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 ?

**[1]**Paul T. Bateman and Roger A. Horn,*A heuristic asymptotic formula concerning the distribution of prime numbers*, Math. Comp.**16**(1962), 363–367. MR**0148632**, https://doi.org/10.1090/S0025-5718-1962-0148632-7**[2]**Paul T. Bateman and Roger A. Horn,*Primes represented by irreducible polynomials in one variable*, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 119–132. MR**0176966****[3]**G. H. Hardy and J. E. Littlewood,*Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes*, Acta Math.**44**(1923), no. 1, 1–70. MR**1555183**, https://doi.org/10.1007/BF02403921**[4]**Daniel Shanks, ``Review of [2],'' RMT 112,*Math. Comp.*, v. 19, 1965, pp. 684-686.**[5]**H. Davenport and A. Schinzel,*A note on certain arithmetical constants*, Illinois J. Math.**10**(1966), 181–185. MR**0188193****[6]**Mohan Lal & Daniel Shanks, ``Distribution of primality in the neighborhood of cubes.'' (To appear.)**[7]**Daniel Shanks and Larry P. Schmid,*Variations on a theorem of Landau. I*, Math. Comp.**20**(1966), 551–569. MR**0210678**, https://doi.org/10.1090/S0025-5718-1966-0210678-1**[8]**Daniel Shanks and John W. Wrench Jr.,*The calculation of certain Dirichlet series*, Math. Comp.**17**(1963), 136–154. MR**0159796**, https://doi.org/10.1090/S0025-5718-1963-0159796-4**[9]**Daniel Shanks,*Lal’s constant and generalizations*, Math. Comp.**21**(1967), 705–707. MR**0223315**, https://doi.org/10.1090/S0025-5718-1967-0223315-8**[10]**Paul T. Bateman and Rosemarie M. Stemmler,*Waring’s problem for algebraic number fields and primes of the form (𝑝^{𝑟}-1)/(𝑝^{𝑑}-1)*, Illinois J. Math.**6**(1962), 142–156. MR**0138616****[11]**Daniel Shanks, ``Corrigendum to [8],''*Math. Comp.*, v. 22, 1968, p. 246.**[12]**Daniel Shanks,*Calculation and applications of Epstein zeta functions*, Math. Comp.**29**(1975), 271–287. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR**0409357**, https://doi.org/10.1090/S0025-5718-1975-0409357-2

Daniel Shanks,*Corrigenda: “Calculation and applications of Epstein zeta functions” (Math. Comp. 29 (1975), 271–287)*, Math. Comp.**29**(1975), no. 132, 1167. MR**0409358**, https://doi.org/10.1090/S0025-5718-1975-0409358-4

Daniel Shanks,*Corrigendum: “Calculation and applications of Epstein zeta functions” (Math. Comp. 29 (1975), 271–287)*, Math. Comp.**30**(1976), no. 136, 900. MR**0409359**, https://doi.org/10.1090/S0025-5718-1976-0409359-7**[13]**Mohan Lai & Daniel Shanks,*Tables of Cubic Residues*. (To appear.)**[14]**Daniel Shanks,*Quadratic residues and the distribution of primes*, Math. Tables Aids Comput.**13**(1959), 272–284. MR**0108470**, https://doi.org/10.1090/S0025-5718-1959-0108470-8**[15]**John W. Wrench Jr.,*Evaluation of Artin’s constant and the twin-prime constant*, Math. Comp.**15**(1961), 396–398. MR**0124305**, https://doi.org/10.1090/S0025-5718-1961-0124305-0**[16]**Daniel Shanks,*On the conjecture of Hardy & Littlewood concerning the number of primes of the form 𝑛²+𝑎*, Math. Comp.**14**(1960), 320–332. MR**0120203**, https://doi.org/10.1090/S0025-5718-1960-0120203-6**[17]**P. Erdös,*On almost primes*, Amer. Math. Monthly**57**(1950), 404–407. MR**0036259**, https://doi.org/10.2307/2307640

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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