What is Σ_(n≥1 )φ(n)/2^(n), where φ(n) is the Euler phi (totient) function, counting the number of integers less than n

Definitions and mathematical statement

Let $\varphi(n)$ be Euler's totient function, \[ \varphi(n)=\#\{a:1\leq a\leq n,\ \gcd(a,n)=1\}, \] so $\varphi(1)=1$ and for $n\geq2$ this is the number of positive integers less than $n$ coprime to $n$. Define \[ S=\sum_{n=1}^{\infty}\frac{\varphi(n)}{2^n}. \] The series converges absolutely since $0\leq\varphi(n)\leq n$ and $\sum n2^{-n}<\infty$. The convention at $n=1$ matters for the numerical value and is fixed by the standard totient function.

Questions attributed to Erdős. Determine $S$, with a closed-form answer desired. Is $S$ irrational, that is, $S\notin{\mathbb{Q}}$? The closed-form request and irrationality question are separate: a numerical approximation or a different infinite-series representation does not by itself provide the requested closed form. The source specifies no class of permitted constants or operations for such a form.

The registry status is open and its classification is OPEN-TRIAGE. The supplied research summary instead claims irrationality known, says no simple elementary closed form is available, and recommends partial-progress wording. It tentatively names Erdős--Borwein--Chamayou with a question mark, later names Borwein--Chamayou, and refers to generalized continued-fraction irrational sums and an unspecified average-order expression. It supplies no precise theorem, article, or actual expression verifying those statements. Therefore the irrationality claim remains explicitly attributed and unverified; it is not treated as a settled conclusion of this draft.

Short English statement

Find a closed form for $\sum_{n\geq1}\varphi(n)/2^n$ and determine whether it is irrational. The standard convention is $\varphi(1)=1$. The summary claims irrationality known but gives tentative attribution and no verifiable theorem reference, conflicting with the open registry status.

Sources

• Local UnsolvedMath registry, permanent ID 3100070, source code catalogue entry, title beginning "What is the sum of phi(n)/2 to the n"; proposer: Erdős. Supplied status open, classification OPEN-TRIAGE, updated 2026-08-06; the research summary separately recommends partial-progress wording.
• Cooper, Combinatorial Problems I Like, October 2020 snapshot, section "Classical Number Theory," source-order bullet 70, https://people.math.sc.edu/cooper/combprob.html. The record reports HTML extraction and access on 2026-07-29; later resolution was not checked item by item. Rights: NEEDS_REVIEW; public access verified but redistribution permission not established.
• The supplied summary mentions Erdős, Borwein, and Chamayou with expressly uncertain attribution and no paper title or theorem; it supplies no actual average-order formula. No bibliographic details are reconstructed.