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

[Aurel Page <aurel.page@normalesup.org>]

Dear Georgi,

Look at the values of (gamma(x)+1)/x at the boundaries of your interval 
and their difference, and you will realise that your function oscillates 
more than 10^14 times in this interval.

Best,
Aurel

On 01/02/2023 08:50, Georgi Guninski wrote:
> The function:
>
> j_1(x)=sin(Pi*x)^2+sin(Pi*(gamma(x)+1)/x)^2
>
> 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
> otherwise.
>
> ? 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))
>
> Questions:
>
> 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.
>
> [1]:
> Note: simple real function with zeros greater than one the primes
> https://www.researchgate.net/publication/367520640_Note_simple_real_function_with_zeros_greater_than_one_the_primes
>
> [2]:
> https://mathoverflow.net/questions/439688/getting-rid-of-complex-zeros-of-function-with-zeros-the-primes