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Riemann-von Mangoldt Formula


RiemannVonMangoldtFormula

In his famous paper of 1859, Riemann stated that the number N(T) of Riemann zeta function zeros sigma+it with 0<t<=T is asymptotically given by

 N(T)=T/(2pi)ln(T/(2pi))-T/(2pi)+O(lnT)
(1)

as T->infty (Edwards 2001, p. 19; Havil 2003, p. 203; Derbyshire 2004, p. 258). This can be written more compactly as

 N(T)=T/(2pi)ln(T/(2pie))+O(lnT).
(2)

This result was proved by von Mangoldt in 1905 and is hence known as the Riemann-von Mangoldt formula.

It follows that the density D(T)=N(T+1)-N(T) of zeros at height T is

 D(T)∼(lnT)/(2pi),
(3)

where, as usual, the asymptotic notation f(n)∼g(n) means that the ratio f(n)/g(n) tends to 1 as n->infty.

Another consequence of this result is that the imaginary parts of consecutive zeta zeros in the upper half-plane 0<t_1<=t_2<=t_3<=... satisfy

 t_n∼(2pin)/(lnn).
(4)

Thus the mean spacing d_n between t_n and t_(n+1) is

 d_n∼(2pi)/(lnn),
(5)

which tends to zero as n->infty.


See also

Landau's Formula, Riemann-Siegel Formula, Riemann Zeta Function, Riemann Zeta Function Zeros

This entry contributed by Jonathan Sondow (author's link)

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References

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, p. 217, 2004.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 138, 2003.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Ivic, A. A. The Riemann Zeta-Function. New York: Wiley, pp. 17-20, 1985.Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859.Reprinted in Das Kontinuum und Andere Monographen (Ed. H. Weyl). New York: Chelsea, 1972.

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Riemann-von Mangoldt Formula

Cite this as:

Sondow, Jonathan. "Riemann-von Mangoldt Formula." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Riemann-vonMangoldtFormula.html

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