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

I’ve derived three additional formulas for each of the four log-step counting functions $T[x]$, $\vartheta[x]$, $\psi[x]$ and $M[x]$ via the following relationships.

$T[x]=\int_{0}^{x}\log[t]\ S'[t]\ dt\\$

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

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

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

I investigated derivation of additional formulas for each of the rational-step counting functions $S[x]$, $\pi[x]$, $J[x]$ and $K[x]$ via the following relationships, but unfortunately these integrals can’t be fully evaluated. However, partial evaluation of these integrals has led to new asymptotics for $\pi[x]$ and $K[x]$.

$S[x]=\int_{0}^{x}\frac{T'[t]}{\log[t]}dt\\$

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

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

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

I’ve derived four additional formulas for each of the counting functions by applying methods 2 and 3 to relationships such as the following.

$J[x]=\frac{1}{2\pi i}\int_{a-\infty \ i}^{a+\infty \ i}\frac{\log \zeta[s]}{s}\ x^s\ ds\\$

$\psi[x]=\frac{1}{2\pi i}\int_{a-\infty \ i}^{a+\infty \ i}\frac{-\zeta'[s]}{s\ \zeta[s]}\ x^s\ ds$

$J[x]=-\frac{1}{2\pi i}\frac{1}{\log x}\int_{a-\infty \ i}^{a+\infty \ i}\frac{d\ \left(\frac{\log \zeta[s]}{s}\right)}{ds}x^s\ ds\\$

$\psi[x]=-\frac{1}{2\pi i}\frac{1}{\log x}\int_{a-\infty \ i}^{a+\infty \ i}\frac{d\ \left(\frac{-\zeta'[s]}{s\ \zeta[s]}\right)}{ds}x^s\ ds$

In summary, I’ve derived a total of 68 formulas for counting functions as follows.

$S[x]$ – 7 formulas

$\pi[x]$ – 7 formulas

$J[x]$ – 7 formulas

$K[x]$ – 7 formulas

$T[x]$ – 10 formulas

$\vartheta[x]$ – 10 formulas

$\psi[x]$ – 10 formulas

$M[x]$ – 10 formulas

Since it would be difficult to illustrate all of these formulas each at multiple evaluation frequency limits, pages 4.2-4.9 below focus on the initial formulas derived via method 1. Each of the pages 4.2-4.9 below illustrates three plots of the Fourier series for the associated counting function for the three evaluations frequencies $f={1, 2, 3}$. The Fourier series for the counting functions are illustrated in blue, and their corresponding estimate functions are illustrated in gray.

Page 4.1 below illustrates slightly different plots for $U[x]$ which are explained on that page.

Page 4.10 below illustrates 3 formulas for $J[x]$ and new asymptotics for $\pi[x]$ and $K[x]$. Page 4.11 below illustrates 6 formulas for $\psi[x]$. All plots on pages 4.10 and 4.11 use an evaluation frequency limit $f=4$.