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

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

If we set ${\displaystyle F(x)={\frac {1}{2}}{\rm {Riesz}}(4\pi ^{2}x)}$ 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

${\displaystyle {\frac {x}{2}}\coth {\frac {x}{2}}=\sum _{n=0}^{\infty }c_{n}x^{n}=1+{\frac {1}{12}}x^{2}-{\frac {1}{720}}x^{4}+\cdots }$

then ${\displaystyle F}$ may be defined as

${\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {x^{k}}{c_{2k}(k-1)!}}=12x-720x^{2}+15120x^{3}-\cdots }$

The values of ${\displaystyle \zeta (2k)}$ 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

${\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {k^{\overline {k+1}}x^{k}}{B_{2k}}}.\ }$

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