Hardy-Littlewood series

Hardy and Littlewood (Acta Mathematica 41 (1918), 119 - 196) showed that the Riemann Hypothesis is equivalent to the estimate

\begin{displaymath}\sum_{k=1}^\infty \frac{(-x)^k}{k!\zeta(2k+1)}\ll x^{-1/4}\end{displaymath}

as $x\to \inftu$.




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