Piltz divisor problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $k\ge2$ and $n\in\mathbb N$, let $d_k(n)$ count ordered positive integer tuples $(n_1,\ldots,n_k)$ with $n_1\cdots n_k=n$. Put $D_k(x)=\sum_{n\le x}d_k(n)$ for real $x\ge1$. Define the polynomial $P_k$ of degree $k-1$ by \[ xP_k(\log x)=\operatorname{Res}_{s=1}\left(\zeta(s)^k\frac{x^s}{s}\right),\qquad \Delta_k(x)=D_k(x)-xP_k(\log x). \] Here $\zeta(s)=\sum_{n\ge1}n^{-s}$ for $\operatorname{Re}s>1$ and then by analytic continuation, and $\operatorname{Res}$ is the Laurent coefficient of $(s-1)^{-1}$. Set \[ \alpha_k=\inf\{a\in\mathbb R:\forall\varepsilon>0,\ \Delta_k(x)=O_{k,\varepsilon}(x^{a+\varepsilon})\text{ as }x\to\infty\}. \] Big-$O$ means an absolute-value bound by a constant, depending on the indicated parameters, for all sufficiently large $x$.

Statement recorded in the supplied source.

Determine $\alpha_k$ for every integer $k\ge2$, equivalently the optimal exponents in the Piltz divisor-error bounds. In particular, is the conjectural value $\alpha_k=(k-1)/(2k)$ correct for every $k\ge2$? [S2]

Short English statement

Find the sharp growth exponents for the errors in all higher divisor-counting sums, including the predicted exponent for each number of factors.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400076 (AMR-093-0076). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, Divisor summatory function, revision 1340008726 (2026) Piltz divisor problem. https://en.wikipedia.org/w/index.php?title=Divisor+summatory+function&oldid=1340008726.
• S3 T. Makoto Minamide, Yoshio Tanigawa and Nigel Watt, On the sum of $\Delta_k(n)$ in the Piltz divisor problem for k3 and4, arXiv2412.11069v1 (2024) Introduction equation1.1 p1. https://arxiv.org/pdf/2412.11069.