On the Use of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions for the Calculation of Infinite Sums and the Analysis of Zeroes of Analytical Functions

Abstract

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. On many occasions, this enables to facilitate the obtaining of known results thus having important methodological meaning. Additionally, some new results, to the best of our knowledge, are also obtained in this way. For example, we established new properties of the sum of inverse zeroes of a digamma function, new formulae for the sums ∑kiρi2 for zeroes ρi of incomplete gamma and Riemann zeta functions having the order ki (These results can be straightforwardly generalized for the sums ∑kiρin with integer n > 2, and so on.)