|Aurel Page on Wed, 01 Feb 2023 09:23:31 +0100|
[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]
|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) 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  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