33B15


Additional Information

M. A. Korolev
Affiliation: Steklov Mathematical Institute, Russian Academy of Sciences, Gubkin str. 8, Moscow 119991, Russia
Email: korolevma@mi.ras.ru

DOI: https://doi.org/10.1090/spmj/1508
Keywords: Riemann zeta function, Hardy function, Riemann--Siegel formula, critical line
Received by editor(s): January 10, 2017
Published electronically: June 1, 2018
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society