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I’ve derived Fourier series for integrals of prime counting functions via three different methods resulting in three formulas for each of the six prime counting functions.

I’ve derived three additional formulas for integrals of each of the three log-step prime counting functions $\vartheta[x]$, $\psi[x]$ and $M[x]$ via the following relationships.

$\int_{0}^{x}\vartheta[y]\ dy=\int_{0}^{x}\ (\int_{0}^{y}\log[t]\ \pi'[t]\ dt)\ dy\\$

$\int_{0}^{x}\psi[y]\ dy=\int_{0}^{x}\ (\int_{0}^{y}\log[t]\ J'[t]\ dt)\ dy\\$

$\int_{0}^{x}M[y]\ dy=\int_{0}^{x}\ (\int_{0}^{y}\log[t]\ K'[t]\ dt)\ dy\\$

I investigated derivation of additional formulas for integrals of each of the rational-step prime counting functions $\pi[x]$, $J[x]$ and $K[x]$ via the following relationships, but unfortunately as indicated on page 4 the inner integrals seem intractable.

$\int_{0}^{x}\pi[y]\ dy=\int_{0}^{x}\ (\int_{0}^{y}\frac{\vartheta'[t]}{\log[t]}\ dt)\ dy\\$

$\int_{0}^{x}J[y]\ dy=\int_{0}^{x}\ (\int_{0}^{y}\frac{\psi'[t]}{\log[t]}\ dt)\ dy\\$

$\int_{0}^{x}K[y]\ dy=\int_{0}^{x}\ (\int_{0}^{y}\frac{M'[t]}{\log[t]}\ dt)\ dy\\$

In summary, I’ve derived a total of 27 formulas for integrals of prime counting functions as follows.

$\int_{0}^{x}\pi[y]\ dy$ – 3 formulas

$\int_{0}^{x}J[y]\ dy$ – 3 formulas

$\int_{0}^{x}K[y]\ dy$ – 3 formulas

$\int_{0}^{x}\vartheta[y]\ dy$ – 6 formulas

$\int_{0}^{x}\psi[y]\ dy$ – 6 formulas

$\int_{0}^{x}M[y]\ dy$ – 6 formulas

Since it would be difficult to illustrate all of these formulas each at multiple evaluation frequency limits, the first two plots below focus on the initial formulas derived via method 1. Since $\int_{0}^{x}\psi[y]\ dy$ can be compared to the asymptotic $\frac{x^2}{2}-\log[2\pi]\ x$, the last plot below illustrates the initial formulas for$\int_{0}^{x}\psi[y]\ dy$ from the three methods, and I’m considering adding plots for the three formulas for $\int_{0}^{x}\psi[y]\ dy$ derived from $\int_{0}^{x}\ (\int_{0}^{y}\log[t]\ J'[t]\ dt)\ dy\\$ in the near future.

The following plot illustrates the integrals of the rational-step prime counting functions $\pi[y]$ (blue), $J[y]$ (orange), and $K[y]$ (green) evaluated at $f$=$1$.

The following plot illustrates the integrals of the log-step prime counting functions $\vartheta[y]$ (blue), $\psi[y]$ (orange), and $M[y]$ (green) evaluated at $f$=$1$. The dashed-gray reference curve is $\frac{x^2}{2}-\log[2\pi]\ x$.

The following plot illustrates the method 1 (blue), method 2 (orange), and method 3 (green) formulas for $\int_{a}^{x}\psi[y]\ dy$ evaluated at f=1 minus the reference function $\frac{x^2}{2}-\log[2\pi]\ x$. The method 1 formula uses a lower integration bound of $a=\frac{1}{2}$, whereas the method 2 and method 3 formulas both use a lower integration bound of $a=0$. The reason for this is an attempt to avoid undesirable effects of the offset associated with the method 1 Fourier series for $\psi[y]$ at $y=0$, whereas evaluation parameters were selected for methods 2 and 3 such that $\psi[0]=0$. Note the evaluations of the three formulas derived from the three methods are all fairly similar.