Problem 1049

../

claims/: The 3 claim pages of Problem 1049, one per claimant's result; the problem's standing derives from them.

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Statement. Let $t>1$ be a rational number. Is

$$\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}$$

irrational, where $\tau(n)$ counts the divisors of $n$?

Status. Open, the site's label (OPEN). The site credits the integer case to Erdős [Er48]; the claim pages are cited in the Current assessment.

Source. erdosproblems.com/1049, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1049, https://www.erdosproblems.com/1049.

References.

• [Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.

Formalization. Statement in formal-conjectures.

Current assessment

The site labels Problem 1049 OPEN and credits Erdős with the integer case. Two accepted partial claims, both refereed, settle part of the question: Erdős 1948 proves irrationality for every integer $t\ge2$, and [[problems/irrationality/E1049/claims/1994_01_01_bundschuh_vaananen|Bundschuh and Väänänen 1994]] prove it, with an irrationality measure, for every $t=a/b$ in lowest terms with $\log b/\log a<\frac12-\frac1{\pi^2}$, which includes the integers and, for example, $7/2$. The pending partial claim Cook 2026, a manuscript with no review, extends the region to $\log b/\log a<\theta^*=0.40568\ldots$, which includes every power of $31/4$. No claim covers $t=3/2$ or any $a/b$ with $\log b/\log a\ge\theta^*$, and the problem is open.

Search scope. 2026-10-07: erdosproblems.com (the page and the forum thread, whose only proof claim is the comment of 11 September 2026), the formal-conjectures statement file, the Numdam record of Bundschuh and Väänänen's paper, and Crossref.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• duverney_tachiya_2019_refinement_chowla_erdos_method_linear_independence_certain_lambert_series
• duverney_tachiya_2019_refinement_chowla_erdos_method_linear_independence_certain_lambert_series / corollary_1_1
• erdos_1948_arithmetical_properties_lambert_series
• vandehey_2012_incomplete_argument_erdos_irrationality_lambert_series
• vandehey_2012_incomplete_argument_erdos_irrationality_lambert_series / theorem_1_1
• vandehey_2012_incomplete_argument_erdos_irrationality_lambert_series / theorem_1_2