Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the standard Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sector and with congruences.


Introduction
Let K be a number field of degree n K , with ring of integers O K , number of real places r 1 , number of complex conjugated places r 2 , regulator R K , class number h K , number of units ω K , discriminant D K and Dedekind zeta function ζ K (see for instance [Nar]).Let I K be the semigroup of nonzero ideals of O K , let ϕ K : I K Ñ N be the Euler function of K, and let N : I K Ñ N be the norm, with ϕ K paq " ϕ K paO K q and Npaq " NpaO K q for every a P O K ´t0u.As usual, p below ranges over prime ideals in I K .The functions Op¨q below depend only on K.
Our first result (see Section 2) is a Mertens formula with congruences for number fields.Though probably well-known at least when m " O K , we provide a proof for lack of reference (compare with [Gro,Satz 2], [Cos,§4.3],[PP1,Theo. 3.1]) since arguments of its proof will be useful for our next result.For every m P I K , let c m " Npmq ź p|m p1 `1 Nppq q .
Theorem 1.1 For every m P I K , if n K ě 2, then as x Ñ `8, we have Assume in the remaining part of this introduction that K is imaginary quadratic and that O K is principal.By Dirichlet's unit theorem, these assumptions are more or less necessary (besides K " Q) for the following sums to be well defined and finite.
We give in Section 3 a version in angular sectors of the Mertens formula given by Theorem 1.1, that will be needed in [PP3].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 : It is important that the function Op¨q in the following result is uniform in m, z and θ.
Theorem 1.2 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 Lastly, we give a uniform asymptotic formula for the sum in angular sectors in C of angle θ of the products of two shifted Euler functions with congruences, that will be needed in [PP3].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 simplicity, we give a version without congruences and without an error term in this introduction, see Section 4 Theorem 4.1 for the general statement.
Theorem 1.3 For all z P C ˆ, θ P s0, 2πs and k P O K , as x Ñ `8, we have Theorems 1.2 and 1.3 are used in [PP3] in order to study the correlations of pairs of complex logarithms of Z-lattice points in the complex line at various scalings, when the weights are defined by the Euler function, proving the existence of pair correlation functions.We prove in op.cit.that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics.A geometric application is given in op.cit. to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi manifolds PSL 2 pO K qzH 3 R .
Recall that I K is the semigroup of nonzero (integral) ideals of the Dedekind ring O K (with unit O K ).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, by rI, Js " I X J the least common ideal multiple of I and J, and by IJ the product ideal of I and J.
We denote by NpIq " CardpO K {Iq the (absolute) norm of 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 |a| 2 if K is imaginary quadratic.
Recall that the Dedekind zeta function ζ K : ts P C : Repsq ą 1u Ñ C of K is defined (see for instance [Nar,§7.1])equivalently by We denote by ϕ K : I K Ñ N the Euler function of K, defined (see for instance [Nar,page 13]) equivalently by For every a P O K ´t0u, we define ϕ K paq " ϕ K paO K q.Note that the Euler function ϕ K is multiplicative2 by the Chinese remainder theorem.We have Npaq " as checked by telescopic sum when a is a power of a prime ideal, and by multiplicativity.We denote by µ K : I K Ñ Z the Möbius function of K, defined by a for some prime ideal p p´1q m if a " p 1 . . .p m for pairwise distinct prime ideals p 1 , . . ., p m and m P N ´t0u .
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 : In particular, since the norm is completely multiplicative and by Equation (2), we have Proof of Theorem 1.1.In this proof, all functions Op¨q depend only on K. Let Recall (see for instance [MO,Theo. 5]) that, as x Ñ `8, we have By Abel's summation formula, as y Ñ `8, we have Furthermore, we have This formula implies since Nppb, mqq ď Npmq that Let us denote by S m pxq the sum on the left hand side in the statement of Theorem 1.1.Note that by the Gauss lemma, for all m, b, c P I K , we have m | bc if and only if mpm, bq ´1 | c.Then by Equation ( 4), by the change of variable c " mpm, bq ´1a, by the complete multiplicativity of the norm, by Equation ( 7) with y " Nppb,mqqx Npbq Npmq , since Nppb, mqq ď Npmq, and by Equation ( 8), we have By decomposing a nonzero integral ideal b into powers of prime ideals, by the definition of the Möbius function, and by the Euler product formula for the Dedekind zeta function, we have Equations ( 9) and ( 5) hence imply Theorem 1.1.l

