Logarithmic derivative

The logarithmic derivative of a nonzero differentiable function $f$ is $f^{\prime }/f$. For $\Re s>1$, the Euler product for the Riemann zeta function gives

Here $\psi ={\Gamma }^{\prime }/\Gamma$ is the digamma function, $B=\log 2+\frac{1}{2}\log \pi -1-\gamma /2$, and $\gamma$ is the Euler--Mascheroni constant. The sum runs over Nontrivial zeros of the Riemann zeta function with multiplicities. Each displayed paired summand is $O_s(|\rho {|}^{-2})$, so it converges locally uniformly away from zeros. Differentiate the genus-one Hadamard factorization of the Riemann xi function and use the Gamma function recurrence. Unlike an unpaired reciprocal-zero sum, this expression does not require an unstated summation convention.

If $f$ is holomorphic near the closed radius-$R$ disc, $f(z_0)\ne 0$, and $|f|\le M$, the displayed estimate holds away from zeros for $|z-z_0|\le R/3$, counting zeros with multiplicity. Factoring local zeros isolates their poles; the remaining logarithm is controlled by disc estimates. If all zeros lie to the left of a point $z$ in real part, their real contributions are nonnegative. This is the disc estimate used by the Landau zero-free-region theorem; a version with general radius ratios is Lemma 24.17 in Montgomery and Vaughan.

Uniformly in a fixed bounded-width disk around height $t$,This follows by taking the logarithmic derivative of the Hadamard product for the completed zeta function and estimating the distant zeros.

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 $\tilde{F}(\sigma +it)\ll e^{-|t{|}^{1/2}}$.

If, for some real $u$,then partial summation continues $-{\zeta }^{\prime }(s-iu)/\zeta (s-iu)$ meromorphically to $\Re s>1/2$ with no pole except at $1+iu$. The zeta function has no zero to the right of the critical line, and its functional equation then implies the Riemann hypothesis.