On the Dirichlet series related to the cubic theta function

N. V. Proskurin

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 212-232

Résumé

The paper studies the function $L(\tau;\cdot)$ defined by the Dirichlet series $$ L(\tau;s)=\sum_\nu\frac{\tau(\nu)}{\|\nu\|^s}, \quad s\in\mathbb C, $$ where $\tau(\nu)$ is the $\nu$th Fourier coefficient of the Kubota?Patterson cubic theta function. For this function, an exact and an approximate functional equations are derived. It is established that the function does not vanish in the halfplane $\operatorname{RE}s\ge 1.3533$ and has no singularities except for a simple pole at the point 5/6. Issues related to computing the coefficients $\tau(\nu)$ and values of the special functions arising in the approximate functional equation are considered. Bibliography: 11 titles.