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]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/ChebyshevFunctions/Inline9.gif) |
(2)
|
 |  |  |
(3)
|
(Hardy and Wright 1979, p. 340), where 
is the 
th prime, 
is the prime counting function, and 
is the primorial.
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 the Mangoldt
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_|)],](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/ChebyshevFunctions/NumberedEquation3.gif) |
(8)
|
i.e., the logarithm of the least 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, ... (OEIS A003418; 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 the prime counting function 
since they
deal with multiplication of primes instead of the counting of them.
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. Canad 21, 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
in An
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 in Gamma: 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. Sequence A003418/M1590 in "The On-Line Encyclopedia of Integer Sequences."
Weisstein, Eric W. "Chebyshev Functions." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ChebyshevFunctions.html
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