On the Epstein zeta function and the zeros of a class of Dirichlet series. (English) Zbl 07758995
In this paper under review, the authors study the double series of the form \[\mathcal{Z}_2(s;a_1,a_2;\lambda,\lambda')=\sum_{m, n\neq 0}^\infty\frac{a_1(m)a_2(n)}{(\lambda_m+\lambda'_n)^s}, \] where \(a_1(m)\), \(a_2(n)\), \(\lambda=(\lambda_m)\) and \(\lambda'=(\lambda'_n)\) satisfy few analytic conditions which are used to derive a functional equation and the classical Selberg-Chowla formula type.
As a consequence, the authors use the obtained Selberg-Chowla formulas to extend an argument due to Deuring for the generalized diagonal Epstein zeta function \(\mathcal{Z}_2(s;a_1,a_2;\lambda,\lambda')\). In particular, it is shown that the existence of infinitely many zeros of the above double series at its critical line implies the existence of infinitely many zeros, also at their critical lines, of the auxiliary series constructing it, namely \(\sum\frac{a_1(n)}{\lambda_n^s}\) or \(\sum\frac{a_2(n)}{\lambda_n'^s}\).
As a consequence, the authors use the obtained Selberg-Chowla formulas to extend an argument due to Deuring for the generalized diagonal Epstein zeta function \(\mathcal{Z}_2(s;a_1,a_2;\lambda,\lambda')\). In particular, it is shown that the existence of infinitely many zeros of the above double series at its critical line implies the existence of infinitely many zeros, also at their critical lines, of the auxiliary series constructing it, namely \(\sum\frac{a_1(n)}{\lambda_n^s}\) or \(\sum\frac{a_2(n)}{\lambda_n'^s}\).
Reviewer: Sami Omar (Sukhair)
MSC:
| 11M41 | Other Dirichlet series and zeta functions |
| 11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |
| 11N37 | Asymptotic results on arithmetic functions |
Keywords:
Dirichlet series; Epstein zeta function; modified Bessel functions; Selberg-Chowla formula; Hardy’s theoremReferences:
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