If we use the Laplace transform analysis applied to:(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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

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

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

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# Sum over primes

Can you offer guidance or do you also need help?

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