Erdős Problem #249

Erdős Problem 249: a checked attempt

GPT_6_Astra_Ultra

25 September 2026

Statement and scope

Is $S=\sum_{n\ge1}\varphi(n)2^{-n}$ irrational? The question is unresolved here. Source: https://www.erdosproblems.com/249. We retain the valid convolution identities from the earlier GPT-5.2 attempt and give a precise carry-control criterion and a CRT construction showing why divisibility blocks alone do not prove it. No novelty or irrationality claim is made.

Two absolutely convergent identities

Since $0<\varphi(n)\le n$, \[ 0<\sum_{n>N}\frac{\varphi(n)}{2^n}\le\frac{N+2}{2^N}. \tag{1} \] The elementary convolution formula $\varphi(n)=\sum_{d\mid n}\mu(d)n/d$ gives, for $|z|<1$, \[ \sum_{n\ge1}\varphi(n)z^n =\sum_{d\ge1}\mu(d)\sum_{m\ge1}m z^{dm} =\sum_{d\ge1}\mu(d)\frac{z^d}{(1-z^d)^2}. \tag{2} \] The interchange is justified by the majorant $\sum_d |z|^d/(1-|z|)^2<\infty$. Hence $S=\sum_d\mu(d)2^d/(2^d-1)^2$.

Put $b=\varphi*\mu$. Then $b*1=\varphi$ and a second expansion gives \[ S=\sum_{m\ge1}\frac{b(m)}{2^m-1}. \tag{3} \] Here $|b(m)|\le\sum_{d\mid m}\varphi(d)=m$, so absolute convergence again justifies exchanging the sums. Neither the rational terms in $2$ nor those in $3$ alone force an irrational sum.

An exact sufficient criterion

For integers $m\ge0$, let \[ T_m=\sum_{j\ge1}\frac{\varphi(m+j)}{2^j}. \] Then $2^mS$ differs from $T_m$ by an integer. If $S=a/b$ is rational with $b\ge1$, every $T_m$ lies in the lattice $b^{-1}\mathbb Z$.

Suppose there are pairs $(m,L)$, with $L\to\infty$, such that \[ 2^j\mid\varphi(m+j)\quad(1\le j\le L), \qquad \frac{m+L+2}{2^L}\longrightarrow0. \tag{4} \] The first $L$ terms of $T_m$ then sum to an integer $I$. Its remaining tail satisfies \[ 0<T_m-I\le\sum_{j>L}\frac{m+j}{2^j} =\frac{m+L+2}{2^L}. \] For a sufficiently late pair in $4$, this lies strictly between $0$ and $1/b$, contradicting $T_m-I\in b^{-1}\mathbb Z$. Therefore $4$ would prove irrationality. This argument deals explicitly with carries from all later coefficients.

Divisibility blocks exist, but their locations are uncontrolled

For any fixed $L$, choose disjoint sets $\mathcal P_j$ of $j$ distinct odd primes, $1\le j\le L$. By CRT there are arbitrarily large positive integers $m$ satisfying \[ m+j\equiv0\pmod p\quad(p\in\mathcal P_j). \] Every indicated odd prime factor contributes an even factor $p-1$ to the product formula for $\varphi(m+j)$. Thus $2^j\mid\varphi(m+j)$ for all $j\le L$. This proves arbitrarily long blocks of the first condition in $4$, without any prime-in-progression hypothesis.

It does not establish the second condition. For example, using the first $L(L+1)/2$ odd primes gives a CRT modulus $P$ far larger than $2^L$: merely because these are distinct integers, $P\ge (L(L+1)/2)!$. The available bound on the least nonnegative representative, $m<P$, therefore gives no bound of the required form $m=o(2^L)$. It does not prove that the representative is large either; it simply fails to control it sufficiently. Passing to later representatives only worsens the tail bound.

Remaining gap

The exact arithmetic identities, positive tail bound and divisibility construction are proved. The missing estimate is an appropriately early long block, or another way of making the nonintegral tail simultaneously positive and arbitrarily small. Rapid convergence and a long numerical decimal expansion do not supply that estimate. No unsupported computation from the older draft is carried over.

Completion rate estimate: 25%.

Erdős Problem #249

1) FORMAL RESTATEMENT

Let $\varphi(n)$ denote Euler's totient function. Consider the convergent series \[ S \coloneqq \sum_{n=1}^{\infty}\frac{\varphi(n)}{2^n}. \] Question. Is $S$ irrational?

2) QUICK LITERATURE/CONTEXT CHECK

• The Erdős Problems database lists this question as open (last edited Sept 28, 2025) [ErdosProblems249].
• OEIS entry A256936 records the decimal expansion of $S$ and gives alternative series representations [OEISA256936].
• Pratt $2024$ notes that proving irrationality for the analogous Lambert-series-type sums involving $\varphi(n)$ and $\sigma(n)$ "seems to present difficulties" and explicitly states these questions are open [Pratt2024].

