Bateman’s constants reconsidered and the distribution of cubic residues

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 $k\pmod p$?