On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions

Abstract

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions of the index k. The consideration of the case of the derivative of the Weierstrass rho-function, ℘z(z,τ), leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points m+nτ and “demi-lattice” points m+1/2+nτ, m+(n+1/2)τ, m+1/2+(n+1/2)τ. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative θ1′(z|τ) of the elliptic theta-function and the Weierstrass ζ-function, and that far from the origin of coordinates such zeroes of the ζ-function tend to the positions of the simple poles of this function.