## 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

### Keywords:

exponential sums; sums over prime numbers
Full Text:

### References:

 [1] R. C. BAKER et G. HARMAN , On the distribution of {\alpha pk} modulo 1 , (Mathematika, vol. 38, 1991 , p. 170-184). MR 92f:11096 | Zbl 0751.11037 · Zbl 0751.11037 [2] E. BOMBIERI , Exponential sums in finite fields , (Amer. J. Math., vol. 88, 1966 , p. 71-105). MR 34 #166 | Zbl 0171.41504 · Zbl 0171.41504 [3] E. BOMBIERI et S. SPERBER , On the estimation of certain exponential sums , (Acta Arith., vol. 69, 1995 , p. 329-358). Article | MR 96d:11089 | Zbl 0826.11040 · Zbl 0826.11040 [4] H. DAVENPORT , Multiplicative Number Theory (Second Edition), (Graduate Texts in Mathematics, vol. 74, Springer Verlag, Berlin-Heidelberg-New York, 1980 ). MR 82m:10001 | Zbl 0453.10002 · Zbl 0453.10002 [5] H. DAVENPORT et A. SCHINZEL , Two problems concerning polynomials , (J. reine angew. Math., vol. 214-215, 1964 , p. 386-391). Article | MR 29 #93 | Zbl 0152.02303 · Zbl 0152.02303 [6] P. DELIGNE , La Conjecture de Weil I , (Publ. de l’I.H.E.S., vol. 43, 1974 , p. 273-308). Numdam | MR 49 #5013 | Zbl 0287.14001 · Zbl 0287.14001 [7] P. DELIGNE , Applications de la Formule des Traces aux Sommes Trigonométriques , Séminaire de Géométrie Algébrique du Bois-Marie SGA41/2, Cohomologie Étale, (Lecture Notes in Mathematics, vol. 569, Springer Verlag, Berlin-Heidelberg-New York, 1977 , p. 168-232). Zbl 0349.10031 · Zbl 0349.10031 [8] P. DELIGNE , La Conjecture de Weil II , (Publ. de l’I.H.E.S., vol. 52, 1981 , p. 313-428). Numdam [9] E. FOUVRY et H. IWANIEC , The divisor function over arithmetic progressions , (with appendix by N. Katz), (Acta Arith., vol. 61, 1992 , p. 271-287). MR 93g:11089 | Zbl 0764.11040 · Zbl 0764.11040 [10] M. FRIED , The field of definition of function fields and a problem in the reduibility of polynomials in 2 variables , Ill. (J. Math., vol. 17, 1973 , p. 128-144). MR 50 #329 | Zbl 0266.14013 · Zbl 0266.14013 [11] J. FRIEDLANDER et H. IWANIEC , Incomplete Kloosterman sums and a divisor problem , (Annals of Maths., vol. 121, 1985 , p. 319-350). MR 86i:11050 | Zbl 0572.10029 · Zbl 0572.10029 [12] J. FRIEDLANDER et H. IWANIEC , Estimates for Character Sums , (Proc. Amer. Math. Soc., vol. 19, 1993 , p. 365-372). MR 93k:11074 | Zbl 0782.11022 · Zbl 0782.11022 [13] A. GHOSH , The Distribution of \alpha p$$^{2}$$ modulo 1 , (Proc. London Math. Soc., (3), vol. 42, 1981 , p. 252-269). MR 82j:10067 | Zbl 0397.10026 · Zbl 0397.10026 [14] G. HARMAN , Trigonometric Sums over Primes I , (Mathematika, vol. 28, 1981 , p. 249-254). MR 83j:10045 | Zbl 0465.10029 · Zbl 0465.10029 [15] R. HARTSHORNE , Algebraic Geometry , (Graduate Texts in Mathematics, vol. 52, Springer Verlag, Berlin-Heidelberg-New York, 1977 ). MR 57 #3116 | Zbl 0367.14001 · Zbl 0367.14001 [16] C. HOOLEY , On exponential sums and certain of their applications , (Journées Arith. 1980 , J.V. Armitage (ed), p. 92-122, Cambridge, 1982 ). Zbl 0488.10041 · Zbl 0488.10041 [17] L. K. HUA , Additive Theory of Prime Numbers , (Translations of Mathematical Monographs, vol. 13, American Math. Soc., 1965 ). MR 33 #2614 | Zbl 0192.39304 · Zbl 0192.39304 [18] N. M. KATZ , Gauss Sums, Kloosterman Sums and Monodromy Groups , (Annals of Maths. Studies, vol. 116, PUP). MR 91a:11028 | Zbl 0675.14004 · Zbl 0675.14004 [19] N. M. KATZ , Exponential sums and Differential Equations , (Annals of Maths. Studies, vol. 124, PUP). MR 93a:14009 | Zbl 0731.14008 · Zbl 0731.14008 [20] P. MICHEL , Autour des Conjectures de Sato-Tate , (Thèse de Doctorat d’État, Université de Paris-Sud, Orsay, 1995 ). [21] D. MUMFORD , The Red Book of Varieties and Schemes , (Lectures Notes in mathematics, vol. 1358, Springer Verlag, Berlin-Heidelberg-New York, 1988 ). MR 89k:14001 | Zbl 0658.14001 · Zbl 0658.14001 [22] A. SCHINZEL , Reducibility of polynomials in several variables , (Bull. Acad. Pol. Sci. Sér. Sci. Math. Astronom. Phys., vol. 11, 1963 , p. 633-638). MR 28 #3032 | Zbl 0122.01901 · Zbl 0122.01901 [23] A. SCHINZEL , Reducibility of polynomials in several variables. II , (Pacific J. Math., vol. 118, 1985 , p. 531-563). Article | MR 86i:12005 | Zbl 0571.12011 · Zbl 0571.12011 [24] W. SCHMIDT , Equations over finite fields : an elementary approach , (Lecture Notes in Mathematics, vol. 536, Springer Verlag, Berlin-Heidelberg-New York, 1976 ). MR 55 #2744 | Zbl 0329.12001 · Zbl 0329.12001 [25] R. C. VAUGHAN , Mean Value Theorems in Prime Number Theory , (J. London Math. Soc., (2), vol. 10, 1975 , p. 153-162). MR 51 #12742 | Zbl 0314.10028 · Zbl 0314.10028 [26] M. VINOGRADOV , The Method of Trigonometric Sum in the Theory of Numbers , Translated, revised and annotated by A. Davenport and K. F. Roth (Interscience, New York, 1954 ). Zbl 0055.27504 · Zbl 0055.27504
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