EXPONENTIAL SUMS AND THE RIEMANN ZETA FUNCTION IV M. N. HUXLEY [Received 1 June 1991—Revised 16 January 1992] ABSTRACT Let F(x) be a real function with sufficiently many derivatives existing and satisfying certain non-vanishing conditions for 1 =s;t «2. We improve the estimates for the exponential sum 5 = 2 especially when M is close to the square root of T. For the Riemann zeta function, We use the 'discrete Hardy-Littlewood method' of Bombieri and Iwaniec. The improvement lies in comparing rational approximations to the derivatives of F at nearby integers m. 1. Introduction Exponential sums appear when we try to show that two properties are independent on a given sequence of real numbers. We write [[*]] for the nearest integer to the real number x, and ((*)) = x — [[x]\ with —\ < ((x)) «s \. An important problem of this type is whether ((x m )) is distributed uniformly for a given sequence of real numbers {x m }. The rounding error in numerical integration can be analysed in terms of a sequence of fractional parts {(x m )) as in m. This paper continues the work of Bombieri, Iwaniec, Huxley, Watt and Kolesnik [1, 9, 15,10, 8] on the simple exponential sum s- 2 «M=)) (ID m = M \ \M// formed with an arbitrary smooth function F(x). Here e(x) denotes exp 2xix, and M 2 < 2M < T. There are six steps. 1. Divide the range for m into short intervals, the major and minor arcs. Pick an integer m within each short interval for which the second derivative of f{x) = TF(x/M) is close to a rational number 2a/q. Approximate f(m + n) by the cubic Taylor polynomial in n. 2. Take a finite Fourier transform modulo q, using Gauss sums. 3. Take a Fourier transform in n. Estimate trivially if q is small (the major arcs). 4. Write the terms of the transformed sum as e(x.y), where x = (h 2 , h, h 3/2 , h m ) for some integer h, and y is constructed from the coefficients of the Taylor polynomial. Use a large sieve inequality to bound the transformed sums in the mean over y. 5. Count the solutions of some inequalities in sums of x vectors for different integers h. 1991 Mathematics Subject Classification: primary 11L40; secondary 11M06, 11P55. Proc. London Math. Soc. (3) 66 (1993) 1-40.