Mangoldt Function

The function defined by

$$\Lambda(n)\equiv\cases{ \ln p & if $n=p^k$\ for $p$\ a prime\cr 0 & otherwise.\cr}$$ (1)

$\mathop{\rm exp}\nolimits (\Lambda(n))$ is also given by [1, 2, ..., $n$]/[1, 2, ..., $n-1$], where $[a,b,c,\dots]$ denotes the Least Common Multiple. The first few values of $\mathop{\rm exp}\nolimits (\Lambda(n))$ for $n=1$, 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 $\zeta(z)$ by

$$-{\zeta'(s)\over\zeta(s)} = \sum_{n=1}^\infty {\Lambda(n)\over n^s},$$ (2)

where $\Re[s]>1$.

The Summatory Mangoldt function, illustrated above, is defined by

$$\psi(x)\equiv \sum_{n\leq x} \Lambda(n),$$ (3)

where $\Lambda(n)$ is the Mangoldt Function. This has the explicit formula

$$\psi(x)=x-\sum_\rho {x^\rho\over\rho}-\ln(2\pi)-{\textstyle{1\over 2}}\ln(1-x^2),$$ (4)

where the second Sum is over all complex zeros $\rho$ of the Riemann Zeta Function $\zeta(s)$ and interpreted as

$$\lim_{t\to\infty}\sum_{\vert\Im(\rho)\vert<t} {x^\rho\over\rho}.$$ (5)

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

$$\ln([x]!)=\psi(x)+\psi({\textstyle{1\over 2}}x)+\psi({\textstyle{1\over 3}}x)+\ldots,$$ (6)

where $[x]$ is the Nint function and $n!$ is a Factorial.

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

$$\psi(x)=x+{\mathcal O}(xe^{-a\sqrt{\ln x}})$$ (7)

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

$$\psi(x)=x+{\mathcal O}(\sqrt{x}\,(\ln x)^2)$$ (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.