Numerical evaluation of the Riemann Zeta-function
نویسندگان
چکیده
We expose techniques that permit to approximate the Riemann Zeta function in different context. The motivation does not restrict only to the numerical verification of the Riemann hypothesis (see Numerical computations about the zeros of the zeta function) : in [5] for example, Lagarias and Odlyzko obtained the best asymptotic known method to compute π(x), the number of primes less than x, thanks to massive evaluation of the zeta-function; Odlyzko and Te Riele in [7] used approximations of the first 2000 critical zeros of ζ(s) to disprove to Mertens conjecture; finally, computations of ζ(n) at integer values of n also consistute a motivation, together with series expressed in terms of those values. (More details can be found on the motivation of approximating ζ(s) in [2].) The first section is dedicated to numerical evaluations at a given point, the next section deals with multi-evaluation.
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