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Computational Strategies for the Riemann Zeta Function
 Journal of Computational and Applied Mathematics
, 2000
"... We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of th ..."
Abstract

Cited by 46 (9 self)
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We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".
On pseudosquares and pseudopowers
, 712
"... Introduced by Kraitchik and Lehmer, an xpseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least xpseudosquare that improves on a bound that is exponential in x due to Schinzel. We ..."
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Cited by 2 (2 self)
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Introduced by Kraitchik and Lehmer, an xpseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least xpseudosquare that improves on a bound that is exponential in x due to Schinzel. We also obtain an equidistribution result for pseudosquares. An xpseudopower to base g is a positive integer which is not a power of g yet is so modulo p for all primes p ≤ x. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most exp(agx/log x) for a suitable constant ag. A bound of exp(agxlog log x/log x) is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRHconditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares.