The distribution of spacings between quadratic residues(20)

Abstract. We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among oth

16¨KURLBERGANDZEEV´RUDNICKPAR

Proof.Wehavethat

1Rr(C,q)=N h∈sC∩Zr 1N(h,q)

BytheChineseRemainderTheorem,

N(h,q)=N(h,p)

p|q

Werewriteformula(4.3)intheform

N(h,p)=

where

(h,p)=2r reff(h)

Thuswe nd

(5.3)p+a(h,p) (h,p)2r

q q (h,q) a(h,c)qa(h,c) (h,c)=)N(h,q)= (h,2rω(q)c2rω(q)ccc|qc|q

Inserting(5.3)wegetaformulaforRr(C,q):RecallingthatN=q/s,

s 1 q(5.4)Rr(C,q)=rω(q) (h,)a(h,c) (h,c)2ch∈sCcc|qq)=Nextweusetheexpression(4.19)for (h,p)towrite (h, p|q/c (h,p)intheform

q(p)(p)(5.5) (h,)=λ()δ(h,)=λ(G)δ(h,G)cq(p)(p)p|GG= p|qG(p),oneforwherethesumisoveralltuplesofset-partitionsG= p|q

eachprimedividingq,andweputforeachsuchtupleG λ(G):=λ((p))

p|q

and

δ(h,G):=

p|q

1h∈HG(p)(p)δ(h,)=0otherwisemodpforallp|q

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