Asymptotic formula for M_k(d)

Derive, for each fixed integer k > 1, an asymptotic formula for M_k(d) as d tends to infinity.

Background

The paper presents numerical values and curve-fitting experiments for M_k(d), including comparisons with polynomials in log d. The available computations do not determine the eventual asymptotic behavior.

The authors explicitly formulate the derivation of an asymptotic formula for each k as a problem, motivated by the apparent slow but nontrivial growth of M_k(d).

References

It would be very interesting to carry out even further numerics to address the following problem. Problem. For each k, conjecture an asymptotic formula for the M_k(d).

— Do perfect powers repel partition numbers?  (2501.03754 - Merca et al., 7 Jan 2025) in Section “Further Conjectures: Do kth Powers Repel Partition Numbers?”

Accordingly, the asymptotic growth of $M_d$ remains open.

— Execution-transcript privacy for fault-tolerant surface-code memories  (2609.09334 - Shen et al., 8 Sep 2026) in Results, subsection “The order of the basis-label leak, and when it is attained,” paragraph beginning “We stop short of an asymptotic claim”

It is conjectured that \begin{equation}\label{conjfunddisc} \frac{1}{D}\sum_{0<d\leq D}L\left(\frac{1}{2},\chi_d\right)k\sim a_kg_k(\log D){\frac{k(k+1)}{2}, \end{equation} where $a_k$ is an arithmetic factor in the form of a Euler product and $g_k$ is a geometric factor.

conjfunddisc:

1D∑0<d≤DL(12,χd)k∼akgk(log⁡D)k(k+1)2,\frac{1}{D}\sum_{0<d\leq D}L\left(\frac{1}{2},\chi_d\right)^k\sim a_kg_k(\log D)^{\frac{k(k+1)}{2}},

— The Mixed Second Moment of Quadratic Dirichlet $L$-functions with Prime Conductors  (2608.25721 - MacMillan, 26 Aug 2026) in Section 1, Introduction