On special values of Dirichlet series with periodic coefficients

Bharadwaj, Abhishek T.; Pathak, Siddhi

doi:10.1016/j.jnt.2021.10.011

Cited by 2 publications

(2 citation statements)

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“…D d (d, σ −1 e .Proj N d (g))) by(5).Noting that e|N µ(e)σ −1 e e = β, we obtain the result.Let W pr d (N) := {L(1, f ) f ∈ F D pr (N; d) and N a=1 f (a) = 0}.We note that it is a Q-vector space.Corollary 3. W pr d (N) is independent of N. More precisely, W pr d (N) = W pr d (d).…”

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confidence: 60%

“…We require to compute the value of each summand. The author along with S. Pathak considered a character decomposition in [5,Section 4] but this was again used for rearrangement and matching the values L(s, f ) and L(s, g) for different periods. In the proofs of Theorem 4 as well as [15, Theorem 10] re-arrangements and the use of Theorem 1 are unavoidable.…”

Section: Introductionmentioning

confidence: 99%

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On Primitivity and Vanishing of Dirichlet Series

Bharadwaj¹

2022

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For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla, and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdös holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishes at two different positive integers under some additional conditions.

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supporting

confidence: 60%

Section: Introductionmentioning

confidence: 99%