The distribution of spacings between quadratic residues(7)

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

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2.Thepaircorrelation-asketch

Inordertoexplaintheproofofourmaintheorem1,wegiveanoverviewoftheargumentinthespecialcaseofthepaircorrelationfunction.

Letqbeanodd,square-freenumberwithω(q)primefactors,andIaninterval,notcontainingtheorigin.De neasintheintroductionthepaircorrelationfunction

1 R2(I,q)=N(h,q)Nh∈sI∩Z

ω(q)whereNisthenumberofsquaresmoduloq,s=q/N=2/σ 1(q) 1istheirmeanspacing,σ 1(q)=p|q(1+),andN(h,q)isthenumber

ofsolutionsinsquaresmoduloqoftheequation

y1 y2=hmodq

WewillsketchaproofthatR2(I,q)→|I|asω(q)→∞(|I|beingthelengthoftheinterval).InfactwehavethemorepreciseresultTheorem3.Forqodd,square-freewehaveforall >0

R2(I,q)=|I|+O(s 1+ )

Herearethemainstepsintheargument:

Step1: BytheChineseRemainderTheorem,N(h,q)=p|qN(h,p)isaproductoverprimesdividingq.Byelementaryconsiderations,oneseesthat

p+a(h,p)(2.1)N(h,p)= (h,p)4

witha(h,p)=O(1)and 0p|h (h,p)=1+δ(h,p),δ(h,p)=1p|h

q (h,q) a(h,c)(2.2)N(h,q)=4ω(q)cc|q witha(h,c):=p|ca(h,p) c and (h,q)=p|q (h,p).

Step2:

Wedecompose (h,q)= (h,c) (h,q)andrewrite (h,q)ascc q (h,)=(1+δ(h,p))=δ(h,g)cqqp|g|Fromthisweseethat

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