Riemann–von Mangoldt formula

In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, states that the number N(T) of zeros of the Riemann zeta function with imaginary part greater than 0 and less than or equal to T satisfies

$N(T)=\frac{T}{2\pi}\log{\frac{T}{2\pi}}-\frac{T}{2\pi}+O(\log{T}).$

The formula was stated by Riemann in his famous paper On the Number of Primes Less Than a Given Magnitude (1859) and proved by von Mangoldt in 1905.