Aurel Page on Wed, 01 Feb 2023 09:23:31 +0100

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

```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 .
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)

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  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
https://www.researchgate.net/publication/367520640_Note_simple_real_function_with_zeros_greater_than_one_the_primes

:
https://mathoverflow.net/questions/439688/getting-rid-of-complex-zeros-of-function-with-zeros-the-primes
```
```

```