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Chebyshev Functions
The two functions
and
defined below are known as the Chebyshev functions.
The function
is defined by
 |  |  | (1) |
 |  | ![ln[product_(k=1)^(pi(x))p_k]](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fChebyshevFunctions%2fInline9.svg&f=jpg&w=240) | (2) |
 |  |  | (3) |
(Hardy and Wright 1979, p. 340), where
is the
thprime,
is theprime counting function, and
is theprimorial. This function has the limit
 | (4) |
and the asymptotic behavior
 | (5) |
(Bach and Shallit 1996; Hardy 1999, p. 28; Havil 2003, p. 184). The notation
is also commonly used for this function (Hardy 1999, p. 27).
The related function
is defined by
 |  |  | (6) |
 |  |  | (7) |
where
is theMangoldt function (Hardy and Wright 1979, p. 340; Edwards 2001, p. 51). Here, the sum runs over all primes
and positive integers
such that
, and therefore potentially includes some primes multiple times. A simple and beautiful formula for
is given by
![psi(x)=ln[LCM(1,2,3,...,|_x_|)],](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fChebyshevFunctions%2fNumberedEquation3.svg&f=jpg&w=240) | (8) |
i.e., the logarithm of theleast common multiple of the numbers from 1 to
(correcting Havil 2003, p. 184). The values of
for
, 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, ... (OEISA003418; Selmer 1976). For example,
 | (9) |
The function also has asymptotic behavior
 | (10) |
(Hardy 1999, p. 27; Havil 2003, p. 184).
The two functions are related by
 | (11) |
(Havil 2003, p. 184).
Chebyshev showed that
,
, and
(Ingham 1995; Havil 2003, pp. 184-185).
According to Hardy (1999, p. 27), the functions
and
are in some ways more natural than theprime counting function
since they deal with multiplication of primes instead of the counting of them.
See also
Mangoldt Function,Prime Counting Function,Prime Number Theorem,PrimorialExplore with Wolfram|Alpha
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References
Bach, E. and Shallit, J.Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, pp. 206 and 233, 1996.Chebyshev, P. L. "Mémoir sur les nombres premiers."J. math. pures appl.17, 366-390, 1852.Costa Pereira, N. "Estimates for the Chebyshev Function
."Math. Comput.44, 211-221, 1985.Costa Pereira, N. "Corrigendum: Estimates for the Chebyshev Function
."Math. Comput.48, 447, 1987.Costa Pereira, N. "Elementary Estimates for the Chebyshev Function
and for the Möbius Function
."Acta Arith.52, 307-337, 1989.Dusart, P. "Inégalités explicites pour
,
,
et les nombres premiers."C. R. Math. Rep. Acad. Sci. Canad21, 53-59, 1999.Edwards, H. M.Riemann's Zeta Function. New York: Dover, 2001.Hardy, G. H.Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 27, 1999.Hardy, G. H. and Wright, E. M. "The Functions
and
" and "Proof that
and
are of Order
." §22.1-22.2 inAn Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 340-342, 1979.Havil, J. "Enter Chebyshev with Some Good Ideas." §15.11 inGamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 183-186, 2003.Ingham, A. E.The Distribution of Prime Numbers. Cambridge, England: Cambridge University Press, 1995.Nagell, T.Introduction to Number Theory. New York: Wiley, p. 60, 1951.Panaitopol, L. "Several Approximations of
."Math. Ineq. Appl.2, 317-324, 1999.Robin, G. "Estimation de la foction de Tchebychef
sur le
ième nombre premier er grandes valeurs de la fonctions
, nombre de diviseurs premiers de
."Acta Arith.42, 367-389, 1983.Rosser, J. B. and Schoenfeld, L. "Sharper Bounds for Chebyshev Functions
and
."Math. Comput.29, 243-269, 1975.Schoenfeld, L. "Sharper Bounds for Chebyshev Functions
and
, II."Math. Comput.30, 337-360, 1976.Selmer, E. S. "On the Number of Prime Divisors of a Binomial Coefficient."Math. Scand.39, 271-281, 1976.Sloane, N. J. A. SequenceA003418/M1590 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Chebyshev FunctionsCite this as:
Weisstein, Eric W. "Chebyshev Functions."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/ChebyshevFunctions.html