Prime Zeta
Calculus
Level
5
Find the sum of the reciprocals of the analytic continuation of the prime zeta function over negative integers. If it's in the form zeta(2 i x)/primezeta(2 i x), then find 4x^2.
The answer is -9.
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1 solution
It can be proven that zeta(x+2)/primezeta(x+2)=infinite sum n=1 to infinity of primezeta(-nx). I will later add a proof of this statement. We want it at x=1, so the sum is zeta(3)/primezeta(3). Therefore 4(-3/2i)^2=-9.