: Mathématiques: Riesz function

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number > number theory > analytic number theory > L-function > Riesz function

mathematical analysis > complex analysis > meromorphic function > L-function > Riesz function

Riesz function

In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series

${\\displaystyle {\ m {Riesz}}(x)=-\\sum _{k=1}^{\\infty }{\ rac {(-x)^{k}}{(k-1)!\\zeta (2k)}}=x\\sum _{n=1}^{\\infty }{\ rac {\\mu (n)}{n^{2}}}\\exp \\left({\ rac {-x}{n^{2}}}\ ight).}$

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