Ramūnas Garunkštis

The lerch zeta-function ∗

This paper studies the Lerch zeta-function L(λ αs) which was introduced by M. Lerch in 1887. The case of non-integer parameter λ is considered, and then L(λ αs) is an entire function. First, the universality of this function with transcendental parameter a is obtained. From this the functional independence is derived. The third part of the paper is devoted to the zero-distribution of the Lerch zeta-function. Zero-free regions are indicated on the right as well on the left of the imaginary axis when λ ≠ 1/2. If λ = 1/2, then zero-free regions lie in the left half-plane. Finally, estimates for the number of zeros in the critical strip are given.

On the lerch zeta-function

This paper studies the Lerch zeta-function L(λ αs) which was introduced by M. Lerch in 1887. The case of non-integer parameter λ is considered, and then L(λ αs) is an entire function. First, the universality of this function with transcendental parameter a is obtained. From this the functional independence is derived. The third part of the paper is devoted to the zero-distribution of the Lerch zeta-function. Zero-free regions are indicated on the right as well on the left of the imaginary axis when λ ≠ 1/2. If λ = 1/2, then zero-free regions lie in the left half-plane. Finally, estimates for the number of zeros in the critical strip are given.

ON THE ZERO DISTRIBUTIONS OF LERCH ZETA-FUNCTIONS

Abstract. We study the zero distributions of the Lerch zeta-functions L(λ, α, s) = ∑∞ n=0 e(λn) (n+α)s for the parameters 0 < λ, α ≤ 1 . Our observations show some analogies to the Riemann zeta-function (existence and number of trivial and nontrivial zeros) and some differences (asymmetrical distribution of the nontrivial zeros for almost all L(λ, α, s) ). Further, we investigate the distribution of zeros of the derivatives.

On the distribution of zeros of the hurwitz zeta-function

Assuming the Riemann hypothesis, we prove asymptotics for the sum of values of the Hurwitz zeta-function ζ(s, a) taken at the nontrivial zeros of the Riemann zeta-function ζ(s) = ζ(s, 1) when the parameter a either tends to 1/2 and 1, respectively, or is fixed; the case a = 1/2 is of special interest since ζ(s, 1/2) = (2 s - 1)ζ(s). If a is fixed, we improve an older result of Fujii. Besides, we present several computer plots which reflect the dependence of zeros of ζ(s,α) on the parameter a. Inspired by these plots, we call a zero of ζ(s, a) stable if its trajectory starts and ends on the critical line as a varies from 1 to 1/2, and we conjecture an asymptotic formula for these zeros.

THE DISCRETE MEAN SQUARE OF THE DIRICHLET L-FUNCTION AT NONTRIVIAL ZEROS OF ANOTHER DIRICHLET L-FUNCTION

We consider the sum of squared absolute values of the Dirichlet L-function taken at the nontrivial zeros of another Dirichlet L-function.

Euler Gamma-Function

The Euler gamma-function Γ(s) usually plays an important role in the theory of zeta-functions. It is a principal ingredient of functional equations, and therefore the behaviour of zeta-functions are influenced by properties of Γ(s). For the convenience, in this chapter we recall some elements from the theory of the gamma-function.

The Riemann hypothesis and universality of the Riemann zeta-function

Abstract We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s).

Zeros of the Lerch Transcendent Function

We investigate the distribution of zeros of the Lerch transcendent func- tion �(q,s,�) = P 1 n=0 q n (n + �) s . We find an upper and lower estimates of zeros of function �(q,s,�) in any fixed strip �1 1,0 < Im s � 1000} is done also.

Questions around the Nontrivial Zeros of the Riemann Zeta-Function. Computations and Classifications

We study the sequence of nontrivial zeros of the Riemann zeta-function with respect to sequences of zeros of other related functions, namely, the Hurwitz zeta-function and the derivative of Riemanns zeta-function. Finally, we investigate connections of the nontrivial zeros with the periodic zeta-function. On the basis of computation we derive several classifications of the nontrivial zeros of the Riemann zeta-function and stateproblems which mightbe ofinterestfor abetter understanding of the distribution of those zeros.

Sum of the periodic zeta-function over the nontrivial zeros of the Riemann zeta-function

We consider the asymptotic of the sum of values of the periodic zeta-function L(s, λ) over nontrivial zeros of the Riemann zeta-function ζ(s). J. Steuding for rational λ obtained the main term. Assuming the Generalized Riemann Hypothesis we calculate the next term.