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Uniform upper bounds for the complex divisor function

Published online by Cambridge University Press:  24 October 2008

R. R. Hall
R. R. Hall
Affiliation:
Department of Mathematics, University of York, York YO1 5DD
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Extract

The function

has had several applications in the analytic theory of numbers. It was used by Ingham [10] to give a new proof that ζ(1 + iθ) ≠ 0. In [1] we introduced the modern notation on the left above and used τ(n, θ) to show that for almost all n the numbers log d are asymptotically uniformly distributed (mod 1). If we define

then it is known (Hall [2], Kátai[10]) that

(where p.p. means ‘on a sequence of asymptotic density 1’). It is still an open problem whether or not we may strike out the term (log 4/π) = 0.24156 & from the exponent in (3).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 421 - 427
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Hall, R. R.. The divisors of integers I. Acta Arith. 26 (1974), 41–46.CrossRefGoogle Scholar
[2]Hall, R. R.. Sums of imaginary powers of the divisors of integers. J. London Math. Soc. (2) 9 (1975), 571–580.CrossRefGoogle Scholar
[3]Hall, R. R.. The propinquity of divisors. J. London Math. Soc. (2) 19 (1979), 35–40.CrossRefGoogle Scholar
[4]Hall, R. R. and Tenenbaum, G.. On the average and normal orders of Hooley's Δr-function. J. London Math. Soc. (2) 25 (1982), 393–406.Google Scholar
[5]Hall, R. R. and Tenenbaum, G.. The average orders of Hooley's Δr-functions. Mathematika 31 (1984), 98–109.CrossRefGoogle Scholar
[6]Hall, R. R. and Tenenbaum, G.. The average orders of Hooley's Δr-functions II. Compositio Math. 60 (1986), 163–186.Google Scholar
[7]Hall, R. R. and Tenenbaum, G.. Divisors. Cambridge Tracts in Math. no. 90 (Cambridge University Press, 1988).CrossRefGoogle Scholar
[8]Hardy, G. H. and Ramanujan, S.. The normal number of prime factors of a number n. Quart. J. Math. Oxford Ser. 48 (1917), 76–92.Google Scholar
[9]Hooley, C.. On a new technique and its applications to the theory of numbers. Proc. London Math. Soc. (3) 38 (1979), 115–151.CrossRefGoogle Scholar
[10]Ingham, A. E.. Notes on Reimann's ζ-function and Dirichlet's L-functions. J. London Math. Soc. 5 (1930), 107–112.CrossRefGoogle Scholar
[11]Kátai, I.. The distribution of divisors (mod 1). Acta Math. Acad. Sci. Hungar. 27 (1976), 149–152.CrossRefGoogle Scholar
[12]Maier, H. and Tenenbaum, G.. On the set of divisors of an integer. Invent. Math. 76 (1984), 121–128.CrossRefGoogle Scholar