On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Functio
On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function
by Athanassios S. Fokas, Jonatan Lenells
English | 2022 | ISBN: 1470450984 | 128 Pages | True PDF | 1.06 MB
by Athanassios S. Fokas, Jonatan Lenells
English | 2022 | ISBN: 1470450984 | 128 Pages | True PDF | 1.06 MB
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).