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Sub-Weyl subconvexity for Dirichlet $L$ -functions to prime power moduli

Published online by Cambridge University Press:  03 November 2015

Djordje Milićević
Bryn Mawr College, Department of Mathematics, 101 North Merion Avenue, Bryn Mawr, PA 19010, USA email
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We prove a subconvexity bound for the central value $L(\frac{1}{2},{\it\chi})$ of a Dirichlet $L$ -function of a character ${\it\chi}$ to a prime power modulus $q=p^{n}$ of the form $L(\frac{1}{2},{\it\chi})\ll p^{r}q^{{\it\theta}+{\it\epsilon}}$ with a fixed $r$ and ${\it\theta}\approx 0.1645<\frac{1}{6}$ , breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving $p$ -adically analytic phases, which can be naturally seen as a $p$ -adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.

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Barban, M. B., Linnik, Yu. V. and Tshudakov, N. G., On prime numbers in an arithmetic progression with a prime-power difference, Acta Arith. 9 (1964), 375390; MR 0171766 (30 #1993).CrossRefGoogle Scholar
Berndt, B. C., Evans, R. J. and Williams, K. S., Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts (Wiley, New York, 1998); MR 1625181 (99d:11092).Google Scholar
Blomer, V. and Milićević, D., p-adic analytic twists and strong subconvexity, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), 561605; MR 3377053.CrossRefGoogle Scholar
Blomer, V. and Milićević, D., The second moment of twisted modular L-functions, Geom. Funct. Anal. 25 (2015), 453516; doi: 10.1007/s00039-015-0318-7; MR 3334233.CrossRefGoogle Scholar
Bourgain, J., Decoupling, exponential sums and the Riemann zeta function, Preprint (2014),arXiv:1408.5794.Google Scholar
Burgess, D. A., On character sums and L-series. II, Proc. Lond. Math. Soc. (3) 13 (1963), 524536; MR 0148626 (26 #6133).CrossRefGoogle Scholar
Conrey, J. B. and Iwaniec, H., The cubic moment of central values of automorphic L-functions, Ann. of Math. (2) 151 (2000), 11751216; MR 1779567 (2001g:11070).CrossRefGoogle Scholar
Fujii, A., Gallagher, P. X. and Montgomery, H. L., Some hybrid bounds for character sums and Dirichlet L-series, in Topics in number theory (Proc. Colloq., Debrecen, 1974), Colloq. Math. Soc. János Bolyai, vol. 13 (North-Holland, Amsterdam, 1976), 4157; MR 0434987 (55 #7949).Google Scholar
Gallagher, P. X., Primes in progressions to prime-power modulus, Invent. Math. 16 (1972), 191201; MR 0304327 (46 #3462).CrossRefGoogle Scholar
Graham, S. W. and Kolesnik, G., Van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series, vol. 126 (Cambridge University Press, Cambridge, 1991); MR 1145488 (92k:11082).Google Scholar
Heath-Brown, D. R., Hybrid bounds for Dirichlet L-functions, Invent. Math. 47 (1978), 149170; MR 0485727 (58 #5549).CrossRefGoogle Scholar
Hiary, G. A., Computing Dirichlet character sums to a power-full modulus, J. Number Theory 140 (2014), 122146; MR 3181649.CrossRefGoogle Scholar
Holowinsky, R., Munshi, R. and Qi, Z., Character sums of composite moduli and hybrid subconvexity, Contemp. Math., to appear. Preprint (2014), arXiv:1409.3797.Google Scholar
Huxley, M. N., Exponential sums and the Riemann zeta function. V, Proc. Lond. Math. Soc. (3) 90 (2005), 141; MR 2107036 (2005h:11180).CrossRefGoogle Scholar
Iwaniec, H., On zeros of Dirichlet’s L series, Invent. Math. 23 (1974), 97104; MR 0344207 (49 #8947).CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004); MR 2061214 (2005h:11005).Google Scholar
Iwaniec, H. and Sarnak, P., Perspectives on the analytic theory of L-functions, Geom. Funct. Anal. (2000), Special Volume, Part II, 705–741, GAFA 2000 (Tel Aviv, 1999); MR 1826269 (2002b:11117).Google Scholar
Katok, S., p-adic analysis compared with real, Student Mathematical Library, vol. 37 (American Mathematical Society, Providence, RI, 2007); MR 2298943 (2008j:12010).Google Scholar
Michel, P., Analytic number theory and families of automorphic L-functions, in Automorphic forms and applications, IAS/Park City Mathematics Series, vol. 12 (American Mathematical Society, Providence, RI, 2007), 181295; MR 2331346 (2008m:11104).Google Scholar
Michel, P. and Venkatesh, A., The subconvexity problem for GL2, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171271; MR 2653249.CrossRefGoogle Scholar
Nelson, P. D., Pitale, A. and Saha, A., Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels, J. Amer. Math. Soc. 27 (2014), 147191; MR 3110797.CrossRefGoogle Scholar
Phillips, E., The zeta-function of Riemann; further developments of van der Corput’s method, Q. J. Math. 4 (1933), 209225.CrossRefGoogle 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), 1116; MR 0068575 (16,905f).Google Scholar
Rankin, R. A., Van der Corput’s method and the theory of exponent pairs, Q. J. Math. 6 (1955), 147153; MR 0072170 (17,240a).CrossRefGoogle Scholar
Ricotta, G., Universality of convexity breaking exponents, in Problem Sessions: Subconvexity Bounds for $L$-functions (notes), 2006, Scholar
Robert, A. M., A course in p-adic analysis, Graduate Texts in Mathematics, vol. 198 (Springer, New York, 2000); MR 1760253 (2001g:11182).Google Scholar
Salié, H., Über die Kloostermanschen Summen S (u, v; q), Math. Z. 34 (1932), 91109; MR 1545243.CrossRefGoogle Scholar
Templier, N., Large values of modular forms, Camb. J. Math. 2 (2014), 91116; MR 3272013.CrossRefGoogle Scholar
Titchmarsh, E. C., The theory of the Riemann zeta-function, 2nd edition (The Clarendon Press, Oxford University Press, New York, 1986), Edited and with a preface by D. R. Heath-Brown; MR 882550 (88c:11049).Google Scholar
van der Corput, J. G., Verschärfung der Abschätzung beim Teilerproblem, Math. Ann. 87 (1922), 3965.CrossRefGoogle Scholar
Vishe, P., A fast algorithm to compute L (1∕2, f ×𝜒q), J. Number Theory 133 (2013), 15021524; MR 3007119.CrossRefGoogle Scholar
Walfisz, A., Zur Abschätzung von 𝜁(1∕2 + it), Nachr. Ges. Wiss. Göttingen Math.-Physik. Kl. 1924 (1924), 155158 (in German).Google Scholar

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Sub-Weyl subconvexity for Dirichlet $L$ -functions to prime power moduli
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