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: 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]**P. T. Bateman & R. A. Horn, ``A heuristic asymptotic formula concerning the distribution of prime numbers,''*Math. Comp.*, v. 16, 1962, pp. 363-367. MR**26**#6139. MR**0148632 (26:6139)****[2]**P. T. Bateman & R. A. Horn, ``Primes represented by irreducible polynomials in one variable,''*Proc. Sympos. Pure Math.*, vol. 8, Amer. Math. Soc., Providence, R. I., 1965, pp. 119-132. MR**31**#1234. MR**0176966 (31:1234)****[3]**G. H. Hardy & J. E. Littlewood, ``On the expression of a number as a sum of primes,''*Acta Math.*, v. 44, 1923, pp. 1-70. MR**1555183****[4]**Daniel Shanks, ``Review of [2],'' RMT 112,*Math. Comp.*, v. 19, 1965, pp. 684-686.**[5]**H. Davenport & A. Schinzel, ``A note on certain arithmetical constants,''*Illinois J. Math.*, v. 10, 1966, pp. 181-185. MR**32**#5632. MR**0188193 (32:5632)****[6]**Mohan Lal & Daniel Shanks, ``Distribution of primality in the neighborhood of cubes.'' (To appear.)**[7]**Daniel Shanks & Larry P. Schmid, ``Variations on a theorem of Landau,''*Math. Comp.*, v. 20, 1966, pp. 551-569. MR**35**#1564. MR**0210678 (35:1564)****[8]**Daniel Shanks & John W. Wrench, Jr., ``The calculation of certain Dirichlet series,''*Math. Comp.*, v. 17, 1963, pp. 136-154. MR**28**#3012. MR**0159796 (28:3012)****[9]**Daniel Shanks, ``Lal's constant and generalizations,''*Math. Comp.*, v. 21, 1967, pp. 705-707. MR**36**#6363. MR**0223315 (36:6363)****[10]**P. T. Bateman & R. Stemmler, ``Waring's problem for algebraic number fields and primes of the form ,*Illinois J. Math.*, v. 6, 1962, pp. 142-156. MR**25**#2059. MR**0138616 (25:2059)****[11]**Daniel Shanks, ``Corrigendum to [8],''*Math. Comp.*, v. 22, 1968, p. 246.**[12]**Daniel Shanks, ``Calculation and applications of Epstein zeta functions.'' (To appear.) MR**0409357 (53:13114a)****[13]**Mohan Lai & Daniel Shanks,*Tables of Cubic Residues*. (To appear.)**[14]**Daniel Shanks, ``Quadratic residues and the distribution of primes,''*MTAC*, v. 13, 1959, pp. 272-284. MR**21**#7186. MR**0108470 (21:7186)****[15]**John W. Wrench, Jr., ``Evaluation of Artin's constant and the twin-prime constant,''*Math. Comp.*, v. 15, 1961, pp. 396-398. MR**23**#A1619. MR**0124305 (23:A1619)****[16]**Daniel Shanks, ``On the conjecture of Hardy & Littlewood concerning the number of primes of the form ,''*Math. Comp.*, v. 14, 1960, pp. 321-332. MR**22**#10960. MR**0120203 (22:10960)****[17]**P. Erdös, ``On almost primes,''*Amer. Math. Monthly*, v. 57, 1950, pp. 404-407. MR**12**, 80. MR**0036259 (12:80i)**

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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