Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case

Y.-F.S. Pétermann Jean-Luc Rémy 1
1 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : 1. Let $s$ be a real number. We prove that, if $s\ge1/2$, $s\not=1$ and $s$ can be written with $D_s$ bits in base 2, then in order to compute $\zeta(s)$ in any relative precision $P\ge11$, that is, in order to compute a $P-$bit number $\zeta_P(s)$such that $|\zeta_P(s)-\zeta(s)|$ is certified to be smaller than the number $ulp(\zeta_P(s))$ represented by a ``1'' at the $P-$th (and last) significant bit-place of $|\zeta_P(s)|$, it is sufficient to perform all the computations (i.e. additions, subtractions, multiplications, divisions, and computation of $k^s$ for integers $k\ge2$) with an internal precision $$ D= \max\left(D_s,P+\max\left(14, \left\lceil\frac3\logP2\log2+2.71\right\rceil\right)\right) $$ (all this contributing an error less than $ulp(\zeta_P(s)/2$), and then to round to the nearest $P-$bit number. For instance if the wanted precision is $P=1000$ (and if $s$ has no more than 1018 significant bits), then an internal precision $D=1018$ is sufficient. 2. Let $s=\sigma+it$ be a complex non real number. Assume $\sigma\ge1/2$ and $t>0$. First we address the problem of exploiting an error relative to modulus in order to estimate the relative errors of each of the real and imaginary parts of the computed $\zeta(s)^*$. Determining regions of the complex plane where these parts cannot vanish could help.Then we establish an easily computable upper bound for a crucial quantity in the error analysis (for the error relative to modulus), subject to the truth of an open conjecture of Brent on the size of the error committed while computing the Bernoulli numbers; we note that the upper bound one can obtain without this conjecture can become so large that even for certain ``reasonable'' value of $s$ it is of no practical use.
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Y.-F.S. Pétermann, Jean-Luc Rémy. Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case. [Research Report] RR-5852, INRIA. 2006, pp.31. ⟨inria-00070174⟩

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