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  • Ping Xi (a1)


We prove that the exponent of distribution of $\unicode[STIX]{x1D70F}_{3}$ in arithmetic progressions can be as large as $\frac{1}{2}+\frac{1}{34}$ , provided that the moduli is squarefree and has only sufficiently small prime factors. The tools involve arithmetic exponent pairs for algebraic trace functions, as well as a double $q$ -analogue of the van der Corput method for smooth bilinear forms.



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1. Andersson, J., Summation formulae and zeta functions, Doctoral Dissertation, Stockholm University, 2006.
2. Deligne, P., La conjecture de Weil, II. Publ. Math. Inst. Hautes Études Sci. 52 1980, 137252.
3. Fouvry, É., Sur le problème des diviseurs de Titchmarsh. J. Reine Angew. Math. 357 1985, 5176.
4. Fouvry, É., Kowalski, E. and Michel, Ph., On the conductor of cohomological transforms. Preprint, 2013, arXiv:1310.3603 [math.NT].
5. Fouvry, É., Kowalski, E. and Michel, Ph., Algebraic trace functions over the primes. Duke Math. J. 163 2014, 16831736.
6. Fouvry, É., Kowalski, E. and Michel, Ph., On the exponent of distribution of the ternary divisor function. Mathematika 61 2015, 121144.
7. Fouvry, É., Kowalski, E. and Michel, Ph., Algebraic twists of modular forms and Hecke orbits. Geom. Funct. Anal. 25 2015, 580657.
8. Fouvry, É., Kowalski, E. and Michel, Ph., Trace functions over finite fields and their applications. In Colloquium de Giorgi, 2013 and 2014, Scuola Normale Superiore Pisa (2015), 735.
9. Friedlander, J. B. and Iwaniec, H., Incomplete Kloosterman sums and a divisor problem (with an appendix by B. J. Birch and E. Bombieri). Ann. of Math. (2) 121 1985, 319350.
10. Graham, S. W. and Kolesnik, G., Van der Corput Method of Exponential Sums (London Mathematical Society Lecture Note Series 126 ), Cambridge University Press (Cambridge, 1991).
11. Heath-Brown, D. R., The divisor function d 3(n) in arithmetic progressions. Acta Arith. 47 1986, 2956.
12. Hooley, C., An asymptotic formula in the theory of numbers. Proc. Lond. Math. Soc. (3) 7 1957, 396413.
13. Irving, A. J., The divisor function in arithmetic progressions to smooth moduli. Int. Math. Res. Not. IMRN 15 2015, 66756698.
14. Katz, N. M., Exponential Sums and Differential Equations (Annals of Math. Studies 124 ), Princeton University Press (1990).
15. Kuznetsov, N. V., The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums. Mat. Sb. 111(153) 1980, 334383; 479.
16. Linnik, Yu. V., All large numbers are sums of a prime and two squares (A problem of Hardy and Littlewood). II. Mat. Sb. 53(95) 1961, 338.
17. Matthes, R., An elementary proof of a formula of Kuznecov for Kloosterman sums. Results Math. 18 1990, 120124.
18. Polymath, D. H. L., New equidistribution estimates of Zhang type. Algebra Number Theory 8 2014, 20672199.
19. Selberg, A., Lectures on Sieves (Collected Papers II ), Springer (1991), 65247.
20. Smith, R. A., On n-dimensional Kloosterman sums. J. Number Theory 11 1979, 324343.
21. Smith, R. A., A generalization of Kuznietsov’s identity for Kloosterman sums. C. R. Math. Rep. Acad. Sci. Canada 2(6) 1980, 315320.
22. Wu, J. and Xi, P., Arithmetic exponent pairs for algebraic trace functions and applications. Preprint, 2016, arXiv:1603.07060 [math.NT].
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  • Ping Xi (a1)


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