gpt pro 5.2 attempt on Math-5533

Erdős Problem #258

1) FORMAL RESTATEMENT

Let $(a_n)_{n\ge1}$ be a sequence of integers with $a_n\to +\infty$ as $n\to\infty$. (Interpreting the statement literally requires $a_n\neq0$ for all $n$; in practice one assumes $a_n\in\mathbb{N}$ and hence $a_n\ge 1$ with $a_n\ge2$ eventually.) Define \[ T(a_1,a_2,\dots):=\sum_{n=1}^{\infty}\frac{\tau(n)}{a_1a_2\cdots a_n}, \] where $\tau(n)$ is the divisor-counting function. The problem asks whether \[ (\ddagger)\qquad \forall (a_n)\subset\mathbb{Z}\text{ with }a_n\to\infty:\ T(a_1,a_2,\dots)\notin\mathbb{Q}. \]

2) QUICK LITERATURE/CONTEXT CHECK

• Erdős--Straus $1971$ proved $(\ddagger)$ holds if $(a_n)$ is monotone nondecreasing (i.e. $a_{n-1}\le a_n$ for all $n$) [ErSt71].
• Erdős $1948$ proved that for every integer $t\ge2$, the Lambert series $\sum_{n\ge1} d(n)/t^n$ is irrational [Er48]; this is related but does not directly cover $a_n\to\infty$.
• As of late 2025/early 2026 the unrestricted case (no monotonicity) remains listed as open on the Erdős Problems website [ErP258].

3) ATTACK PLAN

1. Recognise $T(a_1,a_2,\dots)$ as a Cantor-series type expansion with "bases" $a_n$ and "digits" $\tau(n)$.
2. Try two directions:
   • Extend the Erdős--Straus proof from monotone $a_n$ to general $a_n\to\infty$.
   • Attempt to construct a counterexample sequence $(a_n)$ forcing $T$ to be rational (e.g. via telescoping or mixed-radix periodicity).

I did not succeed in either direction.

4) WORK

Convergence sanity check.

Since $a_n\to\infty$, there exists $N$ such that $a_n\ge2$ for all $n\ge N$. Then for $n\ge N$, \[ \left\lvert a_1\cdots a_n\right\rvert\ge \left\lvert a_1\cdots a_N\right\rvert\,2^{n-N}. \] Also $\tau(n)\le n$ for all $n$. Hence the tail is bounded by a convergent series: \[ \sum_{n\ge N}\frac{\tau(n)}{\left\lvert a_1\cdots a_n\right\rvert} \le \frac{1}{\left\lvert a_1\cdots a_N\right\rvert}\sum_{n\ge N}\frac{n}{2^{n-N}}<\infty. \] So $T(a_1,a_2,\dots)$ is well-defined.

Tiny numerical experiment (illustrative only).

For the monotone sequence $a_n=n+1$ (which satisfies $a_n\to\infty$), the first $50$ terms give \[ \sum_{n=1}^{50}\frac{\tau(n)}{2\cdot 3\cdots (n+1)}\approx 0.94529965343349825190\dots \] This is consistent with convergence and with irrationality, but of course gives no proof.

Known conditional resolution under monotonicity.

Erdős--Straus [ErSt71] prove that if $a_1\le a_2\le\cdots$ and $a_n\to\infty$, then $T(a_1,a_2,\dots)$ is irrational. The present problem is to remove the monotonicity hypothesis.

A tempting (but invalid here) telescoping construction.

If one could take $a_n=\tau(n)+1$, then \[ \frac{\tau(n)}{\prod_{j\le n}(\tau(j)+1)} =\frac{1}{\prod_{j\le n-1}(\tau(j)+1)}-\frac{1}{\prod_{j\le n}(\tau(j)+1)}, \] so the series would telescope to $1$. However $\tau(n)$ does not tend to $\infty$ (it is merely unbounded), so $a_n=\tau(n)+1$ fails the hypothesis $a_n\to\infty$. This illustrates why some growth/regularity condition on $a_n$ may be necessary to rule out rationality.

