Prime number theorem

The prime number theorem states that $\pi (x)\sim x/\log x$, equivalently ${\sum }_{n\le x}\Lambda (n)\sim x$.

If all nontrivial zeta zeros satisfy $\Re \rho \le 1-c$ with fixed $0<c<1/2$, the Riemann–von Mangoldt formula gives ${\sum }_{|\rho |<T}|\rho {|}^{-1}\ll {\log }^{2}T$. The truncated explicit formula for the second Chebyshev function then bounds the error by $O(x^{1-c}{\log }^{2}T+x{\log }^{2}x/T)$. Choosing $T=x^c$ gives the displayed bound and, for every positive $\epsilon$, $O_{\epsilon }(x^{1-c+\epsilon })$.

// Target: number-theory.bigb

Suppose $A>0$ and every sufficiently high Nontrivial zero of the Riemann zeta function satisfies the displayed gap, with bounded heights also separated from one. The truncated explicit formula for the second Chebyshev function and the local zero count for the Riemann zeta function give errors $x{\log }^{2}x\exp (-c\log x/(\log T{)}^A)$ and $x{\log }^{2}x/T$. Balance them by $\log T\asymp (\log x{)}^{1/(A+1)}$. This explains how the width of a zero-free region of the Riemann zeta function determines the exponential scale in the Prime number theorem error.

The classical zero-free region and a truncated Perron contour givefor some constant $c>0$.

For a smooth compactly supported $F:(0,\infty )\to \mathbb{R}$ whose Mellin transform has suitable vertical decay, Mellin inversion andgiveMoving the contour through the pole at $s=1$ and into the classical zero-free region gives main term $X\tilde{F}(1)$ and error $O(X^{1-c/\log \log X})$ when $\tilde{F}(\sigma +it)\ll e^{-|t{|}^{1/2}}$.