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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

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