4.10 Illustration of Three Formulas for $J[x]$ and New Asymptotics for $\pi[x]$ and $K[x]$

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I’ve defined three methods for derivation of Fourier series for prime number counting functions, and this page illustrates the formulas derived for $J[x]$ via each of these three methods.

I’ve investigated derivation of additional formulas for the rational-step prime counting functions $\pi[x]$, $J[x]$, and $K[x]$ from the following relationships. Unfortunately, these integrals can’t be fully evaluated, but partial evaluation of these integrals has led to new asymptotics for $\pi[x]$ and $K[x]$ which are also illustrated on this page.

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

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

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

(Plots coming soon.)