Low Pseudomoments of the Riemann Zeta Function and Its Powers

Abstract

The $2 q$-th pseudomoment $\Psi _{2q,\alpha }(x)$ of the $\alpha $-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $\zeta ^\alpha $ on the critical line. Using probabilistic methods of Harper, we prove upper and lower bounds for these pseudomoments when $q \le \frac{1}{2}$ and $\alpha \ge 1$. Combined with results of Bondarenko et al., these bounds determine the size of all pseudomoments with $q> 0$ and $\alpha \ge 1$ up to powers of $\log \log x$, where $x$ is the length of the partial sum, and it turns out that there are three different ranges with different growth behaviours. In particular, the results give the order of magnitude of $\Psi _{2 q, 1}(x)$ for all $q> 0$.

Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function

“…These follow in the same way by applying [17,Lemma 8] which in fact holds for uniformly small exponents b and c there (see 'Euler product result 1' of [20]). Applying (15), we find that the expectation of the kth power of (14) is bounded above by…”

“…Outline of modified proof. One can check that the uniform version of [17,Proposition 4] is given by the inequality…”

“…Following [17], we break the range of integration down into various sub-ranges. By symmetry in law, the expectation of the integral over t < 0 is equal to that over t > 0, so we focus on this latter range.…”

“…where c k is an explicitly given constant. The case of real k was considered by Bondarenko-Heap-Seip [11] with refinements in the low moments case coming from Heap [23] and then Gerspach [17] who gave a fairly complete resolution of the problem by applying ideas from Harper's proof of Helson's conjecture [21]. As a result, we know that…”

“…The moments were initially considered by Conrey–Gamburd [13] who proved that for fixed , where is an explicitly given constant. The case of real was considered by Bondarenko–Heap–Seip [11] with refinements in the low moments case coming from Heap [23] and then Gerspach [17] who gave a fairly complete resolution of the problem by applying ideas from Harper's proof of Helson's conjecture [21]. As a result, we know that for all real, fixed .…”