Explicit Formula

The so-called explicit formula

gives an explicit relation between prime numbers and Riemann zeta function zeros for  and  not a prime or prime power. Here,  is the summatory Mangoldt function (also known as the second Chebyshev function), and the second sum is over all nontrivial zeros  of the Riemann zeta function , i.e., those in the critical strip so  (Montgomery 2001).

REFERENCES:

Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. https://www.ams.org/notices/200303/fea-conrey-web.pdf.

Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, p. 104, 1980.

Havil, J. "The von Mangoldt Explicit Formula--And How It Is Used to Prove the Prime Number Theorem." §16.9 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 200-202, 2003.

Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001.