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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2006 Volume 337, Pages 212–232 (Mi znsl189)

This article is cited in 4 papers

On the Dirichlet series related to the cubic theta function

N. V. Proskurin

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: 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.

UDC: 517.3

Received: 22.05.2006


 English version:
Journal of Mathematical Sciences (New York), 2007, 143:3, 3137–3148

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