5) VERIFICATION

• The convergence argument uses only $a_n\ge2$ eventually and the trivial bound $\tau(n)\le n$.
• The telescoping observation is algebraically correct, and it explains the remark (seen in Erdős--Straus) that for other arithmetic functions (e.g. $\phi$ or $\sigma$) one can force a rational sum by choosing $a_n=\phi(n)+1$ or $\sigma(n)+1$; for $\tau(n)$ this exact choice does not satisfy $a_n\to\infty$.

6) FINAL

UNRESOLVED

(i) Strongest fully proved partial result obtained here. The series $T(a_1,a_2,\dots)$ converges for every integer sequence with $a_n\to\infty$ (proved above). The conjectured irrationality is known to hold when $(a_n)$ is monotone nondecreasing by Erdős--Straus [ErSt71].

(ii) First gap. Removing monotonicity breaks the existing proof strategy: one needs a way to control carries / mixed-radix expansions or to adapt divisor-function congruence arguments without assuming $a_{n-1}\le a_n$.

(iii) Top 3 next moves.

1. Re-express $T$ as a Cantor series and apply (or sharpen) known rationality criteria for Cantor-series expansions with varying bases $a_n$, attempting to exploit that $\tau(n)$ is "not too large" compared to $a_n\to\infty$.
2. Search for a counterexample sequence $(a_n)$ by imposing congruences on $a_n$ (and hence on partial products) that might force eventual periodicity of the expansion.
3. Attempt to weaken monotonicity to a growth condition (e.g. $a_{n+1}\ge a_n^{1+\delta}$ often enough) and see whether Erdős--Straus methods extend; this might provide intermediate theorems.

(iv) Minimal counterexample structure. A counterexample would be a sequence $a_n\to\infty$ with $\sum_n \tau(n)/(a_1\cdots a_n)\in\mathbb{Q}$. Because the terms are positive, such a sequence would have to enforce exact mixed-radix "eventual periodicity" or telescoping/cancellation at the level of partial products, without violating $a_n\to\infty$. Any such construction would likely require very non-monotone behaviour (since monotone sequences are known to give irrationality).

7) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 30%.

References

• [ErSz59] P. Erdős and G. Szekeres, On the product $\prod_{k=1}^n(1-z^{a_k})$, Publ. Math. Inst. Hung. Acad. Sci. 4 $1959$, 29--34.
• [At61] F. V. Atkinson, On a problem of Erdős and Szekeres, Canad. Math. Bull. 4 $1961$, 7--12.
• [Od82] A. Odlyzko, Minima of cosine sums and maxima of polynomials on the unit circle, J. London Math. Soc. 26 $1982$, 412--420.
• [BoCh18] J. Bourgain and M.-C. Chang, On a paper of Erdős and Szekeres, J. Anal. Math. 136 $2018$, 253--271.
• [BeKo96] A. S. Belov and S. V. Konyagin, An estimate of the free term of a non-negative trigonometric polynomial with integer coefficients, Izvestiya Math. 60 $1996$, 1123--1182.
• [Tang25] Q. Tang, An improved lower bound for Erdős--Szekeres products, arXiv:2509.14182 $2025$.
• [Er48] P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) $1948$, 63--66.
• [Er68d] P. Erdős, Some problems and results on irrationality, Math. Student 36 $1968$, 222--226.
• [Er88c] P. Erdős, On the irrationality of certain series: problems and results, in New Advances in Transcendence Theory (Cambridge University Press, 1988).
• [KoTa24] V. Kovač and T. Tao, On several irrationality problems for Ahmes series, arXiv:2406.17593 (2024; version cited on arXiv as of 2025).
• [TaTe25] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 $2025$.
• [ErSt71] P. Erdős and E. G. Straus, Some number theoretic results, Pacific J. Math. 36 $1971$, 635--646.
• [ErP258] T. F. Bloom (ed.), Erdős Problem #258 (website summary, accessed 2026-01-17), https://www.erdosproblems.com/258.