3.8 $\psi[x]$: Second Chebyshev Function

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$\psi[x]$ – Second Chebyshev Function that takes a step of $\log[p]$ at prime-powers of the form $x=p^n$.

The second Chebyshev function is defined as follows.

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


The following alternate definition takes advantage of the $MangoldtLambda[i]$ function provided by the Wolfram Language.

$\quad\psi[x]=\sum_{i=1}^{\lfloor x\rfloor}MangoldtLambda[i]$


The following plot shows $\psi[x]$ (blue) and the linear function $x$ (orange) which serves as an estimate for $\psi[x]$.

Second Chebyshev Function (l[x])
Second Chebyshev Function $\psi[x]$

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

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


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

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


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


The following $\vartheta_{est}[x]$ function provides an estimate of the $\vartheta[x]$ function which was illustrated on page 3.4.

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


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

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


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

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


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

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


Riemann’s prime-power counting function $J[x]$ can be recovered from the second Chebyshev function $\psi[x]$ by recovering $\pi[x]$ from $\psi[x]$ as illustrated above, and then recovering $J[x]$ from $\pi[x]$ as illustrated on page 3.2.


The $\psi[x]$ and $J[x]$ functions are related via their first-order derivatives as follows. Note the estimate for $\psi[x]$ is $x$, leading to an estimate for $\psi'[x]$ of $1$, an estimate for $J'[x]$ of $1/\log[x]$, and an estimate for $J[x]$ of $Li[x]$.

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


The second Chebyshev function $\psi[x]$ can be recovered from the first-order derivative $J'[x]$ of Riemann’s prime-power counting function as follows.

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


The following formula for $\psi[x]$ is equivalent to the formula defined by von Mangoldt. The linear $x$ term serves as an estimate as was illustrated in the plot above.

$\quad\psi[x]=x-Re\left[\sum_{i=1}^{\infty}\frac{x^{ZetaZero[i]}}{ZetaZero[i]}+\frac{x^{ZetaZero[-i]}}{ZetaZero[-i]}\right]-\log[2\pi]+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2 n}$


The following plot illustrates von Mangoldt’s formula (orange) approximates $\psi[x]$ (blue) when the two sums in the formula are both carried out to a limit of 100. The linear function $x$ (green) is includes as a reference.

Recovery of l[x] using the lFormula Function
Recovery of $\psi[x]$ using von Mangoldt’s Formula