Pair correlations of logarithms of complex lattice points

We study the correlations of pairs of complex logarithms of $\mathbb Z$-lattice points in the complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi orbifold $\mathrm{PSL}_2(\mathbb Z[i]) \backslash\mathbb H^3_{\mathbb R}$.


Introduction
When studying the asymptotic distribution of a sequence of finite subsets of R, finer information is sometimes given by the statistics of the spacing (or gaps) between pairs or k-tuples of elements, seen at an appropriate scaling.These problems often arise in quantum chaos, including energy level spacings or clusterings, and in statistical physics, including molecular repulsion or interstitial distribution.See for instance [Mon,Ber,RS,BocZ,MaS,LS,HoK,PP3].This paper may be seen as a complex version of our paper [PP2] where we study the pair correlation of logarithms of pairs of natural integers, though new phenomena occur, including the necessity to take limits of the underlying spaces, as we now explain.
The general setting for our study may be described as follows.Let E be an abelian locally compact group.Let F " pF N , ω N q N PN be a sequence of finite subsets F N of E, endowed with a weight function ω N : F N Ñ s 0, `8 r (or multiplicity function when its values are positive integers).When studying the asymptotic distribution of differences of elements of F N , looking at them at various scalings is often desirable.As explained by Gromov (see for instance [Gro1]), scaling some metric space (unless this space has a nice family of homotheties, as the Euclidean space R n does) sometimes requires to change the space, especially at the limit.We thus introduce a sequence pE N q N PN of abelian locally compact groups converging for the pointed Hausdorff-Gromov convergence to an abelian locally compact group E 8 (see for instance [Gro2]).Let Haar E8 be a Haar measure on E 8 .Let ψ : N Þ Ñ ψpN q be a scaling function, that is, for every N P N´t0u, let ψpN q : E Ñ E N be any map, typically a dilating homeomorphism for appropriate distances, that we think of as "scaling" the space E. Let ψ 1 : N´t0u Ñ r1, `8r be an appropriately chosen function, called a renormalising function.The pair correlation measure of F at time N with scaling ψpN q is the measure on E N with finite support where ∆ z denotes the unit Dirac mass at z in any measurable space.When the sequence of measures pR F ,ψ N q N PN , renormalized by ψ 1 pN q, converges (see Section 3 for background definitions) for the pointed Hausdorff-Gromov weak-star convergence to a measure g F ,ψ Haar E8 absolutely continuous with respect to the Haar measure Haar E8 of E 8 , the Radon-Nikodym derivative g F ,ψ is called the asymptotic pair correlation function of F for the scaling ψ and renormalisation ψ 1 .When g F ,ψ is a positive constant, we say that F has a Poissonian behaviour for the scaling ψ and renormalisation ψ 1 .When g F ,ψ vanishes on a neighbourhood of 0 in E 8 , we say that the pair pF , ψq exhibits a strong level repulsion.The standard level repulsion only requires g F ,ψ to vanish at 0.
Recall that a Z-lattice in C is a discrete (free abelian) subgroup of pC, `q generating C as an R-vector space.Let Λ be a Z-grid in C (or affine (Euclidean) lattice in the terminology of [MaS, EBMV]), that is, a translate Λ " a ` Λ of a Z-lattice Λ in the Euclidean space C for some a P C (modulo Λ), see for instance [AES].We denote by covol Λ " VolpC{ Λq the area of a fundamental parallelogram for Λ.We denote by Sys Λ " min |z| : z P Λ ´t0u ( ą 0 the systole of the Z-lattice Λ. Recall that the complex logarithm is an isomorphism of abelian topological groups log : C ˆÑ E " C{p2πiZq.Given N P N ´t0u and a function ψ : N ´t0u Ñ s0, `8r, we again denote by ψpN q the scaling map from E to E N " C{p2πiψpN qZq defined by z mod 2πiZ Þ Ñ ψpN qz mod 2πiψpN qZ.In Sections 2 and 3, we study the pair correlations of the family of the complex logarithms of grid points L Λ " `LN " tlog z : z P Λ, 0 ă |z| ď N u, ω N " 1 ˘NPN without multiplicities.In order to simplify the statements in this introduction, we only consider power scalings ψ : N Þ Ñ N α for α ě 0, and we denote them by id α .
Theorem 1.1 Let α ě 0 and let Λ be a Z-grid.As N Ñ `8, the normalized pair correlation measures 1 N 4´2α R L Λ , id α N on the cylinder E N " C{p2πiN α Zq converge for the pointed Hausdorff-Gromov weak-star convergence to the measure g L Λ , id α Leb E8 on E 8 " C{p2πiZq if α " 0 and E 8 " C otherwise, with pair correlation function given by The convergence is uniform on every compact subset of Z-grids Λ for the Chabauty topology.
The renormalisation by 1 N 4´2α in Theorem 1.1 is naturally chosen in order for the pair correlation function to be finite.We refer to Theorems 2.2 and 3.1 for more complete versions of Theorem 1.1, with more general scaling functions, as well as for error terms.These error terms, as well as the ones in Theorems 5.1 and 6.1, constitute the main technical parts of this paper.
A standard scaling function in dimension n is by the inverse of n-root of the average volume gap, which is the quotient of the volume of the ball of smallest radius containing F N by the number of elements in F N .See for instance [Mon, RS, BocZ, LS, HoK], though these references are in dimension n " 1.For the family L Λ , this average volume gap is equivalent to pln N q 2 N 2 , up to a positive multiplicative constant.As we shall see in Theorem 3.1, the corresponding scaling function ψ : N Þ Ñ N ln N gives, as for ψ : N Þ Ñ N α for 0 ă α ă 1 in the above theorem, a Poissonian behavior (see also [Van, EBMV] for a similar behaviour).
There is a phase transition from a Poissonian behaviour when 0 ă α ă 1 to a total loss of mass at α ą 1.In fact, the support of the measure itself converges to infinity for α ą 1.
The transition occurs at the linear scaling, where an exotic pair correlation function g L Λ , id appears, which has a discontinuity along every circle (centered at 0) through a grid point.
Since g L Λ , id pzq vanishes when z P B p0, Sys Λ q, the pair pL Λ , idq exhibits a strong level repulsion.Hence g L Λ , id has near z " 0 a behaviour similar to the case α ą 1.Note that (see Lemma 2.1) g L Λ , id pzq converges to π 2 covol 2 Λ when z goes to 8, corresponding to the Poissonian behaviour of 0 ă α ă 1.
The figure below gives the graph of the pair correlation function g L Λ , ψ of L Λ for the Z-grid (which is a Z-lattice) Λ " Λ " Zris of the Gaussian integers at the linear scaling ψ : N Þ Ñ N in the ball of center 0 and radius 5.The blue lines on the bounding box represent the limit 2 at `8 of g L Λ , ψ .We refer to the end of Section 3 for further illustrations, also in the case of the Eisenstein integers.
