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A Note on Exponential Sums with a Difference

A Note on Exponential Sums with a Difference M. N. HUXLEY This note is an addendum to the paper [2] of Heath-Brown and the author, in which the method of Bombieri, Iwaniec and Mozzochi [1, 6] was applied to the exponential sum in two variables which is used to investigate the mean square over a short interval of an exponential sum in one variable, and also the error term E(T) in the asymptotic formula for the integral of the modulus squared of the Riemann zeta function along its critical line Re5 = | from \ to | + iT. The method involves two parameters: N, the length of short intervals into which we divide the sum over m, and R, the normal size of the denominator of a rational approximation to TF"(x), related in order of magnitude by 2 3 NR x M /T. For suitable choices of N and R, the bound in [2, Equations (5.5) and (5.9)] was In [2], all suitable choices of N and R satisfied H <^ NR. Recently, step 6 of the method, in which we compare rational approximations to the derivatives of TF(x) at different points x, was improved [3,4], giving an extra factor (R/H) in (1). The best 2 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Note on Exponential Sums with a Difference

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/26.4.325
Publisher site
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Abstract

M. N. HUXLEY This note is an addendum to the paper [2] of Heath-Brown and the author, in which the method of Bombieri, Iwaniec and Mozzochi [1, 6] was applied to the exponential sum in two variables which is used to investigate the mean square over a short interval of an exponential sum in one variable, and also the error term E(T) in the asymptotic formula for the integral of the modulus squared of the Riemann zeta function along its critical line Re5 = | from \ to | + iT. The method involves two parameters: N, the length of short intervals into which we divide the sum over m, and R, the normal size of the denominator of a rational approximation to TF"(x), related in order of magnitude by 2 3 NR x M /T. For suitable choices of N and R, the bound in [2, Equations (5.5) and (5.9)] was In [2], all suitable choices of N and R satisfied H <^ NR. Recently, step 6 of the method, in which we compare rational approximations to the derivatives of TF(x) at different points x, was improved [3,4], giving an extra factor (R/H) in (1). The best 2

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1994

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