The distribution of spacings between quadratic residues(12)

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

10¨KURLBERGANDZEEV´RUDNICKPAR

4.Theprimecase

Letp>2beaprime.Forh=(h1,...hr 1)∈(Z/pZ)r 1,wede neNr(h,p)tobethenumberofsolutionsinsquaresyimodp(includingyi=0)ofthesystem

(4.1)yi yi+1=himodp,1≤i≤r 1

Thisnumberdependscruciallyonthenumberofdistinctyj.Foreachh=(h1,...,hr 1),wede nereff(h)tobethenumberofdistinctyj(notnecessarilysquares)satisfyingthesystem(4.1).Sincethesolu-tionsofthehomogeneoussystemyi yi+1=0modparespannedby(1,...,1),reff(h)iswell-de ned(independentoftheparticularsolu-tionyof(4.1)).

Wede nerootsσij(h),1≤i<j≤rby

(4.2)σij(h)=j 1

k=ihk

1sothatσi,i+1(h)=hi,σij=j

k=iσk,k+1.Thesolutionsof(4.1)areall

distinctofandonlyifσij(h)=0,foralli<j,since

yi yj=j 1

k=iyk yk+1=j 1 k=ihk=σij(h)

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