## Mangoldt Function

The function defined by

 (1)

is also given by [1, 2, ..., ]/[1, 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, ... (Sloane's A014963). The Mangoldt function is related to the Riemann Zeta Function by
 (2)

where .

The Summatory Mangoldt function, illustrated above, is defined by

 (3)

where is the Mangoldt Function. This has the explicit formula
 (4)

where the second Sum is over all complex zeros of the Riemann Zeta Function and interpreted as
 (5)

Vardi (1991, p. 155) also gives the interesting formula
 (6)

where is the Nint function and is a Factorial.

Vallée Poussin's version of the Prime Number Theorem states that

 (7)

for some (Davenport 1980, Vardi 1991). The Riemann Hypothesis is equivalent to
 (8)

(Davenport 1980, p. 114; Vardi 1991).

See also Bombieri's Theorem, Greatest Prime Factor, Lambda Function, Least Common Multiple, Least Prime Factor, Riemann Function

References

Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, p. 110, 1980.

Sloane, N. J. A. Sequence A014963 in The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 146-147, 152-153, and 249, 1991.