Inverse Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytic Functions

Recently, we 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. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this article, we discuss what are, in a sense, inverse applications of this theorem. We first prove a Lemma that if two meromorphic on the whole complex plane functions f(z) and g(z) have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large |z|, then for some integer n, dnln(f(z))dzn=dnln(g(z))dzn. The use of this Lemma enables proofs of many identities between elliptic functions, their transformation and n-tuple product rules. In particular, we show how exactly for any complex number a, ℘(z)-a, where ℘(z) is the Weierstrass ℘ function, can be presented as a product and ratio of three elliptic θ1 functions of certain arguments. We also establish n-tuple rules for some elliptic theta functions.