tailieunhanh - Analytic Number Theory A Tribute to Gauss and Dirichlet Part 13

Tham khảo tài liệu 'analytic number theory a tribute to gauss and dirichlet part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 232 PETER SARNAK In Ho a more precise conjecture is made 80 e x C2 x log x 2 . Kwon Kwo has recently investigated this numerically. To do so she makes an ansatz for the lower order terms in 80 in the form ý e x x c2 logx 2 c1 log x Co O xa with a 1. The computations were carried out for x 107 and she finds that for x 104 the ansatz is accurate with c0 Cl and c2 . It would be interesting to extend these computations and also to extend Hooley s heuristics to see if they lead to the ansatz. The difficulty with 76 lies in the delicate issue of the relative density of Din Dr. See the discussions in Lag80 and Mor90 concerning the solvability of 9 . In R36 the two-component of Fd is studied and used to get lower bounds of the form Fix t a large integer then 81 r 1 and V 1 x l gl gx . v 7 t log x dEdR dED- d x d x On the other hand each of these is bounded above by 1 which by Lan- d eDR d x dau s thesis or the half-dimensional sieve is asymptotic to c3 xf ỵ log x. 81 leads to a corresponding lower bound for G x . The result R36 leading to 81 suggests strongly that the proportion of d E D which lie in D- is in 2 1 In Ste93 a conjecture for the exact proportion is put forth together with some sound reasoning . It seems therefore quite likely that 82 G c4 as x TO with - c4 1. ý M x 2 It follows from 78 and 79 that it is still the case that zero percent of the classes in n are reciprocal when ordered by discriminant though this probability goes to zero much slower than when ordering by trace. On the other hand according to 82 a positive proportion even perhaps more than 1 2 of the reciprocal classes are ambiguous in this ordering unlike when ordering by trace. We end with some comments about the question of the equidistribution of closed geodesics as well as some comments about higher dimensions. To each primitive closed p E n we associate the measure pp on X r H or better still the corresponding measure on the unit tangent bundle r SL 2 R which is arc length .