Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function

Lithuanian Mathematical Journal - We prove that every collection of analytic functions (f1(s), . . . , fr(s)) defined on the right-hand side of the critical strip can be simultaneously approximated...     

A joint universality theorem for periodic Hurwitz zeta-functions. II

We obtain a joint universality theorem for periodic Hurwitz zeta-functions under weaker hypotheses than those in the previous papers of the first author.     

On joint universality of periodic Hurwitz zeta-functions

We prove a joint universality theorem for a collection of periodic Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers.     

Simple zeros of the Riemann zeta-function

By estimating the change in argument of a certain function it has been shown that at least 0.3474 of the nonreal zeros of ζ(s) are simple. It is shown here that a more general function containing a real parameter can be used. An optimal choice of which gives a proportion greater than 0.3532.     

On the universality of the Riemann zeta-function

The research has been partially supported by Grant N LAC000 from the International Science Foundation.     

Generalizations of a discrete universality theorem for Hurwitz zeta-functions

It is well known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that the shifts ζ(s + iτ ), t ? \mathbbR \tau \in \mathbb{R} (continuous case), and ζ(s + imh), m ? \mathbbN è{ 0 } m \in \mathbb{N} \cup \left\{ 0 \right\} , with fixed h > 0 (discrete case) approximate any analytic function. In the paper, the discrete universality is extended for some classes of the functions F(ζ(s, α)).     

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

     

On gaps between zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.     

Joint discrete universality for periodic zeta-functions

Abstract

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.     

Large spaces between the zeros of the Riemann zeta-function and random matrix theory II

We develop the theory set out in Part 1 [R.R. Hall, Large spaces between the zeros of the Riemann zeta-function and random matrix theory, J. Number Theory 109 (2004) 240–265] and in particular provide a lower bound (and almost sure evaluation) for Λ(7). The square of this number is rational, as were the previous values, but still rather surprizing.     

Short series over simple zeros of the Riemann zeta-function

Recently, Garaev showed that the series Σ_?∥?ζ¹(?)∥⁻¹ diverges, where the sum is taken over the simple zeros ? = β + iγ of the Riemann zeta-function ζ(s). More precisely, he proved . Using a mean-value estimate due to Ramachandra and some result on the distribution of simple zeros in short intervals on the critical line, we prove for T^0.552 ≤ H ≤ T. This leads to a slight improvement of Garaev's result in replacing his lower bound by .     

A note on the zeros of the zeta-function of Riemann

Archiv der Mathematik - We consider the qth root number function for the symmetric group. Our aim is to develop an asymptotic formula for the multiplicities of the qth root number function as q...     

Large gaps between consecutive zeros of the Riemann zeta-function

Combining the amplifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.033 times the average spacing.     

Simple zeros and discrete moments of the derivative of the Riemann zeta-function

We prove unconditional upper bounds for the second and fourth discrete moment of the first derivative of the zeta-function at its simple zeros on the critical line.     

The density of zeros of the Riemann zeta-function in the critical strip

A new proof of Ingam’s theorem on the density of zeros of the Riemann zeta-function in the critical strip is given basing on an idea of H. Bohr and F. Carlson. Multiplication of segments of the Dirichlet series for the functions ζ(s) and 1/ζ(s) is used, which permits to simplify the proof.     

Gaps between consecutive zeros of the Riemann zeta-function on the critical line

LetR denote the number of gaps of length at leastV between consecutive zeros of the function ζ(1/2+i t) in the interval [0,T]. It is proved that $$R<< TV^{ - 2} \min (\log T, V^{ - 1} \log ^5 T).$$ The same problem is also discussed for Dirichlet series associated with cusp forms.