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
R i e s z ( x ) = ∑ k = 1 ∞ ( − 1 ) k − 1 x k ( k − 1 ) ! ζ ( 2 k ) = x ∑ n = 1 ∞ μ ( n ) n 2 exp ( − x n 2 ) . {\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 F ( x ) = 1 2 R i e s z ( 4 π 2 x ) {\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
x 2 coth x 2 = ∑ n = 0 ∞ c n x n = 1 + 1 12 x 2 − 1 720 x 4 + ⋯ {\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 F {\displaystyle F} may be defined as
F ( x ) = ∑ k = 1 ∞ x k c 2 k ( k − 1 ) ! = 12 x − 720 x 2 + 15120 x 3 − ⋯ {\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {x^{k}}{c_{2k}(k-1)!}}=12x-720x^{2}+15120x^{3}-\cdots }
The values of ζ ( 2 k ) {\displaystyle \zeta (2k)} approach one for increasing k, and comparing the series for the Riesz function with that for x exp ( − x ) {\displaystyle x\exp(-x)} shows that it defines an entire function. Alternatively, F may be defined as
F ( x ) = ∑ k = 1 ∞ k k + 1 ¯ x k B 2 k . {\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {k^{\overline {k+1}}x^{k}}{B_{2k}}}.\ }
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