Riesz function

Article summary

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 : {\rm Riesz}(x) = \sum_{k=1}^\infty \frac{(-1)^{k-1}x^k}{(k-1)! \zeta(2k)}=x \sum_{n=1}^\infty \frac{\mu(n)}{n^2} \exp\left(\frac{-x}{n^2}\right). If we set F(x) = \frac12 {\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 : \frac{x}{2} \coth \frac{x}{2} = \sum_{n=0}^\infty c_n x^n = 1 + \frac{1}{12} x^2 - \frac{1}{720}x^4 + \cdots then F may be defined as : F(x) = \sum_{k=1}^\infty \frac{x^k}{c_{2k}(k-1)!} = 12x - 720x^2 + 15120x^3 - \cdots The values of \zeta(2k) approach one for increasing k, and comparing the series for the Riesz function with that for x\exp(-x) shows that it defines an entire function. Alternatively, F may be defined as : F(x) = \sum_{k=1}^{\infty}\frac{k^{\overline{k+1}}x^{k}}{B_{2k}}. \ 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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