Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T15:27:04.202Z Has data issue: false hasContentIssue false

On the average number of divisors of quadratic polynomials

Published online by Cambridge University Press:  24 October 2008

James Mckee
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB2 1SB

Extract

Let d(n) denote the number of positive divisors of n, and let f(x) be a polynomial in x with integer coefficients, irreducible over ℤ. Erdös[3] showed that there exist constants λ1, λ2 (depending on f) such that

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bellman, R.. Ramanujan sums and the average value of arithmetic functions. Duke Math. J. 17 (1950), 159168.CrossRefGoogle Scholar
[2]Davenport, H.. The Higher Arithmetic, 6th edition (Cambridge University Press, 1992).Google Scholar
[3]Erdös, P.. The sum σ d{f(k)}. J. Lond.Math. Soc. 27 (1952), 715.CrossRefGoogle Scholar
[4]Hooley, C.. On the representation of a number as the sum of a square and a product. Math. Z. 69 (1958), 211227.CrossRefGoogle Scholar
[5]Hooley, C.. On the number of divisors of quadratic polynomials. Acta Mathematica 110 (1963), 97114.CrossRefGoogle Scholar
[6]Hooley, C.. Applications of sieve methods to the theory of numbers. Cambridge Tracts in Mathematics, 70 (Cambridge University Press, 1976).Google Scholar
[7]Scourfield, E. J.. The divisors of a quadratic polynomial. Proc. Glasgow Math. Soc. 5 (1961), 820.CrossRefGoogle Scholar