Double exponential sums and some applications (Q1817768)

The so-called bilinear sums of type I \[ T(M,N) := \sum_{M<m\leq M'\leq 2M} \sum_{N<n\leq N'\leq 2N} a_m\text{ e}\left(x{m^\alpha n^\beta\over M^\alpha N^\beta}\right) \] with \(X > 0, M\geq 1, N\geq 1, \alpha,\beta \in \mathbb R, \text{ e}(t) = \text{ e}^{2\pi it}, |a_m|\leq 1\) often occur in applications of exponential sum techniques. Diverse methods have been used by various authors to obtain bounds for such sums, which are very sensitive to the size of \(X, M\) and \(N\). The author obtains two theorems, too complicated to be stated here in detail, involving upper bounds for \(T(M,N)\). As an application of his new results he proves the asymptotic formula \[ \sum_{|{\mathcal G}|\leq x}t({\mathcal G}) = A_1x\log x + A_2x + A_3\sqrt{x} + O(x^{c+\varepsilon}) \] with \(c = 45/127 = 0.3543\ldots \), where \(t({\mathcal G})\) denotes the number of unitary factors of a finite abelian group, and \(A_1 (\not= 0), A_2\) and \(A_3\) are effective constants. This improves the earlier result of the author [Acta Arith. 84, 17-29 (1998; Zbl 0899.11048)] who had \(c = 47/131 = 0.3587\ldots \).