We now give some existence results of pair correlation functions of logarithms of lattice points with weights, restricting to integral lattices with an arithmetic weight motivated by geometric applications.Let K be an imaginary quadratic number field K, with discriminant D K , whose ring of integers O K is principal.We fix a nonzero ideal Λ in O K , and we denote by ϕ K : O K ´t0u Ñ N the Euler function a Þ Ñ Card `pO K {aO K q ˆ˘of K.In the products below, p runs over the prime ideals of O K .The following result describes the asymptotic behaviour of the pair correlation measures associated with the family (2) Λ ,1 N on the constant cylinder E " C{p2πiZq, renormalized to be probability measures, weak-star converge to the probability measure g L ϕ K Λ ,1 Leb E , with pair correlation function independent of Λ given by g (2) As N Ñ `8, the normalized pair correlation measures 1 , id 1 N on the varying cylinder E N " C{p2πi N Zq converge for the pointed Hausdorff-Gromov weak-star convergence to the measure We refer to Theorems 5.1 and 6.1 for more complete versions of Theorem 1.2, including possible congruence restrictions, and for error terms.The proof of Theorem 1.2 (2) uses Theorem 1.3 of [PP4] that describes the asymptotic behaviour in angular sectors in C for the Euler function of K.For the readers convenience, we briefly review these results in Section 4. In order to simplify the treatment, we only consider the constant and linear scaling in Theorem 1.2.The pair correlation functions at the linear scaling are radially symmetric by Theorem 1.2 (2).The figure above compares the radial profiles of the pair correlation functions g L ϕ K Λ , id 1 for K " Qpiq and Λ " O K " Zris in blue and K " Qpi ?3q and Λ " O K " Zr 1`i ? 3 2 s in orange.The radial profiles of the pair correlation functions converge to a limit `1 Nppq 2 pNppq 2 ´2q ȃt infinity, where p ranges over the prime ideals of O K by Proposition 6.5.This limit is approximately 0.346 for the blue curve and 0.634 for the orange one.
The radial profiles of the pair correlation functions in the weighted and unweighted cases are similar to certain radial distribution functions in statistical physics, see for example [ZP, Sect. II], [SdH,Fig. 7], [Cha,page 199] or [Boh,page 18].See also [Mat `].The unfolding technique (see for instance [Boh,p. 14] and [MaS,§3,§5]), though guiding the very first step of the proofs of Theorem 1.1 and 1.2, falls short of giving a complete answer, in particular when varying the scalings and weights and for the error term analysis.
As explained in Section 7, our motivation for introducing the weights by the Euler function comes from hyperbolic geometry.We prove in Proposition 7.1 that the pair correlation measures of the lengths (counted with multiplicity) of the common perpendiculars between the maximal Margulis cusp neighbourhood and itself in the (one-cusped) Bianchi orbifold PSL 2 pO K qzH 3 R are closely related to the pair correlation measures of the weighted family L ϕ K O K .Theorem 1.2 implies a pair correlation result for the lengths of common perpendiculars of cusps neighborhoods in the Bianchi orbifold PSL 2 pO K q{H 3 R , see Corollary 7.2 for a precise statement and a version with congruences.
Notation.We introduce here some of the notation used throughout the paper.
All our measures are Borel, positive, regular measures on locally compact spaces.The pushforward of a measure µ by a mapping f is denoted by f ˚µ, and its total mass by }µ}.We denote by Leb K the restriction of Lebesgue's measure of C to any Borel subset K of C. For every smooth manifold with boundary Y and every k P N, we denote by C k c pY q the set of complex-valued C k functions with compact support on Y .
We equivariantly identify the space Grid 2 of Z-grids in the real Euclidean plane C, endowed with the Chabauty topology and the affine action of GL 2 pRq ˙R2 with the homogeneous space pGL 2 pRq ˙R2 q{pGL 2 pZq ˙Z2 q, which smoothly fibers by the map a ` Λ Þ Ñ Λ over the space of Z-lattices GL 2 pRq{ GL 2 pZq, with fibers the elliptic curves C{ Λ.
We will use the following indexing sets in Sections 2, 3 and 5. Given a Z-grid Λ, for every N P N ´t0u, let

Pair correlation of grid points without weight or scaling
In this section, we work on the constant cylinder E " C{p2πiZq, endowed with its quotient Riemann surface structure, with its quotient additive abelian locally compact group structure, and with its Haar measure d Leb E px 1 `iy 1 q " dx 1 dy 1 where x 1 P R and y 1 P R{p2πZq.We endow the multiplicative group C ˆwith its Riemann surface structure as an open subset of C and with the restriction of the Lebesgue measure Leb C of C. The logarithm map log : C ˆÑ E defined by ρ e iθ Þ Ñ ln ρ `iθ is a biholomorphic group isomorphism, whose inverse is the exponential map z 1 " x 1 `iy 1 Þ Ñ exppz 1 q " e x 1 e iy 1 .The real part map Re : E Ñ R defined by x 1 `iy 1 Þ Ñ x 1 is a smooth (trivial) fibration, and Note that for every z P C ´t0u, we have lnp|z| 2 q " 2 Replog zq . (5) Since d Leb C pρe iθ q " ρ dρ dθ, we have Let Λ " a ` Λ be a Z-grid.We choose a Z-basis pv 1 , v 2 q of Λ such that the (weak) fundamental parallelogram for the action of Λ on C has smallest diameter.We then denote by diam Λ " diampF Λ q " maxt|v 1 `v2 |, |v 1 ´v2 |u the diameter of F Λ , which is the length of a longest diagonal of the parallelogram F Λ .We denote by covol Λ " VolpC{ Λq " AreapF Λ q " | detpv 1 , v 2 q | the area of the elliptic curve C{ Λ for the measure induced by the Lebesgue measure on C, or the area of the parallelogram F Λ (which does not depend on the choice of the Z-basis pv 1 , v 2 q of Λ).We will use several times the following well known result, having a more precise error term that we won't need, and we only give a proof in order to explicit the dependence on the parameter Λ.
Lemma 2.1 For every k P N, as x Ñ `8, we have Proof.The case k " 0 is the standard Gauss counting result of lattice points in discs.With A x " tp P Λ : |p| ď xu and B x " Ť pPAx pp `F Λ q, so that AreapB x q " CardpA x q AreapF Λ q, we have so that the result for k " 0 follows by computing the area of the two above discs.
Assume now that k ě 1.We use Abel's summation formula a n ˘f 1 ptq dt applied to the numerical sequence `an " Cardtp P Λ : n ´1 ă |p| ď nu ˘nPN and to the smooth functions f : r1, `8r Ñ R defined by t Þ Ñ t k or t Þ Ñ pt ´1q k .Using the case k " 0, the result when k ą 0 then follows from the estimates For every N P N ´t0u, the (not normalised) pair correlation measure of the logarithms of nonzero grid points in Λ, with trivial multiplicities and with trivial scaling function, is the finite measure on the cylinder E defined by Note that for every k P N ´t0u, we have I kN,kΛ " I N,Λ and ν kN,kΛ " ν N,Λ .
Theorem 2.2 As N Ñ `8, the measures ν N on E, renormalized to be probability measures, weak-star converge to the measure absolutely continuous with respect to the Haar measure Leb E , with Radon-Nikodym derivative the function g L Λ ,1 : z 1 Þ Ñ 1 2π e ´2 |Repz 1 q| , which is independent of Λ. Besides, the convergence is uniform on every compact subset of Λ in the space of Z-grids Grid 2 .Furthermore, for every f P C 1 c pEq, we have This result implies the case α " 0 of Theorem 1.1 in the introduction, since we will prove in Formula ( 14) that lim N Ñ`8 Remark 2.3 Theorem 2.2 is still valid if we allow n " m in the definition of the index set I N (this correspond to removing the condition p ‰ q in the definition below of J q ), see also Remark (2) in [PP3,§3] for a general argument.We will use this comment in Corollary 2.4 and 2.5, as well as in Corollary 7.2.
