Let $B_{k}$ refer to the Bernoulli numbers. Then consider the series

$$\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)\biggr|\sum_{k=0}^{\infty}\frac{c_{2n+1,2k+1}}{2k+2}\log\left(\frac{2k+1}{2k+2}\frac{(-1)^{k}B_{2k+2}(2\pi)^{2k+2}}{2(2k+2)!}\right)\biggr|^{2}$$

where

$$c_{2n+1,2k+1}=\frac{(-1)^{n-k}(2n+2k+2)!}{2^{2n+1}(n-k)!(n+k+1)!(2k+1)!}.$$

It is not known whether this series converges. In fact, its convergence is equivalent to the Riemann Hypothesis.

(There are a lot of ways to make series that converge if and only if RH)

Hope that helps,

Let $B_{k}$ refer to the Bernoulli numbers. Then consider the series

$$\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)\biggr|\sum_{k=0}^{\infty}\frac{c_{2n+1,2k+1}}{2k+2}\log\left(\frac{2k+1}{2k+2}\frac{(-1)^{k}B_{2k+2}(2\pi)^{2k+2}}{2(2k+2)!}\right)\biggr|^{2}$$

where

$$c_{2n+1,2k+1}=\frac{(-1)^{n-k}(2n+2k+2)!}{2^{2n+1}(n-k)!(n+k+1)!(2k+1)!}.$$

It is not known whether this series converges. In fact, its convergence is equivalent to the Riemann Hypothesis.

(There are a lot of ways to make series that converge if and only if RH)

Hope that helps,

Reference:

(1) John Corning Carey's Paper: http://jcarey.best.vwh.net/RHHardy.pdf