M. Riesz series

M. Riesz (Sur l'hypothèse de Riemann, Acta Math. 40 (1916), 185-190) proved that the Riemann Hypothesis is equivalent to the assertion that

$$\sum_{k=1}^\infty \frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}\ll x^{1/2+\epsilon}$$

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