YOUNESS LAMZOURI H 2. The purpose of this note is to improve the error term in this asymptotic formula. H 2 (log log H) 3 ζ(3) H2 + O

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1 ON THE AVERAGE OF THE NUMBER OF IMAGINARY QUADRATIC FIELDS WITH A GIVEN CLASS NUMBER YOUNESS LAMZOURI Abstract Let Fh be the number of imaginary quadratic fields with class number h In this note we imrove the error term in Soundararajan s asymtotic formula for the average of Fh Our argument leads to a similar refinement of the asymtotic for the average of Fh over odd h which was recently obtained by Holmin Jones Kurlberg McLeman and Petersen Introduction An imortant roblem in number theory which goes back to Gauss is to determine all imaginary quadratic fields with a given class number Let Fh be the number of imaginary quadratic fields with class number h Then for instance one has F = 9 which follows from the celebrated solution of Baker-Stark-Heegner to Gauss class number roblem for imaginary quadratic fields In [3] Soundararajan studied the quantity Fh and determined its average order More recisely he roved that for any ɛ > 0 Fh = 3ζ ζ3 H + O ɛ H log H / ɛ The urose of this note is to imrove the error term in this asymtotic formula Theorem We have Fh = 3ζ H log log H 3 ζ3 H + O log H In a recent work [] Holmin Jones Kurlberg Mcleman and Petersen studied statistics of the class numbers of imaginary quadratic fields In articular they used the Cohen-Lenstra heuristics together with the work of Granville and Soundararajan [] on the distribution of values of L χ d to formulate a conjecture on the asymtotic nature of Fh as h through odd values They also obtained the analogue of for the average of Fh over odd values of h conditionally on the generalized Riemann 00 Mathematics Subject Classification Primary R9; Secondary R M0 The author is artially suorted by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada

2 YOUNESS LAMZOURI hyothesis GRH More recisely they showed that assuming GRH Fh = 5 H 4 log H + O H ɛ log H 3/ ɛ h odd Unlike which is unconditional the roof of uses GRH to bound a certain character sum over rimes which aears in this case due to the fact that when d > 8 the class number of Q d is odd recisely when d is rime by genus theory The same argument in our roof of Theorem leads to the following refinement of the asymtotic formula Theorem Assume GRH Then Fh = 5 4 h odd H H log H + O log log H 3 log H Let h d be the class number of Q d The main ingredients in the roofs of and are asymtotic formulas for the comlex moments of h d namely 4 and 0 below Using this aroach the best saving one can hoe for in the error terms of Theorems and will be roughly /L if one can obtain an asymtotic formula for the average of h d s over negative fundamental discriminants d with d H +ε uniformly in s such that s L It is also known see for examle [] that the asymtotic formulas 4 and 0 are no longer valid when s log Hlog log H A for some A > This shows that the saving of log H/log log H 3 in the error terms of Theorems and constitute u to the ower of log log H the limit of Soundararajan s method [3] In articular it would be interesting to imrove the ower of log H in the error terms of these results Proofs of Theorems and Let X := H log log H As in [3] it follows from Theorem 4 of [] concerning the distribution of extreme values of L χ d together with Tatuzawa s refinement of the Landau-Siegel Theorem [4] that Fh = d X h d H H + O A log H A for any A > 0 where indicates that the sum is over fundamental discriminants d To estimate the main term in Soundararajan used the following variant of Perron s formula c+i x s πi s + δ s+ δs + ds = if x + δ /x/δ if + δ x 0 if 0 < x + δ

