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Character analogues of Cohen type identities and related Voronoi summation formulas

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Debika Banerjee, Khyati Khurana

Number Theory · Submitted 21 Jun 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Moreover, we provide a new expression for $L(1, \chi)$ from which we establish the positivity of $L(1, \chi)$ for any real primitive character $\chi$.

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arXiv ID2306.12399 (version 2)
Pith IDZS3ZORLF, Pith's permanent record of this paper and its review
CitationDebika Banerjee, Khyati Khurana. "Character analogues of Cohen type identities and related Voronoi summation formulas." arXiv:2306.12399 (2023). https://pith.science/paper/ZS3ZORLF
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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.47 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; .

In \cite{MR2221114}, B.~C.~Berndt and A.~Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, in a follow-up paper \cite{MR3541702}, S. Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified $K$-Bessel function in light of recent results obtained by the first author and B. Maji \cite{ debika2023}. Moreover, we provide a new expression for $L(1, \chi)$ from which we establish the positivity of $L(1, \chi)$ for any real primitive character $\chi$. In addition, we deduce Cohen-type identities and then exhibit the Vorono\"i-type summation formulas for them.

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s0 In \cite{MR2221114}, B.~C.~Berndt and A.~Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. background 1.00 0.03 0.03 asserted 1.00
s1 Later, in a follow-up paper \cite{MR3541702}, S. background 1.00 0.05 0.02 unclear 0.68
s2 Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. background 1.00 0.05 0.03 asserted 1.00
s3 In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified $K$-Bessel function in light of recent results obtained by the first author and B. Maji \cite{ debika2023}. contribution 1.00 0.92 0.21 asserted 1.00
s4 Moreover, we provide a new expression for $L(1, \chi)$ from which we establish the positivity of $L(1, \chi)$ for any real primitive character $\chi$. result 0.93 0.96 0.34 asserted 1.00
s5 In addition, we deduce Cohen-type identities and then exhibit the Vorono\"i-type summation formulas for them. result 0.51 0.95 0.36 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

In \cite{MR2221114}, B.~C.~Berndt and A.~Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, in a follow-up paper \cite{MR3541702}, S. Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified $K$-Bessel function in light of recent results obtained by the first author and B. Maji \cite{ debika2023}. Moreover, we provide a new expression for $L(1, \chi)$ from which we establish the positivity of $L(1, \chi)$ for any real primitive character $\chi$. In addition, we deduce Cohen-type identities and then exhibit the Vorono\"i-type summation formulas for them.

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