Georgi Guninski on Wed, 01 Feb 2023 08:52:26 +0100

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Numerical instability of j_1(x)=sin(Pi*x)^2+sin(Pi*(gamma(x)+1)/x)^2

The function:


has real zeros at the primes for x>1.
This is essentially Wilson's theorem over the reals.
For some properties of j_1 check the note [1].
For integer x, j_1(x)=0 if x is prime and j_1(x)=sin(Pi/x)^2

? j_1(40.0+20.*I)
fails in pari, but I don't care about this.

The following plot looks like a mess to me:

? ploth(x=20-1/100,20+1/100,j_1(x))


1. Is the messy plot correctly computed or is it a bug?
2. Can we get rid of the complex zeros with Re(x)>1
to remain only the primes as zeros?
Outline of attacks is at [2] on mathoverflow.

mpmath also gives messy plots but appears to compute it
for larger argument.

Note: simple real function with zeros greater than one the primes