In mathematics, van der Corput's method generates estimates for exponential sums.
The method applies two processes, the van der Corput processes A and B which relate the sums into simpler sums which are easier to estimate.
The processes apply to exponential sums of the form where f is a sufficiently smooth function and e(x) denotes exp(2πix).
To apply process A, write the first difference fh(x) for f(x+h)−f(x).
Assume there is H ≤ b−a such that Then Process B transforms the sum involving f into one involving a function g defined in terms of the derivative of f. Suppose that f' is monotone increasing with f'(a) = α, f'(b) = β.
Then f' is invertible on [α,β] with inverse u say.
The method of exponent pairs gives a class of estimates for functions with a particular smoothness property.
We consider functions f defined on an interval [N,2N] which are R times continuously differentiable, satisfying uniformly on [a,b] for 0 ≤ r < R. We say that a pair of real numbers (k,l) with 0 ≤ k ≤ 1/2 ≤ l ≤ 1 is an exponent pair if for each σ > 0 there exists δ and R depending on k,l,σ such that uniformly in f. By Process A we find that if (k,l) is an exponent pair then so is
A trivial bound shows that (0,1) is an exponent pair.
It is known that if (k,l) is an exponent pair then the Riemann zeta function on the critical line satisfies