14 Fourier Bessel Series Expansions of Zeta Zero Terms

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This page illustrates Fourier-Bessel series expansions of the following zeta zero terms related to von Mangoldt’s explicit formula for the second Chebyshev function $\psi(x)$ where $\rho_k=ZetaZero[k]$.

(1) $\quad\psi_{\rho_k}(x)=-\frac{x^{\,\rho_k}}{\rho_k}$

(2) $\quad\psi_{\rho_k}'(x)=-x^{\,\rho_k-1}$


The Fourier-Bessel series expansion of a function $f(x)$ is intended to apply to a finite interval such as $x\in(0,\,1)$, but I’ve used the following reflection formulas to derive Fourier-Bessel series expansions of $\psi_{\rho_k}(x)$ and $\psi_{\rho_k}'(x)$ in the interval $x\in(1,\infty)$ as well as $x\in(0,\,1)$.

(3) $\quad\frac{x^{\,\rho_k}}{\rho_k}=\frac{\left(\frac{1}{x}\right)^{\rho_{-k}-1}}{1-\rho_{-k}}$

(4) $\quad x^{\,\rho_k-1}=\frac{\left(\frac{1}{x}\right)^{\rho_{-k}-1}}{x}$


The following two plots illustrate $\psi_{\rho_1}(x)$ in blue and the Fourier-Bessel series expansion of $\psi_{\rho_1}(x)$ in orange for $0<x<1$ and $1<x<10$ respectively where the expansion is evaluated over the first $200$ terms.

Fourier Bessel Series Representation of $\psi_{\rho_1}(x)$ (orange curve) and associated reference function (blue curve)
Fourier Bessel Series Representation of $\psi_{\rho_1}(x)$ (orange curve) and associated reference function (blue curve)

The following two plots illustrate $\psi_{\rho_1}'(x)$ in blue and the Fourier-Bessel series expansion of $\psi_{\rho_1}'(x)$ in orange for $0<x<1$ and $1<x<10$ respectively where the expansion is evaluated over the first $200$ terms.

Fourier Bessel Series Representation of $\psi_{\rho_1}'(x)$ (orange curve) and associated reference function (blue curve)
Fourier Bessel Series Representation of $\psi_{\rho_1}'(x)$ (orange curve) and associated reference function (blue curve)