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arXiv · 2211.02954

Riesz type criteria for $L$-functions in the Selberg class

Shivajee Gupta·Akshaa Vatwani
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Abstract

We formulate a generalization of Riesz-type criteria in the setting of $L$-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for $L$-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for $L$-functions in this subclass. Identities of Ramanujan-Hardy-Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer $G$-function of the type $G^{n \ 0}_{0 \ n}$.

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BibTeXRIS

Shivajee Gupta, Akshaa Vatwani. 2022-11-05. Riesz type criteria for $L$-functions in the Selberg class. https://arxiv.org/abs/2211.02954

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