# On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions

On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions In this paper, transformation formulas for a large class of Eisenstein series defined for $${\text {Re}}(s)>2$$ Re ( s ) > 2 and $${\text {Im}}(z)>0$$ Im ( z ) > 0 by [Equation not available: see fulltext.]are investigated for $$s=1-r$$ s = 1 - r , $$r\in {\mathbb {N}}$$ r ∈ N . Here $$\left\{ f(n)\right\}$$ f ( n ) and $$\left\{ f^{*}(n)\right\}$$ f ∗ ( n ) , $$-\infty<n<\infty$$ - ∞ < n < ∞ are sequences of complex numbers with period $$k>0$$ k > 0 , and $$A_{\alpha }=\left\{ f(\alpha n)\right\}$$ A α = f ( α n ) and $$B_{\beta }=\left\{ f^{*}(\beta n)\right\}$$ B β = f ∗ ( β n ) , $$\alpha ,\beta \in {\mathbb {Z}}$$ α , β ∈ Z . Appearing in the transformation formulas are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity law is proved for periodic Apostol–Dedekind sum outside of the context of the transformation formulas. Furthermore, transformation formulas are presented for $$G(z,s;A_{\alpha },I;r_{1},r_{2})$$ G ( z , s ; A α , I ; r 1 , r 2 ) and $$G(z,s;I,A_{\alpha };r_{1},r_{2})$$ G ( z , s ; I , A α ; r 1 , r 2 ) , where $$I=\left\{ 1\right\}$$ I = 1 . As an application of these formulas, analogues of Ramanujan’s formula for periodic zeta-functions are derived. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

# On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions

, Volume 184 (1) – Jan 30, 2017
27 pages
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/lp/springer_journal/on-generalized-eisenstein-series-and-ramanujan-s-formula-for-periodic-01UPpe0zAG
Publisher
Springer Vienna
Copyright
Copyright © 2017 by Springer-Verlag Wien
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-017-1020-7
Publisher site
See Article on Publisher Site

### Abstract

In this paper, transformation formulas for a large class of Eisenstein series defined for $${\text {Re}}(s)>2$$ Re ( s ) > 2 and $${\text {Im}}(z)>0$$ Im ( z ) > 0 by [Equation not available: see fulltext.]are investigated for $$s=1-r$$ s = 1 - r , $$r\in {\mathbb {N}}$$ r ∈ N . Here $$\left\{ f(n)\right\}$$ f ( n ) and $$\left\{ f^{*}(n)\right\}$$ f ∗ ( n ) , $$-\infty<n<\infty$$ - ∞ < n < ∞ are sequences of complex numbers with period $$k>0$$ k > 0 , and $$A_{\alpha }=\left\{ f(\alpha n)\right\}$$ A α = f ( α n ) and $$B_{\beta }=\left\{ f^{*}(\beta n)\right\}$$ B β = f ∗ ( β n ) , $$\alpha ,\beta \in {\mathbb {Z}}$$ α , β ∈ Z . Appearing in the transformation formulas are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity law is proved for periodic Apostol–Dedekind sum outside of the context of the transformation formulas. Furthermore, transformation formulas are presented for $$G(z,s;A_{\alpha },I;r_{1},r_{2})$$ G ( z , s ; A α , I ; r 1 , r 2 ) and $$G(z,s;I,A_{\alpha };r_{1},r_{2})$$ G ( z , s ; I , A α ; r 1 , r 2 ) , where $$I=\left\{ 1\right\}$$ I = 1 . As an application of these formulas, analogues of Ramanujan’s formula for periodic zeta-functions are derived.

### Journal

Monatshefte f�r MathematikSpringer Journals

Published: Jan 30, 2017

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