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