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arXiv · 1501.04185

On a conjecture of Erdös and certain Dirichlet series

Tapas Chatterjee·M. Ram Murty
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Abstract

Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erdös conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved the conjecture. We show that this conjecture is true for $82\%$ of the remaining integers $q\equiv 1$ ( mod $4$).

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Tapas Chatterjee, M. Ram Murty. 2015-01-17. On a conjecture of Erdös and certain Dirichlet series. https://arxiv.org/abs/1501.04185

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