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Chebyshev Functions


The two functions theta(x) and psi(x) defined below are known as the Chebyshev functions.

ChebyshevFunctionTheta

The function theta(x) is defined by

theta(x)=sum_(k=1)^(pi(x))lnp_k
(1)
=ln[product_(k=1)^(pi(x))p_k]
(2)
=lnx#
(3)

(Hardy and Wright 1979, p. 340), where p_k is the kth prime, pi(x) is the prime counting function, and x# is the primorial. This function has the limit

 lim_(x->infty)x/(theta(x))=1
(4)

and the asymptotic behavior

 theta(n)∼n
(5)

(Bach and Shallit 1996; Hardy 1999, p. 28; Havil 2003, p. 184). The notation theta(n) is also commonly used for this function (Hardy 1999, p. 27).

ChebyshevFunctionPsi

The related function psi(x) is defined by

psi(x)=sum_(p^nu<=x)lnp
(6)
=sum_(n<=x)Lambda(n),
(7)

where Lambda(n) is the Mangoldt function (Hardy and Wright 1979, p. 340; Edwards 2001, p. 51). Here, the sum runs over all primes p and positive integers nu such that p^nu<=x, and therefore potentially includes some primes multiple times. A simple and beautiful formula for psi(x) is given by

 psi(x)=ln[LCM(1,2,3,...,|_x_|)],
(8)

i.e., the logarithm of the least common multiple of the numbers from 1 to n (correcting Havil 2003, p. 184). The values of LCM(1,2,...,n) for n=1, 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, ... (OEIS A003418; Selmer 1976). For example,

 psi(10)=ln2520=3ln2+2ln3+ln5+ln7.
(9)

The function also has asymptotic behavior

 psi(x)∼x
(10)

(Hardy 1999, p. 27; Havil 2003, p. 184).

The two functions are related by

 psi(x)=sum_(k=1)^(|_log_2x_|)theta(x^(1/k))
(11)

(Havil 2003, p. 184).

Chebyshev showed that pi(x)/(x/lnx), theta(x)/x, and psi(x)/x∼1 (Ingham 1995; Havil 2003, pp. 184-185).

According to Hardy (1999, p. 27), the functions theta(n) and psi(n) are in some ways more natural than the prime counting function pi(x) since they deal with multiplication of primes instead of the counting of them.


See also

Mangoldt Function, Prime Counting Function, Prime Number Theorem, Primorial

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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 psi(x)-theta(x)." Math. Comput. 44, 211-221, 1985.Costa Pereira, N. "Corrigendum: Estimates for the Chebyshev Function psi(x)-theta(x)." Math. Comput. 48, 447, 1987.Costa Pereira, N. "Elementary Estimates for the Chebyshev Function psi(x) and for the Möbius Function M(x)." Acta Arith. 52, 307-337, 1989.Dusart, P. "Inégalités explicites pour psi(X), theta(X), pi(X) 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 theta(x) and psi(x)" and "Proof that theta(x) and psi(x) are of Order x." §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 pi(x)." Math. Ineq. Appl. 2, 317-324, 1999.Robin, G. "Estimation de la foction de Tchebychef theta sur le kième nombre premier er grandes valeurs de la fonctions omega(n), nombre de diviseurs premiers de n." Acta Arith. 42, 367-389, 1983.Rosser, J. B. and Schoenfeld, L. "Sharper Bounds for Chebyshev Functions theta(x) and psi(x)." Math. Comput. 29, 243-269, 1975.Schoenfeld, L. "Sharper Bounds for Chebyshev Functions theta(x) and psi(x), 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."

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Chebyshev Functions

Cite this as:

Weisstein, Eric W. "Chebyshev Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChebyshevFunctions.html

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