Erdős Problem #69

Erdős Problem 69: a prime-divisor series

1) Statement and current resolution

Let $\omega(n)$ count the distinct prime divisors of $n$, with $\omega(1)=0$. Is \[ S=\sum_{n\geq1}\frac{\omega(n)}{2^n} \] irrational? Tao and Teräväinen proved this unconditionally in Theorem 1.3 of their 2025 paper. The present attempt proves exact identities, certified approximation bounds and a necessary rationality condition. It does not reconstruct their correlation argument, so its own proof is partial despite the established affirmative resolution.

2) A convergent prime expansion

The product of the distinct prime divisors of $n$ is at least $2^{\omega(n)}$ and divides $n$. Thus $\omega(n)\leq\log_2 n$, and $S$ converges absolutely. Since all terms are nonnegative, summing first over prime divisors is justified and gives \[ S=\sum_p\sum_{k\geq1}2^{-kp} =\sum_p\frac1{2^p-1}. \] This is a sum over all multiples of a prime, not merely its powers: confusing the two would replace $\omega$ by a different arithmetic function. For any integer $P\geq2$, the prime truncation has the rigorous error bound \[ 0<S-\sum_{p\leq P}\frac1{2^p-1} \leq 2\sum_{m>P}2^{-m}=2^{1-P}. \] The strict positivity follows from the infinitude of primes.

3) What rationality would force

Suppose $S=a/b$, with integers $a,b>0$. For each $N\geq1$, set \[ T_N=\sum_{h\geq1}\frac{\omega(N+h)}{2^h}. \] Multiplying the original series by $2^N$ gives \[ T_N=2^N\frac ab-\sum_{n=1}^N\omega(n)2^{N-n} \in \frac1b\mathbb Z. \] In particular every $T_N$ belongs to one fixed discrete lattice, although it need not be bounded. This is a useful exact obstruction for an irrationality proof; a mere large value of $\omega(N+h)$ does not contradict it.

Truncation can be controlled uniformly. Concavity of logarithm gives \[ \log_2(N+h)\leq\log_2 N+\frac{h}{N\log 2}. \] Using $\sum_{h>H}2^{-h}=2^{-H}$ and $\sum_{h>H}h2^{-h}=(H+2)2^{-H}$, we obtain \[ 0\leq T_N-\sum_{h=1}^H\frac{\omega(N+h)}{2^h} \leq 2^{-H}\left(\log_2N+\frac{H+2}{N\log2}\right). \] Thus rationality implies that every displayed finite weighted block is within this explicit error of $b^{-1}\mathbb Z$. It would suffice to find, for every fixed $b$, some $N,H$ whose weighted block has distance from that lattice larger than the error bound.

4) The remaining gap

No argument here produces such blocks for every denominator $b$. The coefficients $\omega(n)$ are not binary digits: they exceed $1$, so binary carries prevent an inference from nonperiodicity of $\omega(n)$ to irrationality of $S$. Likewise, arbitrarily accurate rational approximations with uncontrolled denominators prove nothing about irrationality. The known solution obtains the needed control using quantitative correlations of multiplicative functions; that external proof is cited for the problem's status, not silently counted as a proof supplied by this attempt.

Outcome of this attempt: UNRESOLVED; exact reductions and error estimates proved. No new mathematical advance is claimed. The batch verification checks the divisor/prime rearrangement on finite sums using exact rational arithmetic.

5) Sources

• T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, Theorem 1.3 and Section 5, https://arxiv.org/abs/2512.01739.
• T. F. Bloom, Problem 69, https://www.erdosproblems.com/69, checked 23 September 2026.

The percentage is a conservative estimate of the proof supplied here, not of the confidence in the published resolution.

6) COMPLETION ESTIMATE

COMPLETION: 10%