# Welcome to primefourierseries.com!

The plot in the header above illustrates the Fourier series for the first-order derivative of the first Chebyshev function which takes a step of $\log p$ at each prime the form $x=p$. The plot in the header was generated using the minimum frequency evaluation limit $f=1$, where $f$ is assumed to be a positive integer. The reference function shown in orange in the plot is $2\ f\log x$. The plot in the header illustrates an example of a more general result, which is that for positive integer values of $x$ the Fourier series for the first-order derivative of a prime counting function evaluates to $2\ f$ times the step size of the associated prime counting function at the positive integer $x$.