Bateman’s constants reconsidered and the distribution of cubic residues
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- by Daniel Shanks and Mohan Lal PDF
- Math. Comp. 26 (1972), 265-285 Request permission
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 $k\pmod p$?References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 265-285
- MSC: Primary 10H35
- DOI: https://doi.org/10.1090/S0025-5718-1972-0302590-7
- MathSciNet review: 0302590