SteveC

If you're interested in prime number theory and the Riemann hypothesis, you'll probably be interested in the following website which illustrates Fourier series for several prime counting functions and their first and second-order derivatives. Probably the most interesting result illustrated is that at positive integer values of x the Fourier series for the first-order derivative of a prime counting function always evaluates to 2f times the step size of the prime counting function at the integer x even at the minimum evaluation frequency f=1 (where f is assumed to be a positive integer).

primefourierseries.com

v8archie

Fourier Series are periodic. The picture you have on the front page is not. What are you really doing?

SteveC

Of course you are correct that Fourier series are normally used to represent periodic functions. This is really an infinite set of Fourier series, each with an infinite set of harmonics. Since the range of x values illustrated on the web sites is small, I only needed to evaluate a small number of the infinite set of Fourier series to obtain a reasonable approximation. As the range of x values being illustrated increases, the number of Fourier series being evaluated needs to increase as well to obtain a reasonable approximation.

SteveC

I have now illustrated the evolution of the zeta zeros from the Mellin transform of the Fourier series for the first-order derivative of the second Chebyshev function.
10.1 – Evolution of Zeta Zeros from l'[x] – Fourier Series for Prime Counting Functions