Moments of L-functions — Moments of the Riemann zeta function — Problem 30

Definitions and mathematical statement

For each integer $k\ge 3$, write formally \[ E_{k,2}(x,h):=D_{k,2}(x,h)-M_{k,2}(x,h), \] where the supplied source describes $M_{k,2}(x,h)$ as the main term in an asymptotic formula for $D_{k,2}(x,h)$.

Requested statement (N. Ng, Lethbridge). Prove an omega theorem for $E_{k,2}(x,h)$ for $k\ge 3$.

Unresolved qualifications. The supplied record does not define $D_{k,2}(x,h)$, specify the main term $M_{k,2}(x,h)$, give the domains or ranges of $x$ and $h$, or state whether $h$ is fixed or varies with $x$. It also does not specify an omega scale or whether an unsigned or signed omega result is intended. Thus the request cannot be converted into a uniquely specified mathematical proposition from this record alone.

For clarity, a possible unsigned omega assertion, once the missing data are supplied, has the following meaning. For an unbounded set $X\subseteq{\mathbb{R}}$, functions $E:X\to{\mathbb{C}}$ and $B:X\to(0,\infty)$, the assertion \[ E(x)=\Omega(B(x))\qquad(x\to\infty,\ x\in X) \] means that there exist $c>0$ and a sequence $(x_j)_{j\ge1}$ in $X$, tending to infinity, such that \[ |E(x_j)|\ge cB(x_j)\qquad\text{for every }j\ge1. \] This definition is an explanatory template, not a selection of the intended domain, scale, sign convention, or parameter quantifiers for the source problem. Any dependence of constants and sequences on $k$ and $h$, and any required uniformity, remains unspecified.

Short English statement

N. Ng asks for an omega theorem for the error after subtracting the main term from $D_{k,2}(x,h)$, for integers $k\ge3$. The supplied record omits the definitions, parameter ranges, main term, and intended omega scale and sign, so a complete precise formulation requires those details.

Sources

• Local UnsolvedMath registry, permanent ID $800179$, exact source code catalogue entry, supplied complete source record; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels this problem open; this is a supplied status label, not an independently verified mathematical assessment.
• Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 30, in the supplied category "Moments of the Riemann zeta function": https://arxiv.org/abs/2405.17800. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these provenance details have not been independently verified here.
• The problem text attributes the request to N. Ng, Lethbridge. The registry records Hamieh, Kadiri, Ng as proposers and 2024 as the proposed year.
• The supplied statement credits Topacogullari, references [T1] and [T3], and Drappeau, reference [Dr], with asymptotic formulae for $D_{k,2}(x,h)$. This is a report of the source's background claim. Bibliographic entries, theorem statements, hypotheses, and links for these three reference labels are not supplied.