Large deviations of sums of random variables

Abstract. In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some of the results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences p, χ(p), χd(p), λf (p), and Klq(a−n, b); where p ranges over the primes, t varies in a large interval, χ varies among all characters modulo q, χd varies over quadratic characters attached to fundamental discriminants |d| 6 x, λf (n) are the Fourier coefficients of holomorphic cusp forms f of (a large) weight k for the full modular group, and Klq(a, b) are the normalized Kloosterman sums modulo a large prime q, where a, b vary in (Fq) .

[1]  D. R. Heath-Brown.  A mean value estimate for real character sums , 1995 .

[2]  Corentin Perret-Gentil.  Gaussian distribution of short sums of trace functions over finite fields , 2016, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Andrew Granville,et al.  Large character sums , 1999, math/9903196.

[4]  V. V. Petrov,et al.  On Large Deviations of Sums of Independent Random Variables , 2007 .

[5]  P. Michel,et al.  On the complex moments of symmetric power $L$-functions at $s = 1$ , 2004 .

[6]  H. Davenport.  Multiplicative Number Theory , 1967 .

[7]  Lower bounds for moments of L-functions: symplectic and orthogonal examples , 2006, math/0601498.

[8]  Andrew Odlyzko,et al.  Large deviations of sums of independent random variables , 1988 .

[9]  Extreme values of $|ζ(1+it)|$ , 2005 .

[10]  Henryk Iwaniec,et al.  Topics in classical automorphic forms , 1997 .

[11]  D. R. Heath-Brown,et al.  The Theory of the Riemann Zeta-Function , 1987 .

[12]  Youness Lamzouri,et al.  The distribution of the maximum of partial sums of Kloosterman sums and other trace functions , 2021, Compositio Mathematica.

[13]  E. Kowalski,et al.  The analytic rank of $J_0 ( q )$ and zeros of automorphic $L$-functions , 1999 .

[14]  A. Granville,et al.  The distribution of values of L(1, χd) , 2003 .

[15]  S. Graham,et al.  Lower Bounds for Least Quadratic Non-Residues , 1990 .

[16]  E. Kowalski,et al.  Kloosterman paths and the shape of exponential sums , 2014, Compositio Mathematica.

[17]  H. Montgomery.  Topics in Multiplicative Number Theory , 1971 .

[18]  Jessika Eichel,et al.  Introduction To Analytic And Probabilistic Number Theory , 2016 .

[19]  Youness Lamzouri.  ON THE DISTRIBUTION OF THE MAXIMUM OF CUBIC EXPONENTIAL SUMS , 2018, Journal of the Institute of Mathematics of Jussieu.

[20]  The Distribution of Values of L(1; ) , 2007 .

[21]  Jie Wu,et al.  On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1 , 2006, math/0604334.