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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Bateman’s constants reconsidered and the distribution of cubic residues

Authors: Daniel Shanks and Mohan Lal
Journal: Math. Comp. 26 (1972), 265-285
MSC: Primary 10H35
MathSciNet review: 0302590
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Abstract: We analyze the computation of certain slowly convergent infinite products involving cubic characters. A first-order analysis gives a $2{\text {D}}$ or $3{\text {D}}$ answer immediately, but extensive computation of cubic residues only improves this to ${\text {5D}}$ or ${\text {6D}}$. To do better, one must examine the distribution of cubic residues or evaluate certain Dedekind Zeta functions. Both are done. The constants thus obtained are used to examine a variant of the Hardy-Littlewood Conjecture ${\text {K}}$ concerning primes of the form ${n^3} + a$. Some related mathematics needed and developed includes an answer to this: Which $p$, satisfying ${x^3} \equiv a\pmod p$, have two solutions $x$ that differ by $k\pmod p$?

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Keywords: Distribution of cubic residues, distribution of primes, number-theoretic products, Bateman-Horn conjecture, Dedekind Zeta functions
Article copyright: © Copyright 1972 American Mathematical Society