Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-04T17:33:46.676Z Has data issue: false hasContentIssue false
  • English
  • Français

On the location of the roots of polynomial congruences

Published online by Cambridge University Press:  18 May 2009

C. Hooley
C. Hooley
Affiliation:
School of Mathematics, University of Wales College of Cardiff, Cardiff, Great Britain.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We have indicated in our tract [9] that several interesting problems in the theory of numbers are related to results about the evenness of the distribution of the roots v of a polynomial congruence

where f(x) = a0xn + … + an is an irreducible polynomial having integral coefficients and degree n≧2. We alluded, for example, to our work on the Chebyshev problem of the greatest prime factor of n2 – D [8], in which an essential component was our earlier demonstration [6] of the uniform distribution, modulo 1, of v/k when f(x) = x2 – D. But, having pointed out that the quantitative descriptions of such uniformity had to be very sharp for substantial applications, we then noted with regret that little more than mere uniform distribution was obtained in our generalization [7] of [6] to congruences of higher degree. Indeed, it has only been for certain cubic polynomials that results have been produced that are comparable in power with those for quadratic polynomials, and even these depend on the assumption of the unproved hypothesis R* regarding the size of incomplete Kloosterman sums [10].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 3 , September 1990 , pp. 309 - 316
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Danicic, I., An extension of a theorem of Heilbronn, Mathematika, 5 (1958), 30–37.CrossRefGoogle Scholar
2.Deshouillers, J. M. and Iwaniec, H., On the greatest prime factor of n 2 + 1, Ann. Inst. Fourier (Grenoble) 32 no 4 (1982), 1–11.CrossRefGoogle Scholar
3.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, third edition (Oxford, 1954).Google Scholar
4.Harman, G., Trigonometric sums over primes II, Glasgow Math. J. 24 (1983), 23–37.CrossRefGoogle Scholar
5.Heilbronn, H. A., On the distribution of the sequence θ2n (mod 1), Quart. J. Math. Oxford (2), 19 (1948), 249–256.CrossRefGoogle Scholar
6.Hooley, C., On the number of divisors of quadratic polynomials, Acta Math., 110 (1963), 97–114.CrossRefGoogle Scholar
7.Hooley, C., On the distribution of the roots of polynomial congruences, Mathematika, 11 (1964), 39–49.CrossRefGoogle Scholar
8.Hooley, C., On the greatest prime factor of a quadratic polynomial, Acta Math., 117 (1967), 281–299.CrossRefGoogle Scholar
9.Hooley, C., Applications of sieve methods to the theory of numbers (Cambridge, 1976).Google Scholar
10.Hooley, C., On the greatest prime factor of a cubic polynomial, J. reine angew. Math. 303/304 (1978), 21–50.Google Scholar
11.Hooley, C., On a problem in Diophantine approximation (to appear).Google Scholar