Moments of L-functions — Moments of the Riemann zeta function — Problem 27

Definitions and mathematical statement

Let $\zeta(s)$ be the meromorphic continuation of the Riemann zeta function $\sum_{n=1}^{\infty}n^{-s}$, initially defined for $\operatorname{Re}s>1$. The Riemann Hypothesis asserts that every nontrivial zero of $\zeta$ has real part $1/2$. Use natural logarithms below.

The supplied source attributes to Chandee and Soundararajan [CS] the following conditional estimate: assuming the Riemann Hypothesis, \[ \left|\zeta\!\left(\tfrac12+it\right)\right| \ll \exp\!\left(\left(C_0+o(1)\right)\frac{\log t}{\log\log t}\right) \qquad(t\to+\infty), \qquad C_0=\frac{\log 2}{2}. \] Here $t$ is real, $o(1)$ tends to zero as $t\to+\infty$, and $\ll$ denotes an inequality up to a positive multiplicative constant independent of $t$.

The requested improvement, using the source's exact symbolic value of $C_0$, is to prove that there exists a real constant \[ C<\frac{\log 2}{2} \] such that, assuming the Riemann Hypothesis, for every $\varepsilon>0$ there exist constants $A_{\varepsilon}>0$ and $T_{\varepsilon}>e$ for which every real $t\ge T_{\varepsilon}$ satisfies \[ \left|\zeta\!\left(\tfrac12+it\right)\right| \le A_{\varepsilon}\exp\!\left((C+\varepsilon)\frac{\log t}{\log\log t}\right). \] The constants may depend on the chosen $C$ and on $\varepsilon$, but not on $t$.

Unresolved source discrepancy. The source also writes $C_0=0.1505\ldots$, whereas with natural logarithms $(\log 2)/2=0.346573\ldots$. These values are incompatible. The statement above provisionally follows the exact symbolic threshold; the intended numerical threshold requires source clarification.

Short English statement

Assuming the Riemann Hypothesis, improve the leading constant in the stated upper bound for $\lvert\zeta(1/2+it)\rvert$ as real $t\to+\infty$. The source's symbolic constant $(\log 2)/2$ conflicts with its decimal approximation $0.1505\ldots$; this draft provisionally uses the symbolic value and requires review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800176$, exact source problem code catalogue entry; title: "Moments of L-functions --- Moments of the Riemann zeta function --- Problem 27." Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 27: https://arxiv.org/abs/2405.17800. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance annotations, not a claim of independent source access or verification.
• The registry lists Hamieh, Kadiri, Ng as proposers, with proposed year 2024. The extracted problem itself carries the attribution "N. Ng, Lethbridge."
• The extracted statement attributes the existing conditional estimate to Chandee and Soundararajan, reference label [CS]. No full bibliographic entry or separate link for [CS] is supplied.
• The local record labels the problem open. This is a supplied registry classification, not an independently verified assessment of the literature.