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

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

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 > 2:

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

Under the Lindelöf and Riemann hypotheses the error term can be improved to ${\displaystyle o(\log {T})}$ and ${\displaystyle O(\log {T}/\log {\log {T}})}$ respectively.^[1]

Similarly, for any primitive Dirichlet character χ modulo q, we have

${\displaystyle N(T,\chi )={\frac {T}{\pi }}\log {\frac {qT}{2\pi e}}+O(\log {qT}),}$

where N(T,χ) denotes the number of zeros of L(s,χ) with imaginary part between -T and T.

Notes

1. Titchmarsh (1986), Theorems 13.6(A) and 14.13.

References

• Edwards, H.M. (1974). Riemann's zeta function. Pure and Applied Mathematics. Vol. 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. Vol. 196. Cambridge: Cambridge University Press. ISBN 978-1-107-02883-8. Zbl 1269.11075.
• Patterson, S.J. (1988). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics. Vol. 14. Cambridge: Cambridge University Press. ISBN 0-521-33535-3. Zbl 0641.10029.
• Titchmarsh, Edward Charles (1986), The theory of the Riemann zeta-function (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853369-6, MR 0882550