3 THE NUMBER OF IMAGINARY QUADRATIC FIELDS WITH A GIVEN CLASS NUMBER 3 Our imrovement comes from using a different smooth cut-off function namely I cλn y := e y s λs N ds πi λs s where c λ > 0 are real numbers and N is a ositive integer Lemma Let λ c > 0 be real numbers and N be a ositive integer Then we have = if y > I cλn y [0 ] if e λn y = 0 if 0 < y < e λn Proof First we recall Perron s formula Then we observe that πi πi N ds s = λ N e y s λs λs yet + +t N s ds y s ds s = λ 0 if y > if y = 0 if 0 < y < λ 0 πi ye t + +t N s ds s dt dt N By πi s [0 ] for all values of t i and hence I cλn y [0 ] for all y > 0 The lemma follows from uon noting that ye t + +t N > for all t i [0 λ] if y > and ye t + +t N < for all t i [0 λ] if 0 < y < e λn Proof of Theorem Let c = / log H N be a ositive integer and 0 < λ be a real number to be chosen later By and Lemma we obtain 3 Fh πi d X H s h d s e λs λs N ds H s + O A log H A h e λn H Let T := log X/0 4 log log X Then it follows from equation 5 of [3] that 4 h d s = 3π s E L X s X x s/ dx + O X ex log X 5 log log X d X for all comlex numbers s with Res = c and s T where 5 L X = X Fh and {X} is a sequence of indeendent random variables taking the value with robability /+ 0 with robability /+ and with robability /+ Note that EX = 0 and EX and hence the random roduct 5 converges almost surely by Kolmogorov s three series theorems

4 4 YOUNESS LAMZOURI Since e λs 3 if H is large enough and h d it follows that the contribution of the region s > T to the integral in 3 is X N 3 λ s >T Res=c ds s X N 3 N+ N λt Moreover note that e λs /λs 3 if H is large enough Therefore if follows from 4 that the integral in 3 equals 6 πi s T Res=c 3 π E L X s X e x s/ dx πh s λs N ds λs s + E where E X N N 3 + 3N T λt c X ex log X 5 log log X Choosing λ = 0/T and N = [A log log H] where A > is a constant imlies that 7 E A H log H A Furthermore extending the main term of 6 to 8 c+i πi + O A = 3 π E X 3 π E L X s X E L X c X N 3 λt I cλn πh x L X dx shows that this integral equals e x s/ dx πh s λs λs N + O A H log H A Now it follows from Lemma that for any x X we have πh = if xl X < πh I cλn x L X [0 ] if πh xl X e λn πh = 0 if xl X > πhe λn Thus we obtain 9 X E πh I cλn x L X dx = E = E N ds s π H H min L X X e λn + O L X π H H min L X X log log H 3 + O log H

5 THE NUMBER OF IMAGINARY QUADRATIC FIELDS WITH A GIVEN CLASS NUMBER 5 Finally using Proosition of [] which states that the robability that L X π /6e γ τ is ex e τ C /τ + o for some exlicit constant C we obtain π H E min L X X = π H E L X H + O A log H A = π ζ H ζ3 H + O A log H A Combining this estimate with equations and noting that He λn H H log log H 3 / log H comletes the roof Proof of Theorem Let X T and c be as in the roof of Theorem Let Dx = { x : 3 mod 4} Then similarly to one has see equation 36 of [] Fh = H + O A log H A h odd DX h H Let {Y} be indeendent random variables taking the values and with equal robabilities / and define L Y = Y To obtain the authors of [] rove that assuming GRH see Theorem 33 of [] we have L χ z = Dx E L Y z + O ɛ x /+ɛ Dx uniformly for all comlex numbers z such that z log x/500log log x where Ls χ is the Dirichlet L-function attached to the Kronecker symbol χ = Then by artial summation together with Dirichlet s class number formula they deduced that see [] 9 X 0 h s = π s E L Y s x s/ d Dx + O ɛ X /+ɛ DX for all comlex numbers s with Res = c and s T The roof of Theorem then follows along the same lines of the roof of Theorem by using 0 instead of 4 References [] A Granville and K Soundararajan The distribution of values of L χ d Geom Funct Anal no [] S Holmin N Jones P Kurlberg C McLeman K L Petersen Missing class grous and class number statistics for imaginary quadratic fields Prerint 8 ages arxiv:

6 6 YOUNESS LAMZOURI [3] K Soundararajan The number of imaginary quadratic fields with a given class number Hardy-Ramanujan J [4] T Tatuzawa On a theorem of Siegel Ja J Math Deartment of Mathematics and Statistics York University 4700 Keele Street Toronto ON M3JP3 Canada address: lamzouri@mathstatyorkuca