TOPICS
Search

Explicit Formula

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The so-called explicit formula

psi(x)=x-sum_(rho)(x^rho)/rho-ln(2pi)-1/2ln(1-x^(-2))

gives an explicit relation between prime numbers and Riemann zeta function zeros for x>1 and x not a prime or prime power. Here, psi(x) is the summatory Mangoldt function (also known as the second Chebyshev function), and the second sum is over all nontrivial zeros rho of the Riemann zeta function zeta(s), i.e., those in the critical strip so 0<R[rho]<1 (Montgomery 2001).

See also

Chebyshev Functions, Mangoldt Function, Prime Number Theorem, Riemann-von Mangoldt Formula, Riemann Zeta Function

Explore with Wolfram|Alpha

References

Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. http://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.

Referenced on Wolfram|Alpha

Explicit Formula

Cite this as:

Weisstein, Eric W. "Explicit Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExplicitFormula.html

Subject classifications