Proof.For all N P N and q P Λ with 0 ă |q| ď N , let J q " tp P Λ : 0 ă |p| ď |q|, p ‰ qu and ω q " which is a finitely supported measure on the closed unit disc D of C. Note that the assumptions 0 ă |p| and 0 ă |q| are automatic when 0 R Λ, that is, when Λ is not a Z-lattice.As q Ñ `8, by Equation ( 7) with k " 0, its total mass, which is nonzero since ´q P J q , satisfies for some Op¨q uniform in Λ.Note that we need to remove 0 if 0 P Λ and q from the counting of Equation ( 7), but this is taken care of by the above Op¨q.We hence have for some Op¨q uniform in Λ.We denote by ω q " ωq }ωq} the renormalisation of ω q to a probability measure on D.
Note that the symmetric difference pD ´Cq q q Y p Cq q ´Dq is contained in the union of the annulus Bp0, 1 Also note that By the mean value inequality, for all p P J q and z P p`F Λ q , we have Therefore In particular, as q Ñ `8, we have ω q which is a finitely supported measure on D. By Equations ( 10) and ( 7) with k " 2 and k " 1, its total mass is equal to It follows that 1 Let f P C 1 pDq.By Equations ( 11), ( 12), and (7) with k " 1, we have, as N Ñ `8, Let E ˘" p˘r0, 8r `iRq{p2πiZq so that E " E ´Y E `.Note that log : D ´t0u Ñ E ánd log : C ´D Ñ E `are homeomorphisms.Let us define a measure with finite support on E ˘by ν N " ´, and }ν Ń } " }µ Ń }.For every f P C 1 c pE ´q, we have f ˝log P C 1 c pD ´t0uq (hence f ˝log may be extended to a C 1 function on D which vanishes on a neighborhood of 0) and, by Equations ( 13) and (6), Let sg : E Ñ E be the horizontal change of sign map x 1 `iy 1 Þ Ñ ´x1 `iy 1 , which maps E to E `.Then ν Ǹ " sg ˚νŃ and ν N " ν Ń `νǸ .Since E ´X E `has zero measure for the Haar measure Leb E and since }ν N } " 1 2 }ν N } `Opdiam Λ N 3 q, the last claim of Theorem 2.2 follows.Note that, as needed just after the statement of Theorem 2.2, as N Ñ `8, we have The first claim follows by approximating continuous functions with compact support by C 1 ones.The uniformity of the convergence on compact subsets of lattices follows from the uniformity of the functions Op¨q and the fact that the constants covol Λ and diam Λ vary in a compact subset of s0, `8r when Λ varies in a compact subset of Grid 2 .l The following picture illustrates the convergence in Formula ( 8) in Theorem 2.2 when Λ " Λ " Zris is the ring of Gaussian integers and N " 20, using as horizontal coordinates px 1 , y 1 q P E with x 1 P R and y 1 P r´π, πr.A smooth histogram scaled to a probability density is displayed in gold, and the limiting distribution in grey.

Arithmetic applications.
(1) Let K be an imaginary quadratic number field, with discriminant D K , ring of integers O K and Dedekind zeta function ζ K .We denote by I K the semigroup of nonzero (integral) ideals of the Dedekind ring O K (with unit O K ).We denote by NpIq " CardpO K {Iq the norm of an ideal I P I K , which is completely multiplicative.The norm of a P O K ´t0u is It coincides with the (relative) norm N K{Q paq of a (see for instance [Nar]), and in particular is equal to In particular, the Gauss ball counting argument of Equation ( 7) with k " 0 and x " ?N gives, as Hence Theorem 2.2 implies the existence of a pair correlation function (independent of m) for the family of the complex logarithms of nonzero elements of m without weights or scaling, as stated in the following result, using Remark 2.3.
Corollary 2.4 For every f P C 1 c pEq, as N 1 Ñ `8, we have (2) For every positive integer d, let r 2,d : N ´t0u Ñ N be the integral function where is the number of integral solutions of the Diophantine equation x 2 `d y 2 " n, for every n P N. In particular, if d " 1, then r 2,d " r 2 is the well known function counting the sum of two squares representatives of a given positive integer (see for instance [Cox] or [HaW,Sect. 16.9]).The following result proves that the map on R is the pair correlation function for the family of the logarithms of the nonzero natural integers, without scaling but with weights given by r 2,d (removing the zero weights).Other weights have been considered in [PP2] (including the one given by the Euler function ϕ).Note that the following corollary holds also when r 2,d pnq is replaced by the number of representations of n by the norm form of any imaginary quadratic number field, evaluated on any order of their ring of integers (as for instance the norm form px, yq Þ Ñ x 2 ´xy `y2 of the Eisenstein integers).
Corollary 2.5 As N Ñ `8, we have . By the linearity of p2 Req ˚and 2 Re, and by Equation ( 5), for every N P N ´t0u, we have The pushforward map p2 Req ˚preserves the total mass and is continuous for the weak-star topology, since the map 2 Re : E Ñ R is proper.Hence by Formulas ( 8) and ( 4), we have Corollary 2.5 follows.l 3 Pair correlation of grid points with scaling without weight In this section, we study the pair correlations of complex logarithms of grid points at various scaling.We fix a positive scaling function ψ : N ´t0u Ñ s0, `8r such that lim `8 ψ " `8.We consider a normalisation function ψ 1 : N ´t0u Ñ s0, `8r depending on ψ, which will be made precise later on, but which in most cases will not yield the renormalisation to a probability measure.
We will work on the following family pE N q N PN´t0u of varying cylinders.For every N P N´t0u, we consider E N " C{p2πi ψpN q Zq, endowed with its quotient Riemann surface structure and its quotient additive abelian locally compact group structure.Since a real number θ is well defined modulo 2πZ if and only if ψpN qθ is well defined modulo 2πψpN qZ, the scaled logarithm map ψpN q log : C ˆÑ E N defined by ρ e iθ Þ Ñ ψpN q ln ρ `iψpN qθ is a biholomorphic group isomorphism, whose inverse is the rescaled exponential map ψpN q e i y 1 ψpN q .The real part map Re : C Ñ R induces a map again denoted by Re : E N Ñ R, which is a trivial smooth bundle map with fibers iR{p2πiψpN qZq, such that for every z P E, RepψpN qzq " ψpN q Repzq . (16) We consider also E N as a pointed metric space, with distance the quotient of the Euclidean distance on C and base point its (additive) identity element 0. Note that E N is a proper metric space.As lim `8 ψ " `8, for every R ą 0, there exists N R P N ´t0u such that for every N ě N R , the closed ball Bp0, Rq in C injects isometrically by the canonical projection p N : C Ñ E N .Hence the sequence pE N q N PN´t0u of proper pointed metric spaces converges to the proper metric space C pointed at 0 for the pointed Hausdorff-Gromov convergence (see [Gro2] for background).
Any function f P C 0 c pCq defines for all N large enough a function f N P C 0 c pE N q as follows.Let R f ą 0 be such that the support of f is contained in Bp0, R f q.Then for every N ě N R f , the function f N P C 0 c pE N q is the function which vanishes outside p N pBp0, R f qq and coincides with We say that a sequence pµ N q N PN´t0u of measures µ N on E N converges to a measure µ 8 on C for the pointed Hausdorff-Gromov weak-star convergence if for every f P C 0 c pCq, the sequence pµ N pf N qq N ěN R f converges in C to µ 8 pf 8 q (see [Gro2, Chap. 3 1 2 ] for background).We again use the symbol á in order to denote this convergence.
