A proof of the prime number summation formula without assuming the Riemann hypothesis (Q2543210)

lt has lang been known that there is a connection between the prime numbers and the non-trivial zeros of the Riemann zeta function. Riemann himself noted such a connection in 1859, and Landau proved other results of the same kind in 1909. These results have come to be called ``the explicit formulae of the prime number theory''. In 1945, A. P. Guinand proved a general summation formula, analogous to the Poisson summation formula, of which the results of the explicit formulae can be shown as particular cases. This general formula, which we shall refer to as ``The Prime Number Summation Formula'', connects the imaginary parts of the zeros of the Riemann zeta function and the logarithms of the powers of primes as arguments of the functions concerned. In the course of the proof of this summation formula the Riemann hypothesis was assumed to be correct. It is, however, known that certain special cases of this summation formulae correspond to the ``explicit formulae'' which do not require the assumption of the Riemann hypothesis. This suggests that it is possible to give a proof of the prime number summation formula for a class of functions without assuming the Riemann hypothesis. In this paper the method of contour integration is used to achieve this. With appropriate conditions on the functions concerned, the prime number summation formula that we shall prove in this paper can be written as follows: \[ \begin{aligned}\sum_{p,m} \frac{\log p}{p^{(1/2)n}} \Phi(m \log p) & - \int_0^\infty \Phi(u)e^{(1/2)u}\,du - \frac12 \int_0^\infty \left(\frac{1}{u} - \frac{e^{-(3/2)u}}{\sin u}\right) \Phi(u)\,du \\ & = -\sqrt{2\pi}\left\{ \sum_{\Re(\gamma)>0} \psi(\gamma) - \frac {1}{2\pi} \int_0^\infty \psi(u) \log\frac{u}{2\pi}\,du\right\}, \end{aligned}\] where \(\Phi(x) = \sqrt{2/\pi}\int_0^\infty \psi(t) \cos xt\,dt\), \(p\) runs through the prime numbers, \(m\) through the positive integers and \(1/2 + i\gamma\) through the complex zeros of \(\zeta(s)\), the Riemann zeta function.