The numerical evaluation of the Riesz function
Abstract
The behaviour of the generalised Riesz function defined by \[S_{m,p}(x)=\sum_{k=0}^\infty \frac{()^{k1}x^k}{k! \zeta(mk+p)}\qquad (m\geq 1,\ p\geq 1)\] 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 $x^{1/4}$ and $x^{3/4}$ decay, superimposed on an oscillatory structure, required for the truth of the Riemann hypothesis.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.02800
 Bibcode:
 2021arXiv210702800P
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 26A12;
 41A60;
 30B10;
 30E15;
 33E20
 EPrint:
 8 pages, 5 figures