3.7 $\vartheta[x]$: First Chebyshev Function

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$\vartheta[x]$ – First Chebyshev Function that takes a step of $\log[p]$ at each prime integer $x=p$.

The first Chebyshev function is defined as follows.

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


The following two plots illustrates the $\vartheta[x]$ function (blue) grows slower than the linear function $x$ (orange). The $\vartheta_{est}[x]$ function (defined on page 3.5) is also included as a reference (green).

First Chebyshev Function (c[x]).
First Chebyshev Function $\vartheta[x]$.
First Chebyshev Function (c[x]).
First Chebyshev Function $\vartheta[x]$.


The base prime counting function $\pi[x]$ can be recovered from the first Chebyshev function $\vartheta[x]$ as follows.

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


The prime counting functions $J[x]$ and $K[x]$ can be recovered from $\vartheta[x]$ by recovering $\pi[x]$ from $\vartheta[x]$ 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 $\vartheta[x]$ and $\pi[x]$ functions are related via their first-order derivatives as follows.

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


The first Chebyshev function $\vartheta[x]$ can be recovered from the first-order derivative $\pi'[x]$ of the base prime counting function as follows.

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