Many values of the Riemann zeta function at odd integers are irrational

Values of the Riemann zeta function at integers

Riemann Zeta Function with Odd Arguments

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in ther...

Distribution of irrational zeta values

In this paper we refine Ball-Rivoal’s theorem by proving that for any odd integer a sufficiently large in terms of ε > 0, there exist [ (1−ε) log a 1+log 2 ] odd integers s between 3 and a, with distance at least aε from one another, at which Riemann zeta function takes Q-linearly independent values. As a consequence, if there are very few integers s such that ζ(s) is irrational, then they are ...

Irrationality of values of the Riemann zeta function

The paper deals with a generalization of Rivoal’s construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function ζ(s) only at odd points. We prove theorems on the irrationality of the number ζ(s) for some odd integers s in a given segment of the set of positive integers. Using certain refined arithmetical estimates, we strengthen Rivoal’s origin...

Evaluation of zeta function of the simplest cubic field at negative odd integers

In this paper, we are interested in the evaluation of the zeta function of the simplest cubic field. We first introduce Siegel’s formula for values of the zeta function of a totally real number field at negative odd integers. Next, we will develop a method of computing the sum of a divisor function for ideals, and will give a full description for a Siegel lattice of the simplest cubic field. Us...