The Friedlander–iwaniec Character Sum
نویسندگان
چکیده
As we observed in [3], one can also interpret the sums S(α;β) as special cases of the correlation sums arising in our work on algebraic twists of modular forms, and we also noted that our more general results and techniques had, as special case, this theorem of Birch and Bombieri. Very recently, Y. Zhang [12, Lemma 12] made a crucial use of this estimate in his work on bounded gaps between primes. In view of this renewed interest, we spell out the argument of [3] for this special case, and make it more precise. This gives a conceptual proof of the result which is quite short, but which depends on two deep results of Deligne: the general form of the Riemann Hypothesis over finite fields [2], and the construction of Kloosterman sheaves (explained and studied by Katz in [8]). Proposition 2. For any prime p and any α, β ∈ Fp , we have |S(α, β)| 6 8p.
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