Landau proved, for any fixed $x>1$, that $$\sum_{0<\gamma\leq T} x^\rho = -\frac{T}{2\pi} \Lambda(x)+ O(\log T) \quad \text{for} \quad T\to \infty,$$ where $\rho$ runs over the non-trivial zeros of the Riemann zeta function $\zeta(s)$ and $\Lambda(x) =\log p$ if $x=p^m,\ p$ prime and $\Lambda(x) =0$ otherwise. Recently Gonek has obtained a form of the previous formula which is uniform
A note on Landau's formula
PERELLI, ALBERTO
2000-01-01
Abstract
Landau proved, for any fixed $x>1$, that $$\sum_{0<\gamma\leq T} x^\rho = -\frac{T}{2\pi} \Lambda(x)+ O(\log T) \quad \text{for} \quad T\to \infty,$$ where $\rho$ runs over the non-trivial zeros of the Riemann zeta function $\zeta(s)$ and $\Lambda(x) =\log p$ if $x=p^m,\ p$ prime and $\Lambda(x) =0$ otherwise. Recently Gonek has obtained a form of the previous formula which is uniformFile in questo prodotto:
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