The Mangoldt function is the function defined by
 |
(1)
|
sometimes also called the lambda function. 
has the explicit representation
 |
(2)
|
where 
denotes the least common multiple. The first
few values of 
for 
,
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 
is the Möbius function (Hardy and Wright
1979, p. 254).
The Mangoldt function is related to the Riemann zeta function 
by
 |
(7)
|
where ![Re[s]>1](/images/equations/MangoldtFunction/Inline19.svg)
(Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).
Its Mangoldt summatory function is denoted 
.
See also
Bombieri's Theorem,
Chebyshev Functions,
Explicit Formula,
Greatest
Prime Factor,
Lambda Function,
Landau's
Formula,
Least Common Multiple,
Least
Prime Factor,
Mangoldt Summatory Function,
Riemann Function
Explore with Wolfram|Alpha
References
Edwards, H. M. "Derivation of von Mangoldt's Formula for 
."
§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.Sloane,
N. J. A. Sequence A014963 in "The
On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Mangoldt Function
Cite this as:
Weisstein, Eric W. "Mangoldt Function."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MangoldtFunction.html
Subject classifications
