The mean value formula for the approximate functional equation of ζ~(s) in the critical strip

摘要

Let s = σ + it (0 ≤ σ ≤ 1, t ≤ 1) be complex valuable, d(n) the number of positive divisors of n and ζ(s) the Riemann zeta-function. We define the error term R(s;x) in the approximate functional equation for ζ~2(s) as R(s;x) = ζ~2(s) - ∑ ' from n≤x of d(n)n~(-s) - X~2(s) ∑' from n≤x of d(n)n~(s-1), where (1) X(s) = 2~sπ~(s-1)sin(πs/2)Γ(1-s) and x = t(2π).