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 Riesz function ： ウィキペディア英語版
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
:$\left(x\right) = -\sum_^\infty \frac.$
If we set $F\left(x\right) = \frac12 \left(4 \pi^2 x\right)$ 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
:$\frac \coth \frac = \sum_^\infty c_n x^n = 1 + \frac x^2 - \fracx^4 + \cdots$
then ''F'' may be defined as
:$F\left(x\right) = \sum_^\infty \frac = 12x - 720x^2 + 15120x^3 - \cdots$
The values of ζ(2k) approach one for increasing k, and comparing the series for the Riesz function with that for $x\ \exp\left(-x\right)$ shows that it defines an entire function. Alternatively, ''F'' may be defined as
:$F\left(x\right) = \sum_^\frac\right\}\right\}$ 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.
==Riesz criterion==
It can be shown that
:$\operatorname\left(x\right) = O\left(x^e\right)\qquad \left(\textx\to\infty\right)$
for any exponent ''e'' larger than 1/2, where this is big O notation; taking values both positive and negative. Riesz showed that the Riemann hypothesis is equivalent to the claim that the above is true for any ''e'' larger than 1/4.〔M. Riesz, «Sur l'hypothèse de Riemann», ''Acta Mathematica'', 40 (1916), pp.185-90.». For English translation look (here )〕 In the same paper, he added a slightly pessimistic note too: «''Je ne sais pas encore decider si cette condition facilitera la vérification de l'hypothèse''».

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