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 $n$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write (t>1) and expand each term as a geometric series: [ \frac{1}{t^n-1}=\frac{t^{-n}}{1-t^{-n}}=\sum_{k\ge 1} t^{-nk}. ] Summing over (n\ge 1) and regrouping by $m=nk$ gives [ \sum_{n=1}^\infty \frac{1}{t^n-1} =\sum_{n=1}^\infty\sum_{k\ge 1} t^{-nk} =\sum_{m=1}^\infty \frac{|\\{(n,k):nk=m\\}|}{t^m} =\sum_{m=1}^\infty…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.