# Hypotheses for exponent pairs

The theory of exponent pairs provides bounds for $$\sum_{N<n<2N} e(f(n)),$$ where f behaves like a monomial. Precise formulations of this are in Graham and Kolesnik (GK) which seems to be what is cited in the literature when one wants to apply the A and B processes.

The problem is that for a function such as $f(n) = n^{3/2}$, it does not satisfy equation 3.3.3 in GK. Nevertheless, one should typically be able to apply the A and B processes.

I am asking for a reference that I can cite which allows one to apply the A and B processes to such functions.

See Montgomery's book, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, specifically the examples on p.55. He considers variants of your sum, namely $\sum_{n \leq N} e((n/3)^{3/2})$, and $\sum_{n \leq N} e((2n/3)^{3/2})$, and shows that they have very different asymptotic behavior. The former is asymptotically $c N^{3/4}$ while the latter is $O(N^{1/4})$. These examples illustrate why the hypothesis (3.3.3) in [GK] is assumed.
• Thanks for your response! But, for instance if we consider $e(tn^{3/2})$ and t is transcendental, one should always be able to apply the A and B processes, with some basic assumptions avoiding the case where the derivatives of the resulting amplitude function become too small. Do you know if this is the case? – George Shakan Oct 4 '17 at 16:58