Let Λ be a Z-grid in C. For every N P N ´t0u, the (not normalised, empirical) pair correlation measure of the complex logarithms of points in Λ at time N with trivial weights and with scaling ψpN q is the measure with finite support in E N defined by and the normalized one is 1 N on E N , normalized by ψ 1 pN q as given below, converge for the pointed Hausdorff-Gromov weak-star convergence to a measure g L Λ ,ψ Leb C on C, absolutely continuous with respect to the Lebesgue measure on C, with Radon-Nikodym derivative the function The convergence is uniform on every compact subset of Z-grids Λ in the space Grid 2 .Furthermore, if λ ψ ‰ 0, `8, for all A ě 1 and f P C 1 c pCq with support contained in Bp0, Aq, we have Note that the pair correlation function g L Λ ,ψ depends on Λ but is independent of a.The above result shows in particular that renormalizing to probability measures (taking ) is inappropriate, as the limiting measure would always be 0. We will see during the proof that the above result implies the cases α ą 0 of Theorem 1.1 in the introduction.
The fact that g L Λ ,ψ vanishes when λ ψ " `8 means that the sequence of measures `1 ψ 1 pN q R L Λ ,ψ N ˘NPN´t0u on pE N q N PN´t0u has a total loss of mass at infinity.For error terms when λ ψ " `8 and λ ψ " 0, see respectively Equation (36) and Equation (39).
Proof.Let Λ " a ` Λ be a Z-grid in C. We may assume that a P F Λ .Let N P N ´t0u.Let E N " p˘r0, 8r `iRq{p2πi ψpN q Zq (which contains the base point 0) so that E N " E Ń YE Ǹ .Note that pE N q N PN´t0u converges for the pointed Hausdorff-Gromov convergence to the closed halfplane C ˘" ˘r0, 8r `iR and that C ´X C `has measure 0 for any measure absolutely continuous with respect to the Lebesgue measure on C. Note that if f P C 1 c pC ˘q, then for N large enough, we have We will thus only study the convergence of the measures 1 N on E Ǹ , and deduce the global result by the symmetry of g L Λ ,ψ under sg.
For every p P Λ ´t0u, let and let ω p, N " ÿ qPJ p, N ∆ ψpN q p q and µ Ǹ " Note that ω p, N is a measure on C with finite support, which vanishes if |p| ą 2N by the triangle inequality, hence µ Ǹ is also a measure on C with finite support.
Proof.We may assume that |p| ď 2N .Note that J p, N is the finite set of nonzero grid points in the intersection of the disc Bp´p, N q of radius N centered at ´p with the closed halfplane containing 0 with boundary the perpendicular bisector of 0 and ´p (see the picture below).
Since r C p, N is contained in a halfdisc of radius N and contains the complement in this halfdisc of its intersection with a rectangle of length 2N and height |p| 2 , we have By a Gauss counting argument similar to the one in the proof of Equation ( 7) with k " 0, we have The lemma follows.l Lemma 3.3 For every A ą 0 and for every f P C 1 c pC `q with support contained in Bp0, Aq, as N Ñ `8 and uniformly on Λ varying in a compact subset of Grid 2 , we have Proof.Let A and f be as in the statement of this lemma.Note that since ψpN q ą 0 and by Equation ( 5), for every pm, nq P I N , we have pm, nq P I Ǹ , that is |n| ď |m|, if and only if ψpN q log m´ψpN q log n P E Ǹ .Hence by the change of variable pp, qq Þ Ñ pm " p`q, n " qq (which is a bijection from Λ ˆΛ to Λ ˆΛ), we have By the assumption on the support of f , if an index pp, qq contributes to the above sum, then RepψpN q logpp `qq ´ψpN q log qq ď A. Hence by Equations ( 16) and ( 5), we have ln ˇˇ1 `p q ˇˇď A ψpN q , which tends to 0 as N Ñ `8, since lim `8 ψ " `8.In particular, using the assumption on q, we have so that ˇˇp q ˇˇă 1 if N is large enough.This allows to use the principal branch, again denoted by log, of the complex logarithm in the open ball of center 1 and radius 1.By the analytic expansion of this branch, we have The mean value theorem hence implies that f N pψpN q logpp `qq ´ψpN q log qq " f `ψpN q logp1 `p q q " f pψpN q By Lemma 3.2 and Equation ( 7) with k " 0, we have Similarly, if an index pp, qq contributes to the sum then Equation ( 23) holds.By summing Equation ( 24) on the set of elements pp, qq P Λ ˆΛ such that 0 ă |q| ď |p `q| ď N and |p| " O `AN ψpN q ˘, and by using Equation ( 25), Lemma 3.3 follows.l Let us now study the convergence properties (after renormalization) of the measures ω p, N and of their sums µ Ǹ as N Ñ `8.We assume in what follows that |p| ă N (which is possible if N is large enough since we will have |p| " O `AN ψpN q ˘).Let ι : By the equation on the left in Formula (20), we have When q varies in J p, N , as seen in the proof of Lemma 3.2, the above Dirac masses are exactly at the nonzero points of the Z-grid Λ p,N " 1 ψpN qp Λ that belong to the set By Equation ( 21), the set r Y p,N is the intersection of the disc Bp´1 ψpN q , N ψpN q|p| q with the closed halfplane containing 0 with boundary the perpendicular bisector of 0 and ´1 ψpN q .Let us define The symmetric difference of r Y p,N and Z p,N , that we denote by r YZ p,N , is contained in the union of the rectangle ‰ and the half-annulus so that, as in the proof of Lemma 3.2, the symmetric difference of Y p,N and r Y p,N has area O `Ndiam Λ ψpN q 2 |p| 2 ˘.The symmetric difference of Y p,N and Z p,N , that we denote by YZ p,N , hence has area Leb C pYZ p,N q " O `Npdiam Λ `|p|q ψpN q 2 |p| 2 ˘.In particular, for every φ P C 1 c pC `´t0uq, since Z p,N Ă Bp0, N ψpN q|p| q and Y p,N Ă Bp0, By Equations ( 30), ( 22), ( 27) and ( 28), by the mean value theorem and by Lemma 3.2, we have ˇˇż Hence by Equation ( 31), we have Let f P C 1 c pC `´t0uq with support contained in Bp0, Aq.Note that since the support of f is contained in Bp0, Aq.The change of variable by ι in the integral of Equation (32) applied with φ " f ˝ι, together with Equations ( 29) and ( 26), hence give For every z P C `´t0u, let Note that if z and N are fixed, then for |p| large enough, we have |z| ă ψpN q|p| N , thus the above sum has only finitely many nonzero terms.Let θ N p0q " 0.
Note that θ N pzq vanishes if and only if |z| ă ψpN q Sys Λ N , by the definition of the systole of Λ.
