On the Asymptotics to all Orders of the Riemann Zeta...

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Mathematics - Number Theory
On the Asymptotics to all Orders of the...

On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

Name: On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function
ISBN: 9781470450984

Athanassios S. Fokas, Jonatan Lenells

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We present several formulae for the large t asymptotics of the Riemann zeta function ζ(s), s = σ + it, 0 ≤ σ ≤ 1, t > 0, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum b a n − s for certain ranges of a and b. In addition, we present precise estimates relating this sum with the sum d c n s − 1 for certain ranges of a, b, c, d. We also study a two-parameter generalization of the Riemann zeta function which we denote by Φ(u, v, β), u ∈ C , v ∈ C , β ∈ R . Generalizing the methodology used in the study of ζ(s), we derive asymptotic formulae for Φ(u, v, β).

Categories:

Mathematics - Number Theory

Year:

2022

Publisher:

American Mathematical Society

Language:

english

Pages:

128

ISBN 10:

1470450984