Von Mangoldt Function - Riesz Mean

Riesz Mean

The Riesz mean of the von Mangoldt function is given by

\sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta \Lambda(n) = - \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)} \frac{\zeta^\prime(s)}{\zeta(s)} \lambda^s ds
= \frac{\lambda}{1+\delta} + \sum_\rho \frac {\Gamma(1+\delta)\Gamma(\rho)}{\Gamma(1+\delta+\rho)} +\sum_n c_n \lambda^{-n}.

Here, and are numbers characterizing the Riesz mean. One must take . The sum over is the sum over the zeroes of the Riemann zeta function, and

can be shown to be a convergent series for .

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