The Mean Square of the Hurwitz Zeta-Function in Short Intervals

Abstract

The Hurwitz zeta-function ζ(s,α), s=σ+it, with parameter 0<α⩽1 is a generalization of the Riemann zeta-function ζ(s) (ζ(s,1)=ζ(s)) and was introduced at the end of the 19th century. The function ζ(s,α) plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function ζ(s,α) is the main example of zeta-functions without Euler’s product (except for the cases α=1, α=1/2), and its value distribution is governed by arithmetical properties of α. For the majority of zeta-functions, ζ(s,α) for some α is universal, i.e., its shifts ζ(s+iτ,α), τ∈R, approximate every analytic function defined in the strip {s:1/2<σ<1}. For needs of effectivization of the universality property for ζ(s,α), the interval for τ must be as short as possible, and this can be achieved by using the mean square estimate for ζ(σ+it,α) in short intervals. In this paper, we obtain the bound O(H) for that mean square over the interval [T−H,T+H], with T27/82⩽H⩽Tσ and 1/2<σ⩽7/12. This is the first result on the mean square for ζ(s,α) in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for ζ(s,α) and other zeta-functions in short intervals.