Riemann–von Mangoldt formula
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 greater than 2:
- 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.