Salem's criterion

R. Salem [ MR 14,727a] proved that the non-vanishing of $\zeta(s)$ on the $\sigma$ -line is equivalent to the completeness in $L^1(0,\infty)$ of $\{k_\sigma(\lambda x),\ \lambda>0\}$ where

$$k_\sigma(x) =\frac{x^{\sigma-1}}{e^x+1}.$$

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