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On some exponential sums over primes. (Sur certaines sommes d’exponentielles sur les nombres premiers.) (French) Zbl 0915.11045

The authors consider exponential sums of the type \[ S=\sum_{p\le x} e(f(p)/q), \] where \(p\) runs over primes, \(q\) is a large prime, \(x\le q\), and \(f(x)=P(x)/Q(x)\) is a non-constant rational function over \(\mathbb Z\) which is not a linear polynomial. If \(f\) is a polynomial, then the classical method of I. M. Vinogradov, or its simplified version due to R. C. Vaughan, yields a nontrivial estimate for \(S\), but the result becomes weaker if the degree of the polynomial increases. On the other hand, the main results in the present paper, namely that \(S=O (q^{3/16+\varepsilon }x^{25/32})\), is largely independent of the shape of \(f\), for the implied constant depends only on \(\varepsilon \) and the degrees of \(P\) and \(Q\); this is a remarkable breakthrough indeed. Note that the estimate is nontrivial for \(q^{6/7+\varepsilon }\le x \le q\).
As usual, the sum over primes is first reduced to exponential sums of the “convolution” type. It is methodically interesting that deep results from algebraic geometry play an essential role in the subsequent argument.
The following arithmetical application is given. Let \(f_1\),…, \(f_{33}\) be rational functions as described above. Then for a large prime \(q\), the number of solutions of the congruence \(N \equiv f_1(p_1)+ \cdots +f_{33}(p_{33}) \pmod{q}\) in primes \(p_1, \dots, p_{33}\) is \((\pi (q)^{33}/q)(1+ O(q^{-1/33}))\).

MSC:

11L07 Estimates on exponential sums
11L20 Sums over primes
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References:

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