9.2 First Zeta Zero Influence on $\vartheta[x]$, $\psi[x]$, and $M[x]$

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The following four plots illustrate the influence of the first zeta zero (dashed red) on the first harmonics  of the Fourier series for $\vartheta[x]$ (blue), $\psi[x]$ (orange), and $M[x]$ (green) for four ranges of $x$ values including $0-20$, $0-100$, $0-500$, and $0-2500$. The zeta zero reference function (dashed red) is $2\ Re[x^{ZetaZero[1]}]$ and the positive and negative envelope functions (dashed gray) have a magnitude of $2\ x^{\frac{1}{2}}$.


Influence of First Zeta Zero (dashed red) on c[x] (blue),l[x] (orange), and m[x] (green).
Influence of First Zeta Zero (dashed red) on $\vartheta[x]$ (blue), $\psi[x]$ (orange), and $M[x]$ (green).

Influence of First Zeta Zero (dashed red) on c[x] (blue),l[x] (orange), and m[x] (green).
Influence of First Zeta Zero (dashed red) on $\vartheta[x]$ (blue), $\psi[x]$ (orange), and $M[x]$ (green).

Influence of First Zeta Zero (dashed red) on c[x] (blue),l[x] (orange), and m[x] (green).
Influence of First Zeta Zero (dashed red) on $\vartheta[x]$ (blue), $\psi[x]$ (orange), and $M[x]$ (green).

Influence of First Zeta Zero (dashed red) on c[x] (blue),l[x] (orange), and m[x] (green).
Influence of First Zeta Zero (dashed red) on $\vartheta[x]$ (blue), $\psi[x]$ (orange), and $M[x]$ (green).