A modification of Legendre's formula for the prime counting function . It starts with
where
is the floor function, is the number of integers with , and is the number of integers with , and so on.
Identities satisfied by the s include
|
(2)
|
for
and
Meissel's formula is
|
(5)
|
where
Taking the derivation one step further yields Lehmer's
formula.
See also
Legendre's Formula,
Lehmer's
Formula,
Prime Counting Function
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References
Gram, J. "Rapport sur quelques calculs entrepris par M. Bertelsen et concernant les nombres premiers." Acta Math. 17,
301-314, 1893.Hardy, G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, p. 46, 1999.Mathews, G. B. Ch. 10 in Theory
of Numbers. New York: Chelsea, 1961.Meissel, E. D. F.
"Berechnung der Menge von Primzahlen, welche innerhalb der ersten Milliarde
naturlicher Zahlen vorkommen." Math. Ann. 25, 251-257, 1885.Riesel,
H. "Meissel's Formula." Prime
Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser,
pp. 12-13, 1994.Séroul, R. "Meissel's Formula."
§8.7.3 in Programming
for Mathematicians. Berlin: Springer-Verlag, pp. 179-181, 2000.Referenced
on Wolfram|Alpha
Meissel's Formula
Cite this as:
Weisstein, Eric W. "Meissel's Formula."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MeisselsFormula.html
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