# Sum over primes

1. May 21, 2007

### tpm

If we use the Laplace transform analysis applied to:

$$\sum_{p} exp(-sp) = \int_{0}^{\infty}dt \frac{d\pi (t)}{dt}e^{-st}$$

then using the properties of Laplace transform (i have consulted to "MATHEMATICAL HANDBOOK OF FORMULA AND TABLES" by Spiegel & Avellanas) we find the relations:

$$s^{-n-1} \sum_{p} exp(-p/s) \rightarrow t^{n/2} \sum_{p}p^{-n/2}J_{n} (2\sqrt{tp}$$

$$s^{-1} \sum_{p} exp(-s^{1/2} p) \rightarrow (t \pi )^{-1/2} \sum_{p} exp(-p^{2}/4t)$$

amazingly the Laplace transform of a sum over primes is just another sum over primes