3.9 $M[x]$: Third “Chebyshev-like” Function

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$M[x]$- Third “Chebyshev-like” Function that takes a step of $n\log[p]$ at prime-powers of the form $x=p^n$.

The third “Chebyshev-like” function is defined as follows.

$\quad M[x]=\sum_{i=1}^{\lfloor x\rfloor}If[PrimePowerQ[i],\ \log[i],\ 0]$


The following two plots illustrates the $M[x]$ function (blue) grows faster than the linear function $x$ (orange). The $M_{est}[x]$ function (defined below) is also included as a reference (green).


Third "Chebyshev-like" function $M(x)$.
Third “Chebyshev-like” function $M[x]$.

Third "Chebyshev-like" function $M(x)$.
Third “Chebyshev-like” function $M[x]$.

The following formula recovers $M[x]$ from $\vartheta[x]$.

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


The following formula recovers $\vartheta[x]$ from $M[x]$.

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


Note the conversion coefficients for $\vartheta[x]\leftrightarrow M[x]$ are $n$ times the conversion coefficients for $\pi[x]\leftrightarrow K[x]$.


The following formula also recovers $\vartheta[x]$ from $M[x]$.

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


The following formula recovers $M[x]$ from $\psi[x]$.

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


The following formula recovers $\psi[x]$ from $M[x]$.

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


Note the conversion coefficients for $\psi[x]\leftrightarrow M[x]$ are $n$ times the conversion coefficients for $J[x]\leftrightarrow K[x]$.


The following $M{est}[x]$ function provides an estimate of the $M[x]$ function which was illustrated in the two plots above.

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


The following formula recovers $\pi[x]$ from $M[x]$.

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


The prime counting functions $J[x]$ and $K[x]$ can be recovered from $M[x]$ by recovering $\pi[x]$ from $M[x]$ as illustrated above, and then recovering $J[x]$ and $K[x]$ from $\pi[x]$ as was illustrated on pages 3.2 and 3.3.


The $M[x]$ and $K[x]$ functions are related via their first-order derivatives as follows.

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


The third “Chebyshev-like” function $M[x]$ can be recovered from the first-order derivative $K'[x]$ of the simple prime-power counting function as follows.

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