Kloosterman sums with prime variable. (English) Zbl 1315.11069

Baker, Roger C.

Acta Arith. 156, No. 4, 351-372 (2012).

From the introduction: We are concerned with the exponential sum
\[ S_q(a;x)= \sum_{\substack{ x<p\leq 2x\\ (p,q)=1 }} e\left(\frac{a\overline{p}}{q} \right), \]
where \(x\geq x\); \(q\geq 2\) is an integer, \((a,q)=1\) and \(\overline{w}\) denotes the inverse of \(w\) modulo \(q\). As usual \(e(\theta)= e^{2\pi i\theta}\) and \(e_q(\theta) = e(\theta/q)\). The sum is taken over primes \(p\).
Using bounds for multidimensional exponential sums coming from algebraic geometry, É. Fouvry and P. Michel [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 1, 93–130 (1998; Zbl 0915.11045)] showed that
\[ \sum_{\substack{ x<p\leq 2x\\ (p,q)=1 }} e\left(\frac{f(p)}{q} \right) \ll_{f,\varepsilon} q^{3/16+\varepsilon} x^{25/32} \]
for \(q\) prime, \(2 \leq x \leq q\), and \(f (X)\) a rational function with integer coefficients, not of the form \(cX + d\). Fouvry and Michel showed, that for every \(\delta >0\), there exists \(\eta= \eta(\delta) >0\) such that
\[ S_q(a;x) \ll_\delta x^{1-\eta} \] for \(q\) prime, \((a, q) = 1\) and \(q^{3/4+\delta} \leq x\leq q\). This was sharpened by J. Bourgain [Int. J. Number Theory 1, No. 1, 1–32 (2005; Zbl 1173.11310)], using an ingenious elementary method that is discussed in the paper.
We extend Bourgain’s result, but with a limitation on the multiplicative structure of \(q\).

MSC:

11L20	Sums over primes
11N36	Applications of sieve methods

Keywords:

Kloosterman sums; finite Fourier transforms; counting solutions of congruences; linear sieve

Citations:

Zbl 0915.11045; Zbl 1173.11310