The distribution of spacings between quadratic residues(25)

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

20¨KURLBERGANDZEEV´RUDNICKPAR

WebeginbynotingthatthenumberofGwithsupp(G)=gisO(g ),i.e.,

(6.6)1 g ,

supp(G)=g

Sincewesumoversupp(G) sr(r 1)/2in(6.5),wehavesupp(G) s ,andthus(6.5)isboundedby1 1/2+ csc|q

g sr(r 1)/2g|qg s 1+ c|qc 1/2+

g sr(r 1)/2g|q1.

ByLemma18,thenumberofdivisorsofq/cwhicharelessthansr(r 1)/2isatmosts ,sothistermisboundedby

1+ sc 1/2+ .

c|q

Since

c|qc 1/2+ =(1+p|qp ) (1+1) s1/2 1p|q

thecontributionof(6.5)isatmostO(s 1+ ).

Itnowremainstobound(6.4).We rstconsiderthetermsforwhichcsupp(G)>s.Now,disc(G)≥supp(G),soifcsupp(G)>sthencertainlycdisc(G)>s,andsumofthecorrespondingtermsin(6.4)isboundedby1 1/2+ csc|q supp(G)|q

csupp(G)>ssupp(G) ssupp(G)

1,1 gq= c|qc 1/2+ g|cg>s 11 qg supp(G)=g1 c|qc 1/2+ g|cg>sby(6.6).Changingvariabletod=cg,whichisadivisorofqsatisfyingd>s,thisisboundedby

c 1/2+ 1 =c1/2+

1 1 (d/c)dd|qc|d

d>sd|qd>sc|d

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