3.5 $J[x]$: Riemann’s Prime-Power Counting Function

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$J[x]$ – Riemann’s Prime-Power Counting Function that takes a step of $1\ /\ n$ at prime-powers of the form $x=p^n$.

Riemann’s prime-power counting function is defined as follows.

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

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

Riemann's Prime-Power Counting Function (j[x])
Riemann’s Prime-Power Counting Function $J[x]$

Riemann defined $J[x]$ in terms of $\pi[x]$ as follows.

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

Riemann defined the following Moebius inversion formula to recover $\pi[x]$ from $J[x]$.

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

The $Li[x]$ function is the modern estimate for $J[x]$, and is used by the $RiemannR[x]$ function to create an estimate for $\pi[x]$.

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

Riemann defined a formula for recovering $J[x]$ from the zeta zeros equivalent to the following. The $Li[x]$ term serves as an estimate as was illustrated in the plot above.

$\quad J[x]=Li[x]-Re\left[\sum_{i=1}^{\infty}Ei[\log[x]ZetaZero[i]]+Ei[\log[x]ZetaZero[-i]]\right]-\log[2]+\int_{x}^{\infty}\frac{dt}{t\ (t^2-1)\log[t]}$

The following plot illustrates approximation of the $J[x]$ function (blue) using the formula above with an evaluation of the first 100 pairs of zeta zeros. The $Li[x]$ function (green) is also shown as a reference. The integral term at the end of the formula above is ignored in this evaluation as it seems to take a very long time to evaluate, is very small at small values of $x$, and grows smaller as $x$ increases.

Recovery of j[x] using the jFormulaEst Function
Recovery of $J[x]$ using Riemann’s Formula