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Some Calculations Related to Riemann's Prime Number Formula

Hans Riesel and Gunnar Göhl
Mathematics of Computation
Vol. 24, No. 112 (Oct., 1970), pp. 969-983
DOI: 10.2307/2004630
Stable URL: http://www.jstor.org/stable/2004630
Page Count: 15
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Some Calculations Related to Riemann's Prime Number Formula
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Abstract

The objective of this paper is to study the relation of the complex zeros of the Riemann zeta function to the distribution of prime numbers. This relation arises from a formula of Riemann, which is studied here by extensive machine calculations. To establish the validity of the computations, reasonable upper bounds for the various errors involved are deduced. The analysis makes use of a formula, (32), which seems to be quite new. Only the first 29 pairs of complex zeros $\rho = \frac{1}{2} \pm i\alpha(\alpha < 100)$, and the primes in the interval $x < 10^6$ are considered. It turns out that these zeros of ξ(s) lead to an approximation of π(x), the number of primes ≤ x, that gives the integer part correctly up to about x = 1000.

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