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 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 shows that it defines an entire function. Alternatively, F may be defined as

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.