Extreme values of the Riemann zeta-function on short zero intervals

On the Mean Values of the Riemann Zeta-function in Short Intervals

It is proved that, for T ε ≤ G = G(T) ≤

On the Divisor Function and the Riemann Zeta-function in Short Intervals

We obtain, for T ε ≤ U = U(T ) ≤ T 1/2−ε, asymptotic formulas for Z 2T T (E(t+ U)− E(t)) dt, Z 2T T (∆(t+ U)−∆(t)) dt, where ∆(x) is the error term in the classical divisor problem, and E(T ) is the error term in the mean square formula for |ζ( 1 2 + it)|. Upper bounds of the form Oε(T 1+εU2) for the above integrals with biquadrates instead of square are shown to hold for T 3/8 ≤ U = U(T ) ≪ T ...

On the mean square of the Riemann zeta-function in short intervals

It is proved that, for T ε 6 G = G(T ) 6 2 √ T , ∫︁ 2T T (︁ I1(t+G,G)− I1(t, G) )︁2 dt = TG 3 ∑︁ j=0 aj log (︁√ T G )︁ +Oε(T 1+εG1/2 + T 1/2+εG2) with some explicitly computable constants aj (a3 > 0) where, for fixed k ∈ N, Ik(t, G) = 1 √ π ∫︁ ∞ −∞ |ζ( 1 2 + it+ iu)| 2ke−(u/G) 2 du. The generalizations to the mean square of I1(t+U,G)−I1(t, G) over [T, T+H] and the estimation of the mean square ...

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...

Values of the Riemann zeta function at integers