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Mathematics of Computation

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Computing an arithmetic constant related to the ring of Gaussian integers

Authors: F. Gramain and M. Weber
Journal: Math. Comp. 44 (1985), 241-250
MSC: Primary 11D99; Secondary 11J99, 11Y60, 30D15
Corrigendum: Math. Comp. 48 (1987), 854.
MathSciNet review: 771043
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Abstract: We compute the analogue for ${\mathbf {Z}}[i]$ of Euler’s constant, that is $\delta = {\lim _{n \to + \infty }}{\delta _n}$, where ${\delta _n} = ({\Sigma _{2 \leqslant k \leqslant n}}1/\pi r_k^2) - \log n$. For this purpose we give an estimate for \[ {r_k} = \min \left \{ {r \geqslant 0;{\text {there exists}}\;z \in {\mathbf {C}}\;{\text {such that card}}({\mathbf {Z}}[i] \cap \bar D(z,r)) \geqslant k} \right \},\] and we compute a great number of values of ${r_k}$.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1985 American Mathematical Society