Research Problems in Function Theory — Problem 7.31

Definitions and mathematical statement

Let $(a_n)_{n\ge1}$ be real numbers satisfying $a_1>0$ and $0\le a_n\le n$, and define \[ b_n=\sum_{\nu=1}^na_\nu,\qquad c_n=\sum_{\nu=1}^nb_\nu=\sum_{\nu=1}^n(n-\nu+1)a_\nu. \] Thus $b_n,c_n>0$. The source records that \[ \sum_{n\ge1}(a_n/c_n)^\alpha<\infty\qquad(\alpha>\tfrac12). \] Hayman asks which other functions $f$ guarantee convergence of $\sum_{n\ge1}f(c_n/a_n)$ for every sequence satisfying the displayed assumptions, and suggests a relation with $\sum_{n\ge1}f(n^2)$, under unspecified smoothness assumptions on $f$.

Zero terms are allowed: when $a_n=0$, $c_n/a_n$ is not a finite argument. A possible convention is $f(\infty)=0$, as appropriate to the recorded power functions $f(t)=t^{-\alpha}$, and then to write \[ \sum_{n:a_n>0}f(c_n/a_n). \] That convention is not provided by the source. The class, sign, domain, decay, and smoothness of $f$ must also be fixed. The phrase "converges with" may mean that convergence of $\sum f(n^2)$ implies the universal convergence property, or that it characterizes that universal property. It cannot automatically be read as equivalence of convergence for each individual sequence: the permitted finitely supported sequences would give a finite left sum under the zero-term convention regardless of the comparison series. No one interpretation is silently chosen.

The source supplies a Borwein comparison theorem for the different ratio $c_n/b_n$: if $xf(x)$ is positive and nonincreasing for $x\ge a>0$, and $\sum_{n\ge1}f(n)<\infty$, then $\sum_{n\ge1}f(c_n/b_n)<\infty$. This is retained as a supplied theorem statement, not confused with the requested $c_n/a_n$ case. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no definitive resolution located.

Short English statement

Characterize functions preserving convergence when evaluated at the ratio of a twice-summed sequence to its original terms, and compare with the square-index series. Zero terms, the function class, and the intended convergence implication need clarification.

Sources

• Local UnsolvedMath registry, permanent ID $2307031$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 7.31."
• Hayman, Research Problems in Function Theory (2018), Problem 7.31, attributed to W. K. Hayman; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• The source attributes the $c_n/b_n$ comparison theorem to Borwein without a complete reference. The supplied OPEN-TRIAGE summary gives no separate current-resolution citation.