hello my question (now) is how to evaluate the inverse Laplace transform of the function: (a>0)(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \ln[\zeta(sa)]/s [/tex] (1)

firs from number theory and dirichlet function will be exist a Dirichlet series so we have that:

[tex] \frac{d\zeta(as)}{\zeta(as)}=\sum_{n=1}^{\infty}c(n)n^{-as} [/tex]

So integrating respect to the variable s we have the Dirichlet representation in the form:

[tex] \ln[\zeta(sa)]= \sum_{n=1}^{infty}d(n)e^{-asln(n)} [/tex]

Now using the table of Laplace transforms we will have that the inverse transform of Log of riemann zeta function (1) is:

[tex] \sum_{n=1}^{\infty}d(n)\theta(t-aln(n)) [/tex]

of course i know that Riemann got it in a different way..i would like to know this way... here theta means the Heaviside,s step function and [tex] d(n)=ac(n)/ln(n) [/tex] n=2,3,4,5,6,7,........

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

Dismiss Notice

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!

# Evaluation of inverse transform:

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Evaluation inverse transform | Date |
---|---|

I Inverse of the sum of two matrices | Tuesday at 10:20 AM |

Irreducibility of polynomial (need proof evaluation) | May 21, 2015 |

Polynomials over a ring evaluated at a value? | Apr 30, 2015 |

Evaluation Homomorphism | Feb 20, 2013 |

Evaluate this algebraic expression f(a,b) | Sep 28, 2012 |

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