Moments of L-functions — Moments of the Riemann zeta function — Problem 11

Definitions and mathematical statement

The source asks for an omega theorem for the quantity denoted by $I_3(T)$, conditional on the ternary additive divisor conjecture. The supplied record does not define $I_3(T)$, specify a version of that conjecture, or identify the coefficients of the main-term polynomial $\mathcal P_9$. These omissions remain unresolved hypotheses and notation in this draft.

For a function $E(T)$, interpret $E(T)=\Omega(T^\Theta)$ in the magnitude sense: \[ \exists c>0\;\exists (T_n)_{n\geq 1},\qquad T_n>0,\quad T_n\longrightarrow+\infty,\quad |E(T_n)|\geq cT_n^\Theta\quad\text{for every }n\geq1. \] This interpretation does not require errors of both signs; the supplied record does not specify a signed omega convention.

The requested statement is: assuming the ternary additive divisor conjecture intended by the source, prove that there exists a constant $\Theta>0$ such that \[ I_3(T)-T\mathcal P_9(\log T)=\Omega(T^\Theta) \qquad(T\to+\infty), \] where $\mathcal P_9$ is the fixed degree-nine main-term polynomial intended by the source. Equivalently, under the magnitude interpretation above, prove that there exist $\Theta>0$, $c>0$, and positive parameters $T_n\to+\infty$ satisfying \[ \left|I_3(T_n)-T_n\mathcal P_9(\log T_n)\right| \geq cT_n^\Theta \quad\text{for every }n\geq1. \] The polynomial is part of the intended main term, rather than a polynomial that may vary with $T$ or with $n$. A fully specified mathematical statement requires the source's definition and normalization of $I_3$, its precise conjectural hypothesis, and its identification of $\mathcal P_9$.

Short English statement

Assuming the intended ternary additive divisor conjecture, show that the difference between $I_3(T)$ and its stated degree-nine logarithmic main term has magnitude at least a positive power of $T$ along an unbounded sequence. The definitions of $I_3$, the conjecture, and the main-term polynomial are missing from the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $800161$, source problem code catalogue entry; title: Moments of L-functions --- Moments of the Riemann zeta function --- Problem 11. The registry labels this record open; that label is supplied metadata, not an independently verified mathematical or literature assessment.
• Hamieh, Kadiri, and Ng, Moments of L-functions Problem List (2024), Problem 11, https://arxiv.org/abs/2405.17800. This is the source reference supplied by the registry; it has not been independently consulted for this draft.
• The statement attributes the question to Nathan Ng, Lethbridge. The registry's proposer field lists Hamieh, Kadiri, and Ng, with proposed year 2024.
• Supplied provenance: extraction method arxiv_latex; access date 2026-10-16. This date records the supplied source-access provenance and does not represent independent source verification performed for this draft.