## Variations of Li's criterion for an extension of the Selberg class

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##### Date

2012-08-09

##### Authors

Droll, Andrew

##### Keyword

Selberg class , Li's criterion , GRH , RH , Number Theory , zero-free regions , arithmetic formulae , Riemann hypothesis

##### Abstract

In 1997, Xian-Jin Li gave an equivalence to the classical Riemann hypothesis,
now referred to as Li's criterion, in terms of the non-negativity of a particular
infinite sequence of real numbers. We formulate the analogue of Li's criterion as
an equivalence for the generalized quasi-Riemann hypothesis for functions in an
extension of the Selberg class, and give arithmetic formulae for the corresponding
Li coefficients in terms of parameters of the function in question. Moreover, we
give explicit non-negative bounds for certain sums of special values of polygamma
functions, involved in the arithmetic formulae for these Li coefficients, for a wide class of functions. Finally, we discuss an existing result on correspondences between
zero-free regions and the non-negativity of the real parts of finitely many Li
coefficients. This discussion involves identifying some errors in the original source work which seem to render one of its theorems conjectural. Under an appropriate
conjecture, we give a generalization of the result in question to the case of Li coefficients corresponding to the generalized quasi-Riemann hypothesis. We also
give a substantial discussion of research on Li's criterion since its inception, and
some additional new supplementary results, in the first chapter.