As seen in the proof of Lemma 3.3, the only elements p P Λ that give a nonzero contribution to the sum ř pP Λ´t0u ω p, N pf q satisfy p ‰ 0 and |p| " O `AN ψpN q ˘.By Equation ( 7) with k " 0, we have Thus, by the right equality in Formula (20), we have Case 1.Let us first assume that λ ψ " `8, that is, lim N ψpN q " 0. For every A ě 1, if N is large enough (uniformally on Λ varying in a compact subspace of Grid 2 , since then Λ varies in a compact subspace of the space of Z-lattices, on which the systole function Λ Þ Ñ Sys Λ has a positive lower bound), then for every z P Bp0, Aq, we have θ N pzq " 0 by Equation ( 33), and µ Ǹ pf q " 0 by Formulas ( 20) and ( 34), since the sum defining µ Ǹ pf q is an empty sum.Thus, whatever the (positive) normalizing function ψ 1 is, we have a total loss of mass at infinity : Assume that the renormalizing function ψ 1 is such that ψpN q 3 ψ 1 pN q tends to 0 as N tends to 8, for instance ψ 1 " ψ, as assumed in the first case of Equation ( 17).Note that if ψpN q " N α with α ą 1, then we indeed have λ ψ " `8 and if ψ 1 pN q " N 4´2α as in the statement of Theorem 1.1, we do have lim N Ñ`8 N 4 ψpN q 3 ψ 1 pN q " 0. Together with Lemma 3.3, the above centered formula proves Formula (18) when λ ψ " `8, with a convergence which is uniform on every compact subset of Λ in Grid 2 , as well as the case α ą 1 in Theorem 1.1.Furthermore, if follows from the error term in Lemma 3.3 that for every f P C 1 c pCq with support contained in Bp0, Aq, as N Ñ `8 and uniformly on Λ varying in a compact subset of Grid 2 , we have Case 2. Let us now assume that λ ψ " 0, that is, lim N Ñ`8 ψpN q N " 0. For all z P C `´t0u, by Equations ( 33) and (7) for k " 2, we have In particular, if |z| ě , this proves that the function ψpN q 4 N 4 θ N is uniformly bounded on C `´t0u, and pointwise converges to the constant function π 2 covol Λ .Hence by Equation ( 35) and by the Lebesgue dominated convergence theorem, we have, with a convergence which is uniform on every compact subset of Λ in Grid 2 , More precisely, for every A ě 1, for every f P C 1 c pC `´t0uq with support in Bp0, Aq, and for every Λ in a compact subset of Grid 2 , by Equations ( 35) and (37), using the equality ρ dρ dθ " πA in order to integrate the error term in Equation ( 37), and since ψpN q ď N for N large enough, we have If ψ 1 pN q " N 4 ψpN q 2 as assumed in the second case of Equation ( 17), it follows from Formula (38) and Lemma 3.3 by symmetry that This proves Formula (18) when λ ψ " 0, with a convergence which is uniform on every compact subset of Λ in Grid 2 , as well as the case 0 ă α ă 1 in Theorem 1.1.Furthermore, for every f P C 1 c pCq with support contained in Bp0, Aq, as N Ñ `8 and uniformly on Λ varying in a compact subset of Grid 2 , using the error term in Lemma 3.3 with the fact that that Sys Λ ď diam Λ , we have Case 3. Let us finally assume that lim N Ñ`8 ψpN q N " λ ψ belongs to s0, `8r .We consider the function θ 8 : C Ñ r0, `8r defined by where by convention θ 8 p0q " 0, and replacing p P Λ by p P Λ ´t0u makes no difference.
Note that θ 8 vanishes on the open disc B p0, Sys Λ q, is uniformly bounded and tends to π 2 covol Λ as t Ñ `8 by Equation ( 7) with k " 0. Furthermore, θ 8 is piecewise continuous, with discontinuities along each circle Sp0, |p|q centered at 0 passing through a nonzero lattice point p P Λ. See the picture in the introduction representing the graph of θ 8 when Λ " Λ " Zris and λ ψ " 1.
By Equation ( 33), the sequence of uniformly bounded maps pθ N q N PN converges almost everywhere to θ 8 (more precisely, it converges at least outside Ť pP Λ´t0u Sp0, |p|q).Hence by Equation ( 35) and by the Lebesgue dominated convergence theorem, we have Note that if N is large enough, the left term vanishes if |z| ă λ ψ 2 Sys Λ .Let f P C 1 c pC `q with support in Bp0, Aq.By integration on annuli and Equation (7) with k " 3, we have ˇˇż Hence by Equation ( 35), we have If ψ 1 pN q " ψpN q 2 as assumed in the third case of Equation ( 17), it follows from Formula (40) and Lemma 3.3 by symmetry that This proves Formula (18) when λ ψ ‰ 0, 8, with a convergence which is uniform on every compact subset of Λ in Grid 2 , as well as the case α " 1 in Theorem 1.1 (since if ψpN q " N , then λ ψ " 1 and ψ 1 pN q " ψpN q 2 " N 2 " N 4´2α ).Furthermore, for every f P C 1 c pC `q with support contained in Bp0, Aq, as N Ñ `8 and uniformly on Λ varying in a compact subset of Grid 2 , using Equation ( 41) and the error term in Lemma 3.3 with the fact that that Sys Λ ď diam Λ , we have By symmetry, this concludes the proof of Theorem 3.1.l Let us give a numerical illustration of Theorem 3.1 when Λ " Λ " Zris and ψpN q " N .The following figure shows the points 60 log m ´60 log n contained in the ball of radius 5 centered at 0 for pm, nq P I 60 .
The second figure shows an approximation (given by Mathematica and its smoothing process) of the pair correlation function g L Λ ,ψ computed using the empirical measure In this short section, we recall the notation and statements of [PP4] that we will use in Sections 5 and 6.
Let K be an imaginary quadratic number field (with D K , O K , ζ K , I K , N the notation introduced before Corollary 2.4).We assume in Sections 4, 5 and 6 that O K is principal (or equivalently factorial (UFD)).This implies" see for instance [Nar], that D K P t´4, ´8, ´3, ´7, ´11, ´19, ´43, ´67, ´163u.For all I, J P I K , we write J | I if I Ă J, we denote by pI, Jq " I `J the greatest common ideal divisor of I and J, and by IJ the product ideal of I and J.
We denote by ϕ K : I K Ñ N the Euler function of K, defined (see for instance [Nar,page 13]) equivalently by where, here and thereafter, p ranges over the prime ideals of O K .For every a P O K ´t0u, we define ϕ K paq " ϕ K paO K q.
We first give a version in angular sectors of the Mertens formula on the average of the Euler function that will be needed in the proof of Theorem 5.1.For all z P C ˆ, θ P s0, 2πs and R ě 0, we consider the truncated angular sector Cpz, θ, Rq " ρ e it z : Note that for every z 1 P C ˆ, we have It is important that the function Op¨q in the following result is uniform in m, z and θ.For every m P I K , let Lemma 4.1 (A Sectorial Mertens formula) Assume that K is imaginary quadratic with O K principal.For all m P I K , z P C ˆand θ P s0, 2πs, as x Ñ `8, we have Proof.See [PP4,Thm. 1.2].l We now give an asymptotic formula for the sum in angular sectors of the products of shifted Euler functions with congruences, which is used in the proof of Theorems 5.1 and 6.1.When K " Q (the sectorial restriction is then meaningless), this formula is due to Mirsky [Mir,Thm. 9,Eq. (30)] without congruences, and to Fouvry [PP2,Appendix] with congruences.