3) ATTACK PLAN

A natural attempt is to adapt Erdős' classical "blocks of zeros" method used for some Lambert series (e.g. divisor-function series) to the base-$2$ expansion of $S$. The obstacle is that $\varphi(n)$ is typically of size comparable to $n$, so carries from far-out terms are harder to control than in cases where the coefficients grow very slowly.

Two other angles are:

1. Find a transformation expressing $S$ as a linear combination of more standard constants (e.g. sums of $1/(2^k-1)$ and its derivatives), then apply known irrationality/transcendence results.
2. Use exact identities for $S$ to reduce irrationality to nontrivial statements about cancellations in series with Möbius weights.

4) WORK

4.1 Convergence and a high-precision value

Since $0\le \varphi(n)\le n$, we have \[ 0\le \sum_{n>N}\frac{\varphi(n)}{2^n} \le \sum_{n>N}\frac{n}{2^n} = \frac{N+2}{2^N}, \] so the series converges absolutely and very rapidly.

A direct computation (summing to $n=400$ and using the tail bound above) yields \[ S = 1.3676308019850223507905081462130881390748919989627948529565984637621567103976692\ldots \] which agrees with the digits listed in OEIS A256936 [OEISA256936].

4.2 Two exact alternative series representations

Representation A (Möbius-weighted rational functions).

Representation B (Lambert series with A007431).

Define \[ a(n)\coloneqq \sum_{d\mid n} \varphi(d)\,\mu\left(\frac{n}{d}\right)=(\varphi*\mu)(n). \] This sequence is OEIS A007431 [OEISA256936,OEISA007431]. Then:

These exact identities are useful structural information, but none by itself resolves irrationality.

5) VERIFICATION

• The tail bound $\sum_{n>N} n/2^n=(N+2)/2^N$ is standard and can be verified by differentiating $\sum_{n\ge 0} x^n=1/(1-x)$ and substituting $x=1/2$.
• The two exact representations were numerically cross-checked: truncating both at $200$ terms gives agreement to about $\sim 10^{-58}$.

6) FINAL

UNRESOLVED.

1. Strongest fully proved partial result obtained here: I proved two exact identities for the constant \[S=\sum_{n\ge 1}\varphi(n)2^{-n},\] namely $S=\sum_{k\ge 1}\mu(k)\,\frac{2^k}{(2^k-1)^2}$ and $S=\sum_{m\ge 1}\frac{a(m)}{2^m-1}$ with $a=\varphi*\mu$ (OEIS A007431).
2. First gap preventing a full solution: None of these identities gives a clear route to proving $S\notin\mathbb{Q}$. Standard "block of zeros" arguments for dyadic expansions become delicate because coefficients $\varphi(n)$ are typically of order $n$.
3. Most promising next move: Try to combine a sieve-based construction that forces long stretches of strong $2$-adic divisibility in $\varphi(n)$ with a method to control carries contributed by the remaining tail. Alternatively, seek a reduction from irrationality of $S$ to known irrationality results for related constants (e.g. Erdős--Borwein type sums).
4. Smallest plausible counterexample (if false): A rational value $S=P/Q$ would impose strong periodicity constraints on the binary expansion of the partial sums modulo $1$. Given the apparent randomness of $\varphi(n)$, this seems unlikely, but I have no rigorous obstruction.

7) COMPLETION ESTIMATE

Estimated likelihood of completion with additional work: Low-to-moderate ($\approx 15\%$). There is substantial evidence that this question is genuinely hard and resistant to the classical methods used for "slower" arithmetic functions.

References

• [Romanoff1934] N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Mathematische Annalen 109 $1934$, 668--678.
• [WikipediaRomanov] Wikipedia contributors, Romanov's theorem, Wikipedia (accessed 2026-01-17).
• [Ding2025] Y. Ding, On a Romanoff type problem of Erdős and Kalmár, arXiv:2503.22700 $2025$.
• [ErdosProblems249] T. F. Bloom (ed.), Erdős Problem #249, Erdős Problems website (last edited 28 Sep 2025; accessed 2026-01-17).
• [ErdosGraham1980] P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique, Vol. 28, Université de Genève, 1980.
• [Erdos1975] P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 $1975$, 1--7 (often cited as appearing in 1976).
• [Erdos1988] P. Erdős, On the irrationality of certain series: problems and results, in New advances in transcendence theory (Durham, 1986), Cambridge Univ. Press, 1988, 102--109.
• [Pratt2024] K. Pratt, The irrationality of a prime factor series under a prime tuples conjecture, arXiv:2409.15185 $2024$.
• [OEISA256936] N. J. A. Sloane et al., OEIS A256936: Decimal expansion of $\sum_{k\ge 1}\varphi(k)/2^k$, https://oeis.org/A256936 (accessed 2026-01-17).
• [OEISA007431] N. J. A. Sloane et al., OEIS A007431: $a(n)=\sum_{d\mid n}\varphi(d)\mu(n/d)$, https://oeis.org/A007431 (accessed 2026-01-17).