Chebyshev Functions

The two functions and defined below are known as the Chebyshev functions.

ChebyshevFunctionTheta

The function is defined by

(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).

ChebyshevFunctionPsi

The related function is defined by

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

(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.

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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. 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."