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

DOI:
https://doi.org/10.1090/S0025-5718-1972-0302590-7

MathSciNet review:
0302590

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Abstract | References | Similar Articles | Additional Information

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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*Tables of Cubic Residues*. (To appear.)

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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