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Claim. For every integer $t$ with $|t|>1$,

$$\sum_{n=1}^\infty\frac1{t^n-1}=\sum_{n=1}^\infty\frac{\tau(n)}{t^n}$$

is irrational. This is the single Theorem of P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63--66 (card erdos_1948_arithmetical_properties_lambert_series), which proves the same for the sine analog of the series. For $t>0$ the proof uses the Chinese remainder theorem to find, for arbitrarily large $k$, blocks of consecutive integers whose divisor counts force arbitrarily long runs of zeros in the base-$t$ expansion of $\sum\tau(n)t^{-n}$, so the expansion is neither finite nor periodic. The citation gives the year without a month or day, so this page is dated to the first day of that year. J. Vandehey, On an incomplete argument of Erdős on the irrationality of Lambert series, arXiv:1206.0340 (card vandehey_2012), repairs the argument for negative integer bases only, which the problem does not ask about.

Covers. Every integer $t\ge2$ of Problem 1049, and no non-integer rational $t$.

Acceptance. Refereed: the paper is a journal publication in the Journal of the Indian Mathematical Society (New Series), volume 12 (1948), the refereed evidence. The site's curator, Thomas Bloom, credits the integer case to this paper in the problem page's commentary, but the site labels the problem OPEN, so that credit is not reviewed evidence.

Formalization. The theorem erdos_1049_variants_geq_2_integer in research/adapters/FormalConjecturesAdapter.lean of Will Cook's repository plectis-erdos (the formalization link, lines 96--107 at the pinned commit) states the integer case for every $t\ge2$ and derives it from that repository's theorem irrational_erdosSum_full_support. The module attributes the result to Erdős's 1948 paper and claims no novelty for it, so it is a link on this page rather than a claim of its own. The formal-conjectures statement file (the record link, at the commit of 2026-09-23 that added the attribute) names it as the formal_proof of erdos_1049.variants.geq_2_integer. This repository records no build or axiom audit of the development, so the page lists no formalized evidence.

Depends on. Nothing in this wiki.