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EXPLICIT ZERO-COUNTING THEOREM FOR HECKE–LANDAU ZETA-FUNCTIONS

Part of: Algebraic number theory: global fields Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  06 February 2017

MACIEJ GRZEŚKOWIAK
MACIEJ GRZEŚKOWIAK*
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland email [email protected]
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Abstract

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We give an explicit upper bound for the number of zeros of Hecke–Landau zeta-functions in a rectangle.

Keywords

complete latticedistributive latticecomplete congruencecongruence lattice

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11R42: Zeta functions and $L$-functions of number fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 400 - 411
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Backlund, R., ‘Über die Nullstellen der Riemannschen Zetafunction’, Acta Math. 41 (1918), 345–375.CrossRefGoogle Scholar
Fryska, T., ‘An estimate of the order of the Hecke–Landau 𝜁(s, 𝜒)-functions’, Funct. Approx. Comment. Math. 16 (1988), 55–62.Google Scholar
Fryska, T., ‘Some effective estimates for the roots of the Dirichlet L-series, II’, Funct. Approx. Comment. Math. 16 (1988), 21–36.Google Scholar
Grześkowiak, M., ‘An algorithmic construction of finite elliptic curves of order divisible by a large prime’, Fund. Inform. 132(4) (2015), 331–343.Google Scholar
Kadiri, H. and Ng, N., ‘Explicit zero density theorems for Dedekind zeta-functions’, J. Number Theory 132(4) (2012), 748–775.Google Scholar
Landau, E., ‘Über Ideale und Primideale in Idealklassen’, Math. Z. 2 (1918), 135–152.Google Scholar
McCurley, K., ‘Explicit estimates for the error term in the prime number theorem for arithmetic progressions’, Math. Comp. 42(165) (1941), 265–285.Google Scholar
Olver, F. W. J., Asymptotics and Special Functions, Computer Science and Applied Mathematics (Academic Press, New York–London, 1974).Google Scholar
Rosser, J., ‘Explicit bounds for some functions of prime numbers’, Amer. J. Math. 63 (1941), 211–232.Google Scholar
Trudgian, T., ‘An improved upper bound for the argument of the Riemann zeta-function on the critical line’, Math. Comp. 81(278) (2012), 1053–1061.Google Scholar
Trudgian, T., ‘An improved upper bound for the argument of the Riemann zeta-function on the critical line II’, J. Number Theory (134) (2014), 280–292.Google Scholar
Trudgian, T., ‘An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions’, Math. Comp. 84(293) (2015), 1439–1450.Google Scholar