Mangoldt Function


The Mangoldt function is the function defined by

 Lambda(n)={lnp   if n=p^k for p a prime; 0   otherwise,

sometimes also called the lambda function. exp(Lambda(n)) has the explicit representation


where LCM(a,b,...) denotes the least common multiple. The first few values of exp(Lambda(n)) for n=1, 2, ..., plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, ... (OEIS A014963).

The Mangoldt function is implemented in the Wolfram Language as MangoldtLambda[n].

It satisfies the divisor sums


where mu(n) is the Möbius function (Hardy and Wright 1979, p. 254).

The Mangoldt function is related to the Riemann zeta function zeta(z) by


where R[s]>1 (Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).


The summatory Mangoldt function, illustrated above, is defined by


where Lambda(n) is the Mangoldt function, and is also known as the second Chebyshev function (Edwards 2001, p. 51). psi(x) is given by the so-called explicit formula


for x>1 and x not a prime or prime power (Edwards 2001, pp. 49, 51, and 53), and the sum is over all nontrivial zeros rho of the Riemann zeta function zeta(s), i.e., those in the critical strip so 0<R[rho]<1 (Montgomery 2001), and interpreted as


Vallée Poussin's version of the prime number theorem states that


for some a (Davenport 1980, Vardi 1991). The prime number theorem is equivalent to the statement that


as x->infty (Dusart 1999).

Von Mangoldt proved his formula 30 years after Riemann's paper, which contained a related formula that inspired von Mangoldt's. Von Mangoldt's formula was then used to prove the prime number theorem in the equivalent form


The Riemann hypothesis is equivalent to


(Davenport 1980, p. 114; Vardi 1991).

Vardi (1991, p. 155) also gives the interesting formula


where |_x_| is the floor function and n! is a factorial.

See also

Bombieri's Theorem, Chebyshev Functions, Explicit Formula, Greatest Prime Factor, Landau's Formula, Lambda Function, Least Common Multiple, Least Prime Factor, Riemann Function

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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.Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, p. 104, 1980.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. "Derivation of von Mangoldt's Formula for psi(x)." §3.2 in Riemann's Zeta Function. New York: Dover, pp. 50-54, 2001.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 28, 1999.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 109, 2003.Krantz, S. G. "The Lambda Function" and "Relation of the Zeta Function to the Lambda Function." §13.2.10 and 13.2.11 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 161, 1999.Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001.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.Sloane, N. J. A. Sequence A014963 in "The On-Line Encyclopedia of Integer Sequences."Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 146-147, 152-153, and 249, 1991.

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Mangoldt Function

Cite this as:

Weisstein, Eric W. "Mangoldt Function." From MathWorld--A Wolfram Web Resource.

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