Absolutely Convergent Multiplicative Expansions of the von Mangoldt Function

Definitions and mathematical statement

Let ${\mathbb{N}}_{>0}=\{1,2,3,\ldots\}$. The von Mangoldt function is \[ \Lambda(n)=\begin{cases} \log p,&n=p^k\text{ for a prime }p\text{ and an integer }k\geq 1,\\ 0,&\text{otherwise}, \end{cases} \] where $\log$ denotes the natural logarithm; in particular, $\Lambda(1)=0$.

In the standard normalized convention, a function $h$ on ${\mathbb{N}}_{>0}$ is multiplicative if $h(1)=1$ and $h(ab)=h(a)h(b)$ whenever $a,b\in{\mathbb{N}}_{>0}$ satisfy $\gcd(a,b)=1$. This does not require complete multiplicativity.

The supplied question has the following formulation once a scalar field $K$ is specified. Do there exist functions \[ F:{\mathbb{N}}_{>0}\longrightarrow K, \qquad G:{\mathbb{N}}_{>0}\times{\mathbb{N}}_{>0}\longrightarrow K \] such that $F$ is multiplicative, and, for every $n\in{\mathbb{N}}_{>0}$, \[ G(1,n)=1,\qquad G(ab,n)=G(a,n)G(b,n) \quad\text{whenever }\gcd(a,b)=1, \] \[ \sum_{q=1}^{\infty}|G(q,n)|<\infty, \qquad \Lambda(n)=F(n)\sum_{q=1}^{\infty}G(q,n)? \] Absolute convergence here is required separately for every positive integer $n$; no uniform convergence requirement is supplied.

The source does not specify the scalar codomains of $F$ and $G$. Thus the display is a formulation template, for example for $K={\mathbb{R}}$ or $K={\mathbb{C}}$, rather than a source-supported choice between them. The normalized convention for multiplicativity is made explicit above but is not defined in the record. No further restriction on the dependence of $G(q,n)$ on $n$, or any requirement that $G$ have a particular Ramanujan-expansion form, is supplied.

Short English statement

The question asks whether the von Mangoldt function can be expressed as a multiplicative prefactor times an absolutely convergent series whose terms are multiplicative in the summation variable for each fixed positive integer input. The scalar codomains and any intended additional restrictions on the coefficients remain unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004299$, source problem code catalogue entry, titled "Absolutely Convergent Multiplicative Expansions of the von Mangoldt Function." No individual proposer is identified. The recorded proposed year is 2019.
• Supplied report citation: Analytic Number Theory (2020), Oberwolfach Reports, pp. 3141--3205, DOI 10.4171/owr/2019/50. https://doi.org/10.4171/owr/2019/50. The background attributes the collection to workshop open-problem discussions documented in Oberwolfach Reports; no individual problem page is supplied.
• Supplied literature reference: On Ramanujan expansions and primes in arithmetic progressions. https://doi.org/10.1007/s12188-024-00282-4.
• The supplied literature sources also list the "original OWR report": https://doi.org/10.4171/owr/2019/50.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that Hardy's Ramanujan expansion is not absolutely convergent, and that later work on truncated or smoothed Ramanujan expansions does not provide the requested absolutely convergent multiplicative $F,G$ representation. This is supplied dated curation, not an independently verified assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying qualification says that only formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. This metadata does not establish independent mathematical verification of this draft.