For all z P C ˆ, θ P s0, 2πs, k P O K , m P I K and x ě 1, let where (46) For instance, if m " O K then by [PP4, Eq. ( 18)], we have Since it will be useful in Section 6, by [PP4,Lem. 4.2], we have Theorem 4.2 (A Sectorial Mirsky Formula) Assume that K is imaginary quadratic with O K principal.There exists a universal constant C ą 0 such that for all k P O K , m P I K , z P C ˆ, θ P s0, 2πs and x ě 1, we have Proof.See [PP4,Thm. 4.1 and Lem. 4.2].l

Pair correlation of integral lattice points with Euler weight and no scaling
In this section, we fix an imaginary quadratic number field K whose ring of integers O K is principal.We fix a (nonzero integral) ideal Λ P I K .Note that Λ " Λ is a Z-lattice (hence a Z-grid) in C, with covol Λ " NpΛq as seen in Equation ( 15).As in Section 2, we work on the constant cylinder E " C{p2πiZq in this section.
Recall that L ϕ K Λ is the family defined in Equation (2).For every N P N ´t0u, the (not normalised, empirical) pair correlation measure of the logarithms of nonzero elements in Λ, with trivial scaling function and multiplicities given by the Euler function, is the measure on E with finite support defined, with Theorem 5.1 As N Ñ `8, the measures r ν N on E, renormalized to be probability measures, weak-star converge to the measure absolutely continuous with respect to the Lebesgue measure on E, with Radon-Nikodym derivative the function g Furthermore, for all f P C 1 c pEq and α P s0, 1 2 r, with c Λ " NpΛq This result gives the first assertion of Theorem 1.2 in the introduction.
Proof.In this proof, all functions Op¨q are absolute, since there are finitely many K.
The first assertion of Theorem 5.1 follows from the second one, by the density of C 1 c pEq in C 0 c pEq for the uniform convergence.For all N P N and q P Λ with 0 ă |q| ď N , let J q be given by the equation on the left in Formula (9).We now define which is a finitely supported measure on the closed unit disc D " Bp0, 1q of C, and is nonzero since ´q P J q .
Lemma 5.2 As |q| Ñ `8, we have } r Proof.This follows from Lemma 4.1 applied with m " Λ, z " 1, θ " 2π and x " |q|, since ϕ K pqq " OpNpqqq and Lemma 5.3 For all f P C 1 pDq and α P s0, 1 2 r , as |q| Ñ `8, we have Proof.Note that c Λ ě 1 and let us define By Lemma 5.2, as |q| Ñ `8, we have Let Q " t |q| α u ě 1, which tends to `8 as |q| Ñ `8.For all elements m and n in t0, . . ., Q ´1u, let so that D ´t0u is the disjoint union of the sets A n,m for m, n P t0, . . ., Q ´1u.
With the notation of Equation ( 42), we have Note that since n `1 ď Q, as Q tends to `8, we have Hence for every z P A n,m , we have by the mean value theorem Since we have therefore By Equations ( 50) and ( 43), we have By Equations ( 51), ( 49), applying twice Lemma 4.1 with m " Λ, θ " 2π Q and x " |q| n`1 Q , |q| n Q , and using the fact that |q| Q tends to `8 as |q| Ñ `8 since α ă 1, we have, as |q| Ñ `8, Note that qD " Bp0, |q|q.By cutting the sum defining r ω q and the integral over D into Q 2 subparts, by using Equations ( 52) and ( 53), and since n ď Q ď q α , as |q| Ñ `8, we have This proves Lemma 5.3.
l For every N P N ´t0u, let us define which is a finitely supported measure on D. By Lemma 4.1 and Theorem 4.2 both with m " Λ, θ " 2π, x " N and the second one with k " 0, since c Λ ě 1 and c Λ,0 ď 1 by Equation ( 45), and since there are finitely many such fields K), its total mass is ´`ÿ For every f P C 1 pDq, by Lemmas 5.3 and 5.2, again by Lemma 4.1 with m " Λ, θ " 2π and x " N , we have For every N P N ´t0u, let us define which is a measure with finite support on E ˘" p˘r0, 8r `iRq{p2πiZq, so that r ν Ń " log ˚r µ Ń " r ν N | E ´, and }r ν Ń } " }r µ Ń }.For every f P C 1 c pE ´q, the function f ˝log is a C 1 function on D which vanishes on a neighborhood of 0. By Equation (6), we have 2 }r ν N } and the last claim of Theorem 5.1 follows by symmetry.l 6 Pair correlation of integral lattice points with scaling and Euler weight As in Section 5, we fix an imaginary quadratic number field K whose ring of integers O K is principal, and a (nonzero integral) ideal Λ " Λ P I K .We also study the pair correlations of the family L ϕ K Λ defined in the introduction, but now with the linear scaling function ψ " id 1 : N Þ Ñ N .We leave to the reader the study of a general scaling ψ, assumed to converge to `8, proving a Poissonian behaviour for sublinear scalings and total loss of mass behaviour for superlinear scalings.We also leave to the reader a statement similar to Theorem 6.1, replacing the above Z-lattice Λ by a Z-grid a `Λ for any a P O K .
As in Section 3, we work on the family of varying cylinders pE N " C{p2πi N Zqq N PN´t0u .As in Section 3, for every f P C 1 c pCq, for every N large enough such that the support of f is contained in Bp0, πN q, we denote by f N P C 1 c pE N q the map which coincides with f on Bp0, πN q modulo 2πi N Z and vanishes elsewhere.For every N P N ´t0u, we consider the measure on E N with finite support defined with which is the (not normalised) empirical pair correlation measure at time N of the complex logarithms of the elements of Λ with multiplicities given by the Euler function and with linear scaling N .Theorem 6.1 As N Ñ `8, the family `1 N 6 r R N ˘NPN of measures on E N converges (for the pointed Hausdorff-Gromov weak-star convergence) to the measure absolutely continuous with respect to the Lebesgue measure on C, with Radon-Nikodym derivative the function Furthermore, for all A ě 1 and f P C 1 pCq with compact support contained in Bp0, Aq, as N Ñ `8, we have The above result with Λ " O K gives the second assertion of Theorem 1.2 in the introduction, using the value of c O K ,k given in Equation (47).
Note that, as the proof below shows, the total mass of r R N is equivalent to c N 8 as N Ñ `8, for some constant c ą 0. Hence renormalising r R N to be a probability measure would make it converge to the zero measure on C.
Proof.We proceed as in the beginning of the proof of Theorem 3.1 : We only have to prove the second assertion above; We define E N " p˘r0, 8r `iRq{p2πi N Zq; We only study the convergence of the measures 1 N 6 r R N on the half-cylinder E Ǹ to the measure g L ϕ K Λ ,id 1 Leb C `on the half-plane C `" tz P C : Repzq ě 0u as N Ñ `8; And we deduce the global result by the symmetry of g L ϕ K Λ ,id 1 under z Þ Ñ ´z.For all N P N ´t0u and p P Λ ´t0u, let J p, N be given par Equation ( 19).Note that pΛ ´t0uq X Bp0, N ´|p|q Ă J p, N Ă pΛ ´t0uq X Bp0, N q . (54) We now define the key auxiliary measure by .
