Van der Corput's method


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 denotes exp.

Process A

To apply process A, write the first difference fh for f−f.
Assume there is H ≤ b−a such that
Then

Process B

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' = α, f' = β. Then f' is invertible on with inverse u say. Further suppose f ≥ λ > 0. Write
We have
Applying Process B again to the sum involving g returns to the sum over f'' and so yields no further information.

Exponent pairs

The method of exponent pairs gives a class of estimates for functions with a particular smoothness property. Fix parameters N,R,T,s,δ. We consider functions f defined on an interval which are R times continuously differentiable, satisfying
uniformly on for 0 ≤ r < R.
We say that a pair of real numbers 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 is an exponent pair then so is.
By Process B we find that so is.
A trivial bound shows that is an exponent pair.
The set of exponents pairs is convex.
It is known that if is an exponent pair then the Riemann zeta function on the critical line satisfies
where.
The exponent pair conjecture states that for all ε > 0, the pair is an exponent pair. This conjecture implies the Lindelöf hypothesis.