Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6426204 | Indagationes Mathematicae | 2015 | 25 Pages |
Abstract
Let d(n) be the number of divisors of n, let Î(x)âân⩽xd(n)âx(logx+2γâ1) denote the error term in the classical Dirichlet divisor problem, and let ζ(s) denote the Riemann zeta-function. It is shown that â«0TÎ(t)|ζ(12+it)|2dtâªT(logT)4. Further, if 2⩽k⩽8 is a fixed integer, then we prove the asymptotic formula â«1TÎk(t)|ζ(12+it)|2dt=c1(k)T1+k4logT+c2(k)T1+k4+Oε(T1+k4âηk+ε), where c1(k) and c2(k) are explicit constants, and where η2=3/20,η3=η4=1/10,η5=3/80,η6=35/4742,η7=17/6312,η8=8/9433. The results depend on the power moments of Î(t) and E(T), the classical error term in the asymptotic formula for the mean square of |ζ(12+it)|.
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Aleksandar IviÄ, Wenguang Zhai,