Then r ω p, N is a measure with finite support on Bp0, 1 |p| q ´t0u, which is nonzero if N ě 2|p| (which is the case if p is bounded and N Ñ `8), and vanishes if |p| ą 2N .If N ě 2|p|, by Theorem 4.2 with m " Λ, k " p and θ " 2π, by Formula (54), since |p| ě 1, and since c Λ,p ď 1 (see Equation ( 45)), we have In particular, if N ě 2|p|, since c 1 Λ ą 0 by Equation (48), we have The next result implies that the measures r ω p,N , once normalized to be probability measures, weak-star converge to the measure dµpzq " 6 π |p| 6 |z| 4 d Leb Bp0, 1 |p| q pzq on Bp0, 1 |p| q as N Ñ `8, uniformly on p P Λ ´t0u bounded.Lemma 6.2For all p P Λ ´t0u, α P s 0, 1 r and f P C 1 c pCq, as N Ñ `8, we have Proof.As in the proof of Lemma 5.3, we will estimate the difference of the main terms in the above centered formula by cutting the sum defining the renormalized measure r ω p,N and by cutting similarly the integral on Bp0, 1 |p| q.We assume, as we may, that N ě 2|p|.Let Q " t N α u ě 1, which tends to `8 as N Ñ `8.For all m, n P t0, . . ., Q ´1u, let so that Bp0, 1 |p| q ´t0u is the disjoint union of the sets A 1 n,m for m, n P t0, . . ., Q ´1u.With the notation of Equation ( 42), we have Note that diamp A 1 n,m q " O `1 Q |p| ˘.Hence for every z P A 1 n,m , we have by the mean value theorem If |p| ď N 1´α (which is the case if p is bounded and N Ñ `8) and if n ď Q ´2, then Hence for all m, n P t0, . . ., Q ´1u, by Formula (54), if |p| ď N 1´α and if n ‰ Q ´1, we have For all m, n P t0, . . ., Q ´1u, let S n,m " If n ‰ Q ´1, by Equations ( 59) and ( 60) for the first equality, and for the second one, by Equations ( 56), ( 58) and ( 43), by Theorem 4.2 applied twice with m " Λ, k " p, θ " 2 π Q and x " N pn`1q Q , N n Q , we have, as N Ñ `8 (so that in particular N ě 2|p|), Note that by Equations ( 58), ( 54) and ( 43) for the first inequality, and for the second one, by Equations ( 56) and twice (55), as N Ñ `8, we have For all m, n P t0, . . ., Q ´1u, let By Equations ( 59) and ( 57), we have Furthermore, , putting together Equations ( 61), ( 63), ( 62) and ( 64), and since Q " t N α u P r N α 2 , N α s for N large enough, we have This proves Lemma 6.2.l Now, let us introduce the finitely supported measure on C ´t0u defined by where as previously ι : z Þ Ñ 1 z (noting that the measure r ω p, N vanishes if |p| ą 2N and has finite support contained in Bp0, 1 |p| q ´t0u).
Lemma 6.3For all A ě 1 and f P C 1 pC `q with compact support contained in Bp0, Aq, as N Ñ `8, we have Proof.Let us assume that N ą A π , so that the ball Bp0, Aq injects by the canonical projection As in the proof of Lemma 3.3 (see Formulas ( 23) and ( 24)), if a pair pp, qq occurs in the index of the sum defining either r R N pf N q or r µ Ǹ pf q with nonzero corresponding summand, then |p| |q| " O `A N ˘, |p| " OpAq, and Hence, by Equation ( 55), since c Λ,p ď 1 (see Equation ( 45)) and by Lemma 2.1 with k " 0, we have This proves Lemma 6.3.l Lemma 6.4For all A ě 1 and f P C 1 pC `q with compact support contained in Bp0, Aq, as N Ñ `8, we have Proof.Let A and f be as in the statement, let N be large enough, and let α P s0, 1r .Since the support of r ω p, N is contained in Bp0, 1 |p| q, the support of ι ˚r ω p, N is contained in tz P C : |z| ě |p|u.Since a nonzero element of O K has norm, hence absolute value, at least 1, the measures r µ Ǹ and g L ϕ K Λ , id 1 pzq d Leb C pzq both vanish on Bp0, 1q.Hence we may assume that the support of f is contained in tz P C : |z| ě 1u, so that the support of f ˝ι is compact.Note that }f ˝ι} 8 " }f } 8 and as the support of f is contained in Bp0, Aq, that }dpf ˝ιq} 8 ď A 2 }df } 8 .
By Equation ( 55) and by Lemma 6.2, by Equation ( 26), since 1 ď |p| " OpAq and c Λ,p ď 1, as N Ñ `8, we hence have By Lemma 2.1 with k " 0, we hence have Taking α " 1 2 , this proves Lemma 6.4 since c 1 λ ď 1 and A ě 1. l Theorem 6.1 now follows from Lemmas 6.3 and 6.4, as explained in the beginning of the proof.l The following figure illustrates Theorem 6.1 when K " Qp 1`i ? 3 2 q and Λ " O K " Zr 1`i ? 3 2 s.It shows an approximation of the pair correlation function g L ϕ K Λ , id 1 computed using the empirical measure 1 50 6 r R 50 in the ball of radius 5 centered at the origin, to be compared with the orange radial profile of g L ϕ K Λ , id 1 in the second figure of the introduction.
The graph of g L ϕ K Λ , id 1 is bounded by Lemma 2.1 with k " 6 since c Λ,p ď 1.It is asymptotic to a horizontal plane at infinity, by the following result.In its proof, we use the Möbius function µ K : I K Ñ Z of K, defined by @ a P I K , µ K paq " # 0 if p2 | a for some prime ideal p p´1q m if a " p 1 . . .p m for distinct prime ideals p 1 , . . ., p m (in particular µ K pO K q " 1), For every a P O K ´t0u, we define µ K paq " µ K paO K q.We have (see for instance [Sha]) the Möbius inversion formula: for all f, g : Proposition 6.5 We have Proof.Let us consider the multiplicative 2 function on I K defined by Let us prove that uniformly in x ě 1, we have ÿ Applying this with x " |z| 2 , by Equation (3), since the map be the Dirichlet convolution of f with the Möbius function µ K of K. Then g is multiplicative.For every prime ideal p of O K , we have gppq " f ppq µ K pO K q `f pO K q µ K ppq " 1 NppqpNppq 2 ´2q and gpp k q " f pp k q µ K pO K q `f pp k´1 q µ K ppq " 0 for every k ě 2. Therefore, for every b P I K , we have gpbq " µ K pbq 2 Proof.This is immediate if µ K pbq " 0. Otherwise, b " p 1 . . .p k with p 1 , . . ., p k pairwise distinct prime ideals, and The last claim follows from the well known error term in the Dedekind zeta function summation.l By for instance Equation ( 7) with Λ " O K , k " 0 and x " ?y, by Equation ( 15) with m " O K , and again since the map k Þ Ñ kO K is |O K |-to-1, as y Ñ `8, we have Using the Möbius inversion formula (65) for the first equality, Equation (67) for the third equality, Lemma 6.6 for the fifth equality and an Eulerian product (since g is multiplicative and vanishes on ideals divisible by a nontrivial square) for the sixth equality, with Spxq " ř aPI K : Npaqďx f paq, uniformly in x ě 1, we have By summation by parts, we hence have This proves Equation ( 66) and concludes the proof of Proposition 6.5.l 7 Pair correlations of common perpendiculars in the Bianchi manifolds PSLpO K qzH 3

R
We again fix an imaginary quadratic number field K whose ring of integers O K is principal, and a (nonzero integral) ideal Λ " Λ P I K .In this section, we give a geometric motivation for the introduction of the Euler function as multiplicities in the family L ϕ K Λ of complex logarithms of elements of Λ defined in Equation ( 2), and we give a geometric application of the results in Section 5.
