f(p) = 0

[tommy1729]

since this does not relate to tetration , ill post it here.

it is easy to construct functions f(x) such that f(x) is analytic for REAL x > 1 and f(p) = 0 IFF p is a prime.

something looking similar to sin(pi x)^2 + sin( gamma(x)/x pi )^2 for instance.

however , it seems harder to find an entire function f(z) that satisfies f(z) = 0 IFF COMPLEX z = prime.

i heard about zeta functions and L-series , but i was thinking about simpler solutions.

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[tommy1729]

\( \sin(\pi x)^2 + \sin( (\Gamma(x+1) (1 - ((x/2) (\Psi(x/2 + 1/2) - \Psi(x/2)) - 1/2) \sin(\pi x) / (\pi x)))/(x+1) \pi )^2 \)

is nicer ; it satisfies f(p) = 0 and is entire.

but i dont know where are the other zero's ...

regards

tommy1729

Mod note: If you use tex why not use it to make it more readable? I modified your formula accordingly.