Riemann-von Mangoldt formula

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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.