We refer to [PP1,BPP]  D ´, γ r D `q ą 0. The multiplicity multpαq of α is the ratio A{B where ‚ A is the number of elements pγ ´, γ `q P pΓ{Γ D ´q ˆpΓ{Γ γD `q such that r α is the unique shortest arc between γ ´r D ´and γ `γ r D `, and ‚ B is the cardinality of the pointwise stabilizer of r α in Γ.The length λpαq of the common perpendicular α is the length of the geodesic segment r α in r Y .For every in the set OL 6 pD ´, D `q of lengths of common perpendiculars, the length multiplicity of is the sum of the multiplicities of the common perpendiculars between D ´, D `having the length : ωp q " ÿ α common perpendicular beween D ´and D `with λpαq" multpαq .
If PerppD ´, D `q is the set of all common perpendiculars from D ´to D `with multiplicities, then pλpαqq αPPerppD ´, D `q is the marked ortholength spectrum from D ´to D `, and the set OLpD ´, D `q " pOL 6 pD ´, D `q, ωq of the lengths of the common perpendiculars endowed with the length multiplicity ω is the ortholength spectrum from D ´to D `.
As defined in [PP2,§6], the pair correlation measure of the common perpendiculars from D ´to D `is the pair correlation measure of the family `" `pF D ´,D Ǹ " OL 6 pD ´, D `q X r0, 2 ln N sq N PN , ω ˘.
Let us specialize these objets as follows.Let r Y " H 3 R " `tpz " x `iy, tq P C ˆR : t ą 0u, ds 2 " dx 2 `dy 2 `dt 2 t 2 be the upper halfspace model of the real hyperbolic 3-space with constant curvature ´1.We identify as usual its space at infinity B 8 H 3 R " pC ˆt0uq Y t8u with P 1 pCq " C Y t8u.For every b P I K , let Γ 0 rbs be Hecke's congruence subgroup modulo b of the Bianchi group PSL 2 pO K q, which is the preimage of the upper triangular subgroup of PSL 2 pO K {bq under the reduction morphism PSL 2 pO K q Ñ PSL 2 pO K {bq.It acts faithfully on H 3 R by Poincaré's extension, and is a lattice in the isometry group of H 3 R .Let Y b " Γ 0 rbszH 3 R , 37 which is a finite (possibly ramified) cover of the Bianchi orbifold PSL 2 pO K qzH 3 R .Note that since O K is principal, this Bianchi orbifold has only one cusp (the number of cusps being the class number of K, see for instance [EGM]).
Let r D ´" r D `be the horoball H 8 " tpz, tq P H 3 R : t ě 1u in H 3 R , whose image D ´" D `in Y b is a Margulis neighbourhood of a cusp of Y b .In order to emphasize the dependence on the ideal b, we will use the notation F b D ´,D `" F D ´,D `for the family of lengths of common perpendiculars between D ´and D `in Y b .
The following result relates the pair correlation measures of the common perpendiculars from this Margulis cusp neighbourhood to itself to the pair correlation measures of the complex logarithms of the elements of Λ " b, with multiplicities given by the Euler function ϕ K .As explained in Remark 2.3, in the following result, we remove from the index set I N of the summations defining R Proof.The orbit of H 8 under Γ 0 rbs consists, besides H 8 itself, of the Euclidean 3-balls H p q of Euclidean radius 1 2|q| 2 tangent to the horizontal plane C at the rational elements p q with p P O K , q P b ´t0u and pp, qq " 1.
Every common perpendicular between D ´and D `has a vertical representative in H 3 R which starts from a point in C ˆt1u and ends on the boundary of H p q with p q as above.Its hyperbolic length is 2 ln |q|.In particular, the set OL 6 pD ´, D `q is equal to t2 ln |q| " 2 Replog qq : q P b ´t0uu .0 The stabilizer of H 8 , or equivalently of 8, in Γ 0 rbs is the upper triangular subgroup of Γ 0 rbs, hence of PSL 2 pO K q.It contains the upper unipotent subgroup consisting of translations by O K with finite index, equal to |O K | 2 .Hence given a denominator q P b ´t0u, the points at infinity with denominator q of the geodesic lines containing a lift of a common perpendicular between D ´and D `are, modulo translation by O K , exactly the points p q where p ranges over a set of representatives of pO K {qO K q ˆ.
By Equation (5), the map z Þ Ñ " 2 Repzq from L N " tlog q : q P b ´t0u, |q| ď N u to the set F D ´,D Ǹ of lengths of the common perpendiculars between D ´and D `with length at most 2 ln N hence sends the multiplicity 2 |O K | ϕ K ˝expplog qq of log q to the multiplicity ωp q of the corresponding length of common perpendicular .The claim follows.l The following result computes the pair correlation function without scaling of the lengths of the common perpendiculars from the Margulis cusp neighbourhood at infinity to itself in the Hecke-Bianchi orbifold Γ 0 rbszH 2 R .The maps Re : E N Ñ R for N P N being not uniformly proper, the case with scalings requires a new analysis, that we plan to study in another paper.
Corollary 7.2 For every ideal b P I K , as N Ñ `8, the pair correlation measures `, 1 N on R, renormalized to be probability measures, weak-star converge to a measure absolutely continuous with respect to the Lebesgue measure on R, with pair correlation function given by s Þ Ñ e ´2|s| .
Proof.This follows from Theorem 5.1 with Λ " b as in the proof of Corollary 2.5, using Proposition 7.1.l

I
Ń " tpm, nq P I N : |m| ď |n|u and I Ǹ " tpm, nq P I N : |n| ď |m|u .Given a subset b of the set of ambient parameters, for every positive function g of a variable in N ´t0u, we will denote by O b pgq (and Opgq when b is empty) any function f on N ´t0u such that there exists a constant C 1 depending only on the parameters in b and a constant N 0 possibly depending on the all the parameters (including the ones in b) such that for every N ě N 0 , we have |f pN q| ď C |gpN q|.
1 60 2 R L Λ ,ψ 60 in the ball of center 0 and radius 5. We refer to the first picture in the introduction for the graph of the pair correlation function g L Λ ,ψ .The figure below gives on the left the graph of the pair correlation function g L Λ , ψ of the Z-lattice Λ " Λ " Zr 1`i ? 3 2 s of the Eisenstein integers at the linear scaling ψ : N Þ Ñ N in the ball of center 0 and radius 5.The blue lines on the bounding box represent the limit 2 π covol 2 Λ " 2π 3 at `8 of g L Λ , ψ , given by Equation (7) with k " 2. On the right, we have the approximation of the pair correlation function computed with the empirical measure 1 60 2 R L Λ ,ψ 60 4 Mertens and Mirsky formulae for algebraic number fields For every b P I K , we have 0 ď gpbq ď Npbq m ‰ n.Recall that the map 2 Re : E Ñ R is a continuous proper map.Proposition 7.1 For every ideal b P I K , for more information on the following notions.Let Y be a nonelementary geodesically complete connected proper locally CATp´1q good orbispace, so that the underlying space of Y is Γz r Y with r Y a geodesically complete proper CATp´1q space and Γ a discrete group of isometries of r Y preserving no point nor pair of points in r Y YB 8 r Y .Let D ´and D `be connected proper nonempty properly immersed locally convex closed subsets of Y , that is, D ´and D `are locally finite Γ-orbits of proper nonempty closed convex subsets r D ´and r D `of r Y .A common perpendicular α between D ´and D ìs the Γ-orbit of the unique shortest arc r α between r D ´and γ r D `for some γ P Γ such that dp r