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, describes the distribution of the zeros of the Riemann zeta function.
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The formula states that the number N(T) of zeros of the zeta function with imaginary part greater than 0 and less than or equal to T satisfies
The formula was stated by Riemann in his notable paper "On the Number of Primes Less Than a Given Magnitude" (1859) and was finally proved by Mangoldt in 1905.
Backlund gives an explicit form of the error for all T > 2:
Under the Lindelöf and Riemann hypotheses the error term can be improved to and respectively.[1]
Similarly, for any primitive Dirichlet character χ modulo q, we have
where N(T,χ) denotes the number of zeros of L(s,χ) with imaginary part between -T and T.