i was thinking about an integral equation for Riesz function(adsbygoogle = window.adsbygoogle || []).push({});

[tex] e^{-x}-1= \int_{0}^{\infty}\frac{dt}{t}frac(\sqrt (xt)Riesz(1/t) [/tex]

apparently if should work , in fact if i take Mellin transform to both sides it gives me

[tex] \Gamma (s)= \frac{\Gamma(s+1)}{\zeta (-2s)} \frac{\zeta(-2s)}{s} [/tex]

if i make the essay (by professor wolf) for the Riesz function [tex] Riesz(x)= x^{1/4}sin(A- \gamma /2 log(x)) [/tex] , with 'gamma' being the imaginary part of the First non-trivial zero for the Riemman zeta function i get the result

0=1 ¡¡ , and i do not know how to follow.

Euler's formula [tex]2i sin(log(x))=x^{i}+x^{-i} [/tex]

and the fractional part representation (valid for 1>s>0 )

[tex] -\zeta(s)=s\int_{0}^{\infty}dt frac(1/t)t^{s-1} [/tex]

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Integral equation for Riesz function

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Integral equation Riesz | Date |
---|---|

I Finite Integral Domains ... Adkins & Weintraub, Propn 1.5 | Mar 11, 2018 |

I Irreducibles and Primes in Integral Domains ... | Apr 5, 2017 |

Integral equation for Xi-function | Nov 17, 2009 |

Integral equation of second kind and prime number counting function | Jun 21, 2004 |

Cuasi-exact Integral equation for Pi(x)/x 2 | Feb 29, 2004 |

**Physics Forums - The Fusion of Science and Community**