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$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.

This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< s< 1$ (convergence in this interval is essentially the Riemann hypothesis). Convergence at 1 is equivalent to prime number theorem.

Does divergence at 1/2 have similiar implications?

Myshkin
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Koushik
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  • Accurately speaking, it is the analytic continuation, instead of convergence and divergence –  Oct 25 '13 at 07:25
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    Divergence at 1/2 means that the partial sums of the Mobius function cannot exhibit more than square-root cancelation. Or equivalently that the error term in the prime number theorem cannot be substantially smaller than $\sqrt{x}$. – Lucia Oct 25 '13 at 10:02
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    @Lucia: Perhaps you should post your comment as an answer, as it seems to resolve the question. – Eric Naslund Oct 27 '13 at 20:15

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