sum over primes (conjecture)

**zetafunction** · Apr26-09, 08:32 AM

mi idea or conjecture is that

$LaTeX Code: \\sum _{p} exp(-sp) = \\sum_{n>0} \\frac{P(-n)}{n!}(-1)^{n} s^{n}$

where P(-n) are the regularized values of the sum

$LaTeX Code: \\sum _{p}p^{-s}= P(s)$ extended to negative (integers) value of 's'

if the sum where over integers instead of just over primes i believe

$LaTeX Code: \\sum _{0\\le n } exp(-sn) = \\sum_{n>0} \\frac{\\zeta(-n)}{n!}(-1)^{n} s^{n}$

i got the inspiration looking at Euler-maclaurin summation formula.