Riemann–von Mangoldt formula

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.

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

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 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 greater than 2:

\left\vert{ N(T) - \left({\frac{T}{2\pi}\log{\frac{T}{2\pi}}-\frac{T}{2\pi} } - \frac{7}{8}\right)}\right\vert < 0.137 \log T + 0.443 \log\log T + 4.350 \ .

References

Edwards, H.M. (1974). Riemann's zeta function. Pure and Applied Mathematics 58. New York-London: Academic Press. ISBN 0-12-232750-0. Zbl 0315.10035.
Ivić, Aleksandar (2013). The theory of Hardy's Z-function. Cambridge Tracts in Mathematics 196. Cambridge: Cambridge University Press. ISBN 978-1-107-02883-8. Zbl pre06093527.
Patterson, S.J. (1988). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics 14. Cambridge: Cambridge University Press. ISBN 0-521-33535-3. Zbl 0641.10029.

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