Riemann–von Mangoldt formula

The Riemann–von Mangoldt formula counts nontrivial zeros of the Riemann zeta function up to height T. Its leading term is \(\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}\), with an error term of order \(\log T\).

Riemann zeta function: The meromorphic function defined initially by \(\zeta(s)=\sum_{n=1}^{\infty}n^{-s}\), with analytic continuation to the complex plane. Its nontrivial zeros are the objects counted by the formula.
Nontrivial zeros of the Riemann zeta function: Zeros of the analytically continued Riemann zeta function lying in the critical strip \(0<\operatorname{Re}(s)<1\). They are precisely the zeros included in the count.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part \(\tfrac12\). The formula counts zeros in the whole critical strip, while the hypothesis places them all on its center line.
Bernhard Riemann: A German mathematician whose work founded major parts of modern geometry and the analytic theory of prime numbers. His 1859 paper introduced the zeta-function framework behind the zero count.
Argument principle: A theorem relating the change in argument of a holomorphic function around a contour to its zeros and poles inside. A contour integral of the zeta function’s logarithmic derivative yields the zero count.
Critical strip: The region \(0<\operatorname{Re}(s)<1\) in which the nontrivial zeros of the Riemann zeta function lie. It locates the zeros whose imaginary parts the formula counts.
Hardy’s theorem: The theorem that infinitely many nontrivial zeros of the Riemann zeta function lie on the critical line. Its lower bound for critical-line zeros is compared with the total count supplied here.
On the Number of Primes Less Than a Given Magnitude: Bernhard Riemann’s 1859 paper connecting the zeros of the zeta function with the distribution of prime numbers. It established the historical setting for counting zeta zeros.
Riemann–Siegel theta function: A real-valued function governing the phase of the zeta function on the critical line. Its growth supplies the leading terms in the count of critical-line zeros.
Critical line: The vertical line \(\operatorname{Re}(s)=\tfrac12\) in the complex plane. The zero-counting formula applies to all nontrivial zeros, not only those on this line.

Linked from 1 page

Montgomery pair correlation conjectureRelated: It estimates the local mean density needed to normalize zero spacings.