Analytic properties of zeta- and L-functions

Number theory is a branch of pure mathematics concerned with properties of the integers. Algebraic and Analytic number theory are considered branches from number theory, where analytic number theory uses the tools and methods from analysis to solve difficult number-theoretic problems. Take for instance, Dirichlet himself used complex-analytic tools to prove his famous theorem on the density of primes in arithmetic progressions. The Riemann zeta-function occupies a central place in analytic number theory, particularly because of its close correlation with prime numbers via the Euler product. The distribution of zeros of the Riemann zeta function, which is called Riemann hypothesis, is considered one of the greatest unsolved problems in pure mathematics. In 1975, Voronin proved his famous universality theorem for the zeta-function, which states that any non-vanishing holohmorphic function can be approximated uniformly by certain shifts of zeta-function. This stunning approximation property is called universality. It has many critical applications to the theory of distribution of values, the Riemannhypothesis, algebraicnumbertheory,andphysics. Voronin`s theorem attracted many mathematicians, who have had exciting contributions to the universality theory development by improving and extending it in various directions. Unfortunately, the known proofs of universality theorems are ineffective. For this reason, one may say that effective universality is a big challenge in universality theory. The first part of the project is concerned with the universality theorems due to Bagchi and Eminyan, concerned with a character satisfying certain approximation conditions, and providing another type of universality. Roughly speaking, we would like to study the character analogue of the effectivity problem. An important generalization of the notion of zeta-functions is multiple zeta-functions. The history of multiple zeta-functions goes back to the days of Euler, where they have arisen in mathematics and physics. Still, it has only recently appeared how interesting they are. The multiple zeta-functions are connected with several fascinating topics, including knot theory, quantum field theory, and mirror symmetry. Although however, the multiple zeta-functions are still not well understood, since they are complex functions in several variables, and their analytic behaviour is quite a complicated problem. This makes the study of these functions is a big challenge. In the second part of the project, we are interested in the Laurent-Stieltjes constants (which are the coefficients of the Laurent expansion of zeta and L-functions at their pole) of the Euler-Zagier multiple zeta-functions. We first study the sign changes of the Laurent-Stieltjes constants of the Euler-Zagier multiple zeta-functions and then give upper estimates of these constants. We also extend our above purpose for m u l t i p l e