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Riemann-Mangoldt Function

$\displaystyle f(x)$ $\textstyle =$ $\displaystyle \sum_{n>1}^\infty {\pi_0(x^{1/n})\over n}$  
  $\textstyle =$ $\displaystyle \mathop{\rm Li}\nolimits (x)-\sum_{{\rm nontrivial\ }\rho\atop \z...
...\rho)=0}\mathop{\rm ei}(\rho\ln x)-\ln 2+\int_x^\infty {dt\over t(t^2-1)\ln t},$ (1)

where $\zeta(z)$ is the Riemann Zeta Function, $\mathop{\rm Li}\nolimits (x)$ is the Logarithmic Integral and $\mathop{\rm ei}\nolimits (x)$ is the Exponential Integral. The Mangoldt Function is given by
\ln p & if $n=p^m$\ for $(m\geq 1)$\ and $p$\ prime\cr
0 & otherwise\cr}
\end{displaymath} (2)

-{\zeta'(x)\over\zeta(s)} = \sum_{n=1}^\infty {\Lambda(n)\over n^s}
\end{displaymath} (3)

for $\Re[s]>1$.
J(x)=\sum_{n\leq x} {\Lambda(n)\over \ln n}.
\end{displaymath} (4)

The Summatory Riemann-Mangoldt function is defined by

\psi(x)=\sum_{n\leq x} \Lambda(n) = \theta(x)+\theta(x^{1/2})+\ldots.
\end{displaymath} (5)

See also Prime Number Theorem, Riemann Function


Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 364-365, 1991.

© 1996-9 Eric W. Weisstein