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On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions. (English) Zbl 1427.11083

Summary: In this paper, transformation formulas for a large class of Eisenstein series defined for \(\operatorname{Re}(s)>2\) and \(\operatorname{Im}(z)>0\) by
\[ G(z, s; A_\alpha, B_\beta; r_1, r_2) = {\sum^{\infty}_{m, n=-\infty}}{'}\frac{f (\alpha_m) f^\ast (\beta n)} {((m + r_ 1 )z + n + r_ 2)^s} \]
are investigated for \(s=1-r\), \(r\in {\mathbb {N}}\). Here \( \left\{f(n)\right\} \) and \(\left\{f^*(n)\right\} \), \(-\infty<n<\infty \) are sequences of complex numbers with period \(k>0\), and \(A_{\alpha }=\left\{ f(\alpha n)\right\} \) and \(B_{\beta }=\left\{ f^*(\beta n)\right\} \), \(\alpha ,\beta \in {\mathbb {Z}}\). 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 \(G(z,s;A_{\alpha },I;r_{1},r_{2})\) and \(G(z,s;I,A_{\alpha };r_{1},r_{2})\), where \(I=\left\{ 1\right\} \).
As an application of these formulas, analogues of Ramanujan’s formula for periodic zeta-functions are derived.

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11M41 Other Dirichlet series and zeta functions
11F20 Dedekind eta function, Dedekind sums
11B68 Bernoulli and Euler numbers and polynomials
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References:

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