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

If we set ${\displaystyle F(x)={\frac {1}{2}}{\rm {Riesz}}(4\pi ^{2}x)}$ we may define it in terms of the coefficients of the Laurent series development of the hyperbolic (or equivalently, the ordinary) cotangent around zero. If

then F may be defined as

The values of ζ(2k) approach one for increasing k, and comparing the series for the Riesz function with that for ${\displaystyle x\ \exp(-x)}$ shows that it defines an entire function. Alternatively, F may be defined as

${\displaystyle n^{\overline {k}}}$ denotes the rising factorial power in the notation of D. E. Knuth and the number B_n are the Bernoulli number. The series is one of alternating terms and the function quickly tends to minus infinity for increasingly negative values of x. Positive values of x are more interesting and delicate.

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