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 or answer immediately, but extensive computation of cubic residues only improves this to or . 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 concerning primes of the form . Some related mathematics needed and developed includes an answer to this: Which , satisfying , have two solutions that differ by ?

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Additional Information

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

Keywords:
Distribution of cubic residues,
distribution of primes,
number-theoretic products,
Bateman-Horn conjecture,
Dedekind Zeta functions

Article copyright:
© Copyright 1972
American Mathematical Society