Identities and Relations Involving Zeros and Special Values of Riemann’s Zeta and Related Functions (November 2020)

Abstract: This work is mostly concerned with the sums of the nth powers of the reciprocals of zeros of entire and meromorphic functions. In this talk, we first begin with identities that have been obtained for a class of entire and meromorphic functions and that relate the sums of the nth powers of the reciprocals of zeros and poles of these functions with the coefficients of their Taylor series expansions. These identities lead to the recurrence relations involving nontrivial zeros and values at integer arguments of the Riemann zeta function. Based upon these relations, we compute the sums extending over all nontrivial zeros of the Riemann zeta function. We also present recurrence relations for some other zeta and related functions. We establish similar results for the digamma function, Barnes G-function, and its logarithmic derivative. Relying on these, we find the formulae for the values of the Riemann zeta function at odd positive integers. The main tools are the identities mentioned above and Hadamard infinite product representations for entire functions of finite order. Determinant formulae for the sums of the nth powers of the reciprocals of zeros of entire or meromorphic functions as well as for the coefficients of their Taylor series expansions will be presented. We will discuss the main results, meanwhile presenting the ideas of the proofs and providing examples along the way. We will conclude by discussing some further research in this direction

Keywords: Riemann's zeta and related functions, sums of powers of reciprocals of zeros, nontrivial zeros, entire functions, meromorphic functions, recurrence formulae, multiple Γ-functions

About the speaker
Dr. Armen Bagdasaryan is from the American University of the Middle East, Kuwait.

For more information, please contact [email protected].