Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T18:35:12.088Z Has data issue: false hasContentIssue false
  • English
  • Français

CANCELLATIONS BETWEEN KLOOSTERMAN SUMS MODULO A PRIME POWER WITH PRIME ARGUMENTS

Part of: Exponential sums and character sums Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  29 January 2019

Kui Liu ,
Igor E. Shparlinski  and
Tianping Zhang
Kui Liu
Affiliation:
School of Mathematics and Statistics, Qingdao University, No. 308, Ningxia Road, Shinan, Qingdao, Shandong 266071, P.R. China email [email protected]
Igor E. Shparlinski
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
Tianping Zhang
Affiliation:
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119 Shaanxi, P.R. China email [email protected]
Get access

Abstract

We obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.

MSC classification

Primary: 11L05: Gauss and Kloosterman sums; generalizations 11T23: Exponential sums
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 475 - 487
Copyright
Copyright © University College London 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blomer, V., Fouvry, É., Kowalski, E., Michel, P. and Milićević, D., On moments of twisted L-functions. Amer. J. Math. 139 2017, 707–768.Google Scholar
Blomer, V., Fouvry, É., Kowalski, E., Michel, P. and Milićević, D., Some applications of smooth bilinear forms with Kloosterman sums. Proc. Steklov Math. Inst. 296 2017, 18–29.Google Scholar
Blomer, V. and Milićević, D., The second moment of twisted modular L-functions. Geom. Funct. Anal. 25 2015, 453–516.Google Scholar
Davenport, H., Multiplicative Number Theory, 2nd edn., Springer (New York, 1980).Google Scholar
Fouvry, É., Kowalski, E. and Michel, P., Algebraic trace functions over the primes. Duke Math. J. 163 2014, 1683–1736.Google Scholar
Fouvry, É., Michel, P., Rivat, J. and Sárközy, A., On the pseudorandomness of the signs of Kloosterman sums. J. Aust. Math. Soc. 77 2004, 425–436.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society (Providence, RI, 2004).Google Scholar
Khan, R., The divisor function in arithmetic progressions modulo prime powers. Mathematika 62 2016, 898–908.Google Scholar
Koblitz, N., p-adic Numbers, p-adic Analysis, and Zeta-functions (Graduate Texts in Mathematics 58 ), 2nd edn., Springer (New York, 1984).Google Scholar
Kowalski, E., Michel, P. and Sawin, W., Bilinear forms with Kloosterman sums and applications. Ann. of Math. (2) 186 2017, 413–500.Google Scholar
Kowalski, E., Michel, P. and Sawin, W., Bilinear forms with Kloosterman sums, II. Preprint, 2018, arXiv:1802.09849.Google Scholar
Kuznetsov, N. V., The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums. Math. USSR-Sb. 39 1981, 299–342.Google Scholar
Linnik, Y. V., Additive problems and eigenvalues of the modular operators. In Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Institut Mittag–Leffler (Djursholm, 1963), 270–284.Google Scholar
Liu, K., Shparlinski, I. E. and Zhang, T. P., Divisor problem in arithmetic progressions modulo a prime power. Adv. Math. 325 2018, 459–481.Google Scholar
Niederreiter, H., The distribution of values of Kloosterman sums. Arch. Math. 56 1991, 270–277.Google Scholar
Postnikov, A. G., On the sum of characters with respect to a modulus equal to a power of a prime number. Izv. Akad. Nauk SSSR. Ser. Mat. 19 1955, 11–16 (in Russian).Google Scholar
Postnikov, A. G., On Dirichlet L-series with the character modulus equal to the power of a prime number. J. Indian Math. Soc. 20 1956, 217–226.Google Scholar
Sarnak, P. and Tsimerman, J., On Linnik and Selberg’s conjecture about sums of Kloosterman sums. In Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, Vol. II (Progress in Mathematics 270 ), Birkhaüser (Boston, MA, 2009), 619–635.Google Scholar
Shparlinski, I. E., Distribution of inverses and multiples of small integers and the Sato–Tate conjecture on average. Michigan Math. J. 56 2008, 99–111.Google Scholar
Shparlinski, I. E., On sums of Kloosterman and Gauss sums. Trans. Amer. Math. Soc. (to appear).Google Scholar
Shparlinski, I. E. and Zhang, T. P., Cancellations amongst Kloosterman sums. Acta Arith. 176 2016, 201–210.Google Scholar
Stepanov, S. A. and Shparlinski, I. E., Estimation of trigonometric sums with rational and algebraic functions (Russian). In Automorphic Functions and Number Theory, Part I, Akad. Nauk SSSR (Dal’nevostochn. Otdel., Vladivostok, 1989), 5–18.Google Scholar
Stepanov, S. A. and Shparlinski, I. E., An estimate for the incomplete sum of multiplicative characters of polynomials (Russian). Diskret. Mat. 2 1990, 115–119. Engl. transl., Discrete Math. Appl. 2 (1992), 169–174.Google Scholar
Vaughan, R. C., An elementary method in prime number theory. Acta Arith. 37 1980, 111–115.Google Scholar