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arXiv · 1710.05001

Transformation formulas of a character analogue of $\logθ_{2}(z)$

Merve Çelebi Boztaş·Mümün Can
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Abstract

In this paper, transformation formulas for the function \[ A_{1}\left(z,s:χ\right)=\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}χ\left(n\right)χ\left(m\right)\left(-1\right)^{m}n^{s-1}e^{2πimnz/k} \] are obtained. Sums that appear in transformation formulas are generalizations of the Hardy--Berndt sums $s_{j}(d,c),$ $j=1,2,5$. As applications of these transformation formulas, reciprocity formulas for these sums are derived and several series relations are presented.

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Merve Çelebi Boztaş, Mümün Can. 2017-10-13. Transformation formulas of a character analogue of $\logθ_{2}(z)$. https://doi.org/10.1007/s11139-018-0042-7

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