Corpus ID: 235755269

The numerical evaluation of the Riesz function

@inproceedings{Paris2021TheNE,
  title={The numerical evaluation of the Riesz function},
  author={Richard B. Paris},
  year={2021}
}
The behaviour of the generalised Riesz function defined by Sm,p(x) = ∞ ∑ k=0 (−)x k!ζ(mk + p) (m ≥ 1, p ≥ 1) is considered for large positive values of x. A numerical scheme is given to compute this function which enables the visualisation of its asymptotic form. The two cases m = 2, p = 1 and m = p = 2 (introduced respectively by Hardy and Littlewood in 1918 and Riesz in 1915) are examined in detail. It is found on numerical evidence that these functions appear to exhibit the x and x decay… Expand

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SHOWING 1-8 OF 8 REFERENCES
A sequential Riesz-like criterion for the Riemann hypothesis
TLDR
It is proved that the Riemann hypothesis is equivalent to c k ≪ k − 3 / 4 + e for all e > 0 and that c k implies that the zeros of ζ ( s ) are simple. Expand
Asymptotics and Mellin-Barnes Integrals
Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typicallyExpand
On the Riesz and Baez-Duarte criteria for the Riemann Hypothesis
We investigate the relation between the Riesz and the B{\'a}ez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function $R(x)$ appearing in the RieszExpand
The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspectsExpand
Sur l’hypothèse de Riemann
On the growth of perturbations of the exponential series
  • Math. Balkanica 21
  • 2007
A note on the evaluation of the Riesz function
  • Technical Report 04:03, University of Abertay
  • 2004