3.10 Summary of Prime Counting Function Relationships

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The relationships between the rational-step prime counting functions $\pi[x]$, $J[x]$, and $K[x]$ are as follows.

$\quad J[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\frac{1}{n}\ \pi[x^{1/n}]\\$

$\quad\pi[x]=\sum_{1}^{\lfloor\log[2, x]\rfloor}\frac{\mu[n]}{n}\ J[x^{1/n}]\\$
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$\quad K[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\ \pi[x^{1/n}]\\$

$\quad\pi[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\mu[n]\ K[x^{1/n}]\\$
$\\$

$\quad K[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\frac{EulerPhi[gsfd[n]]}{gsfd[n]}\ J[x^{1/n}]\\$

$\quad J[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\frac{\mu[gsfd[n]]\ EulerPhi[n]\ gsfd[n]}{n^2}\ K[x^{1/n}]$


The estimates for $\pi[x]$ and $K[x]$ are based on the coefficients above for recovering  these two functions from $J[x]$ and the $Li[x]$ and $\log[2]$ terms of Riemann’s formula for $J[x]$.

$\quad RiemannR[x]=\sum_{n=1}^{\infty}\frac{\mu[n]}{n}\ Li[x^{1/n}]\\$

$\quad K_{est}[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\frac{EulerPhi[gsfd[n]]}{gsfd[n]}\ (Li[x^{1/n}]-\log[2])$


The relationships between the log-step prime counting functions $\vartheta[x]$, $\psi[x]$, and $M[x]$ are as follows.

$\quad\psi[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\vartheta[x^{1/n}]\\$

$\quad\vartheta[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\mu[n]\ \psi[x^{1/n}]\\$
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$\quad M[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}n\ \vartheta[x^{1/n}]\\$

$\quad\vartheta[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\mu[n] n\ M[x^{1/n}]\\$
$\\$

$\quad M[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\frac{n\ EulerPhi[gsfd[n]]}{gsfd[n]}\ \psi[x^{1/n}]\\$

$\quad\psi[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\frac{\mu[gsfd[n]]\ EulerPhi[n]\ gsfd[n]}{n}\ M[x^{1/n}]$


The estimates for $\vartheta[x]$ and $M[x]$ are based on the coefficients above for recovering  these two functions from $\psi[x]$ and the $x$ and $\log[2\pi]$ terms of von Mangoldt’s formula for $\psi[x]$.

$\quad\vartheta_{est}[x]:=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\mu[n]\ x^{1/n}\\$

$\quad M_{est}[x]=\sum_{n=1}^{\lfloor\log[2, x]\rfloor}\frac{n\ EulerPhi[gsfd[n]]}{gsfd[n]}\ (x^{1/n}-\log[2\pi])$


Note the conversion coefficients for $\vartheta[x]\leftrightarrow\psi[x]$, $\vartheta[x]\leftrightarrow M[x]$, and $\psi[x]\leftrightarrow M[x]$ are $n$ times the conversion coefficients for $\pi[x]\leftrightarrow J[x]$, $\pi[x]\leftrightarrow K[x]$, and $J[x]\leftrightarrow K[x]$ respectively.


The log-step prime counting functions $\vartheta[x]$, $\psi[x]$, and $M[x]$ are also related as follows.

$\quad\vartheta[x]=\log[gsfd[e^{M[x]}]]$

$\quad\vartheta[x]=\log[gsfd[e^{\psi[x]}]]$


The rational-step prime counting functions $\pi[x]$ and $K[x]$ are also related to the log-step prime counting functions $\vartheta[x]$, $\psi[x]$, and $M[x]$ as follows.

$\quad\pi[x]=PrimeNu[e^{\vartheta[x]}]$

$\quad\pi[x]=PrimeNu[e^{\psi[x]}]$

$\quad\pi[x]=PrimeNu[e^{M[x]}]$

$\quad K[x]= PrimeOmega[e^{\psi[x]}]$


The relationships between the log-step functions $T[x]$, $\vartheta[x]$, $\psi[x]$, and $M[x]$ and the rational-step functions $S[x]$, $\pi[x]$, $J[x]$, and $K[x]$ can also be expressed in terms of their first-order derivatives as follows.

$\quad T'[x]=\log[x]\ S'[x]$

$\quad\vartheta'[x]=\log[x]\ \pi'[x]\\$

$\quad\psi'[x]=\log[x]\ J'[x]\\$

$\quad M'[x]=\log[x]\ K'[x]\\$


The log-step functions $T[x]$, $\vartheta[x]$, $\psi[x]$, and $M[x]$ can be recovered from the first-order derivatives of the rational-step functions $S[x]$, $\pi[x]$, $J[x]$, and $K[x]$ as follows.

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

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

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

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