FINE-SCALE DISTRIBUTION OF ROOTS OF QUADRATIC CONGRUENCES

被引:1
作者
Marklof, Jens [1 ]
Welsh, Matthew [1 ]
机构
[1] Univ Bristol, Sch Math, Bristol, England
基金
英国工程与自然科学研究理事会;
关键词
PAIR CORRELATION; EQUIDISTRIBUTION;
D O I
10.1215/00127094-2022-0081
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence mu(2) = D mod m, with D > 0 square-free and D (sic) 1 mod 4. This is achieved by translating the problem to convergence of certain geodesic ran-dom line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.
引用
收藏
页码:2303 / 2364
页数:62
相关论文
共 31 条
[1]  
[Anonymous], 1980, Prime geodesic theorems
[2]   A Poincar, Section for the Horocycle Flow on the Space of Lattices [J].
Athreya, Jayadev S. ;
Cheung, Yitwah .
INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2014, 2014 (10) :2643-2690
[3]   Higher composition laws I: A new view on Gauss composition, and quadratic generalizations [J].
Bhargava, M .
ANNALS OF MATHEMATICS, 2004, 159 (01) :217-250
[4]   Pair correlation of angles between reciprocal geodesics on the modular surface [J].
Boca, Florin P. ;
Pasol, Vicentiu ;
Popa, Alexandru A. ;
Zaharescu, Alexandru .
ALGEBRA & NUMBER THEORY, 2014, 8 (04) :999-1035
[5]   Pair correlation of hyperbolic lattice angles [J].
Boca, Florin P. ;
Popa, Alexandru A. ;
Zaharescu, Alexandru .
INTERNATIONAL JOURNAL OF NUMBER THEORY, 2014, 10 (08) :1955-1989
[6]  
BYKOVSKII V. A., 1984, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), V134, P15
[7]   EQUIDISTRIBUTION OF ROOTS OF A QUADRATIC CONGRUENCE TO PRIME MODULI [J].
DUKE, W ;
FRIEDLANDER, JB ;
IWANIEC, H .
ANNALS OF MATHEMATICS, 1995, 141 (02) :423-441
[8]   Weyl Sums for Quadratic Roots [J].
Duke, W. ;
Friedlander, J. B. ;
Iwaniec, H. .
INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2012, 2012 (11) :2493-2549
[9]   MIXING, COUNTING, AND EQUIDISTRIBUTION IN LIE-GROUPS [J].
ESKIN, A ;
MCMULLEN, C .
DUKE MATHEMATICAL JOURNAL, 1993, 71 (01) :181-209
[10]  
Fouvry E, 1997, ACTA ARITH, V79, P249