Let f be the formal power series over Z/2Z whose nth coefficient is the parity of the divisor function Ă_(0)(n)

Definitions and mathematical statement

Write ${\mathbb{F}}_2={\mathbb{Z}}/2{\mathbb{Z}}$, and for $n\ge1$ let $d(n)=|\{a\in{\mathbb{N}}_{>0}:a\mid n\}|$. For a coefficient sequence $(b_n)_{n\ge0}$ over ${\mathbb{F}}_2$, the proposed density is the natural density \[ \lim_{N\to\infty}\frac1N|\{0\le n<N:b_n=1\}|=\frac1{32}, \] including the assertion that this limit exists. Riasanovsky's question asks for this equality when the $b_n$ are the coefficients of $1/f$, with the positive-index coefficients of $f$ equal to $d(n)\bmod2$.

The constant coefficient is not specified, and this affects the meaning of the question. In fact, \[ f_+(x)=\sum_{n\ge1}(d(n)\bmod2)x^n=\sum_{m\ge1}x^{m^2} \] has constant coefficient zero and leading term $x$. It has no inverse in ${\mathbb{F}}_2[[x]]$. It does have a Laurent-series inverse in ${\mathbb{F}}_2((x))$; one possible precise question is the displayed density assertion for $b_n=[x^n]f_+(x)^{-1}$, $n\ge0$. Alternatively, assigning constant coefficient one gives \[ f_0(x)=1+f_+(x),\qquad f_0^{-1}\in{\mathbb{F}}_2[[x]], \] and gives a different density assertion with $b_n=[x^n]f_0^{-1}$. The source does not select between these conventions, so neither is silently adopted. The corrupted divisor-function symbol in the transcription is read here as the number-of-positive-divisors function described in words. The supplied research summary marks the specific $1/32$ claim open and unverified; its informal theta-series and other analogies provide no precise normalization or resolution.

Short English statement

Does the reciprocal of the divisor-parity series over ${\mathbb{F}}_2$ have coefficient $1$ with natural density $1/32$? The constant coefficient and the intended formal-series ring must be clarified to obtain a unique statement.

Sources

• Local UnsolvedMath registry, permanent ID $3100071$, exact source code catalogue entry; attributed to Riasanovsky.
• Cooper, Combinatorial Problems I Like, October 2020 snapshot, Classical Number Theory, source-order bullet 71; https://people.math.sc.edu/cooper/combprob.html. HTML extraction, recorded access 2026-07-29.
• Supplied research summary, classified OPEN-TRIAGE, describes a divisor-parity/theta-series conjecture and says a resolution could not be verified. No precise primary citation or complete Riasanovsky reference is supplied.