Moments of L-functions — Moments of the Riemann zeta function — Problem 28

Definitions and mathematical statement

Let $\zeta(s)$ denote the Riemann zeta function, defined for $\operatorname{Re}s>1$ by $\sum_{n=1}^{\infty}n^{-s}$ and elsewhere by meromorphic continuation. The Riemann hypothesis (RH) asserts that every zero $\rho$ with $0<\operatorname{Re}\rho<1$ satisfies $\operatorname{Re}\rho=\tfrac12$. All logarithms below are natural.

The source writes $S(t)=\pi^{-1}\arg\zeta(\tfrac12+it)$ without specifying the argument convention or values at zero ordinates. For a precise candidate formulation under RH, for $t>0$ that is not a zero ordinate, define the argument by continuous variation along the line segments from $2$ to $2+it$ and then to $\tfrac12+it$, starting with $\arg\zeta(2)=0$. Set \[ S(t)=\frac{1}{\pi}\arg\zeta\!\left(\tfrac12+it\right). \] At a positive zero ordinate $\gamma$, use the convention \[ S(\gamma)=\frac12\left(\lim_{h\downarrow0}S(\gamma-h)+\lim_{h\downarrow0}S(\gamma+h)\right). \] These argument and boundary conventions are explicit interpretations for this draft and require confirmation against the source.

The supplied record reports that Carneiro, Chandee, and Milinovich [CCM] proved, assuming RH, \[ |S(t)|\leq\left(\frac14+o(1)\right)\frac{\log t}{\log\log t} \qquad(t\to+\infty). \] Requested improvement. Assuming RH, establish the existence of a fixed real constant $C$ with $0\leq C<\tfrac14$ such that, for every $\varepsilon>0$, there is a real number $T_{\varepsilon}>e$ for which \[ |S(t)|\leq(C+\varepsilon)\frac{\log t}{\log\log t} \qquad\text{for every real }t\geq T_{\varepsilon}. \] Thus the requested reduction concerns the leading constant in the asymptotic bound. The source does not specify a particular smaller target constant.

Short English statement

Assuming the Riemann hypothesis, improve the supplied asymptotic bound for the normalized argument of the Riemann zeta function by replacing its leading constant $1/4$ with a fixed smaller constant. The argument normalization and treatment of zero ordinates have been made explicit as draft conventions requiring review.

Sources

• Local UnsolvedMath registry, permanent ID $800177$, exact source problem code catalogue entry; title: Moments of L-functions --- Moments of the Riemann zeta function --- Problem 28. The registry credits Hamieh, Kadiri, and Ng, with proposed year 2024, and labels the problem open; this is a supplied status, not an independently verified assessment.
• Hamieh, Kadiri, and Ng, Moments of L-functions Problem List (2024), Problem 28, https://arxiv.org/abs/2405.17800. The supplied item attributes the question to Nathan Ng, Lethbridge. Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are source-record annotations, not a claim of independent access or verification for this draft.
• Carneiro, Chandee, and Milinovich, reference label [CCM] in the supplied statement, are credited there with the conditional bound having leading constant $1/4$. No full bibliographic entry or separate URL for [CCM] is supplied.