A sectorial Mertens formula
Assume in the remaining part of this paper that K is imaginary quadratic and that O K is principal (or equivalently factorial (UFD)).By Dirichlet's unit theorem, the group of units O K , whose order we denote by |O K |, is finite if and only if pr 1 , r 2 q is equal to p1, 0q or p0, 1q.This justifies our restriction, the case K " Q being well-known.With the notation of the beginning of the introduction, we then have (see for instance [Nar]) D K P t´4, ´8, ´3, ´7, ´11, ´19, ´43, ´67, ´163u, and Given a Z-lattice Λ in the Euclidean space C (that is, a discrete (free abelian) subgroup of pC, `q), we denote by covol Λ " VolpC{ Λq the area of a fundamental parallelogram F Λ for Λ and by diam Λ the diameter of F Λ .Note that every element m With the notation of Equation ( 1), note that for every z 1 P C ˆ, we have Proof of Theorem 1.2.Let z P C ˆ, θ P s0, 2πs and y ą 0. Since AreapCpz, θ, yqq " θ 2 y 2 , the standard Gauss counting argument, the finiteness of the number of imaginary quadratic number fields with class number 1, and the equality on the left of Formula (11) give ¯.
Let us denote by S m,z,θ pxq the sum on the left hand side in the statement of Theorem 1.2.Then by Equation ( 4), we have The proof then proceeds exactly as in the proof of Theorem 1.1.l

A sectorial Mirsky formula
We now give a uniform asymptotic formula for the sum in angular sectors of the products of shifted Euler functions 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 Theorem 4.1 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 and m P I K , there exists c m,k P s0, 1s such that for all z P C ˆ, θ P s0, 2πs and x ě 1, we have ˇˇSz,θ,k,mpxq We will prove Theorem 4.1 at the end of this Section after giving a number of Lemmas required for the proof.We fix k P O K and m " mO K P I K , and we define h " kO K , which is a possibly zero integral ideal.We start by giving the first definition and a simpler formula for the constant c m,k that appears in the statement of Theorem 4.1.We define and Lemma 4.2 The series in Equation (15) defining c m,k converges absolutely.We have c m,k ď 1 and c 1 m ą 0. Furthermore, we have where In the special case m " O K , Equation ( 16) becomes Theorem 1.3 in the introduction follows from Theorem 4.1 and the above computation.
Proof.Let us prove that uniformly in x ě 1, we have This implies, by taking x " 1, that the first claim of Lemma 4.2 is satisfied, since the Möbius function has values in t0, ˘1u.Let us denote by Z m,h pxq the above sum.Since N `pcpb, mq, mpb, cqq ˘ď Npmpb, cqq, we have Equation ( 19) follows, since there are only finitely many fields K satisfying the assumptions of Theorem 4.1.
The proof of Equation ( 16) that we now give is similar to Fouvry's proof of Equation ( 21) in [PP2,Appendix].
For every b P I K , let χ b : I K Ñ t0, 1u be the characteristic function of the set of elements c P I K such that pc, bq | h.Let us define a map ψ b : Note that the assertion pcpb, mq, mpb, cqq | b h is equivalent to the assertion For every b P I K , let χ b : I K Ñ t0, 1u be the characteristic function of the set of elements c P I K such that the above divisibility assertion is satisfied.Let us finally define a map C ˚: I K Ñ R (which depends on m and h) by By the absolute convergence property, Equation ( 15) then becomes In order to transform the series C ˚pbq defined by Formula ( 21) into an Eulerian product and in order to analyse it, we will use the following two lemmas.
Lemma 4.3 For every b P I K , the maps χ b , χ b and ψ b on I K are multiplicative.
Proof.We have ψ b pO K q " O K and χ b pO K q " χ b pO K q " 1.Let I, J P I K be coprime.The equality pIJ, bq " pI, bqpJ, bq and the fact that pI, bq and pJ, bq are coprime imply that χ b pIJq " χ b pIqχ b pJq.
In order to prove the multiplicativity of the map ψ b , we write Since I is coprime to pJ, bq and since J is coprime to pI, bq, we obtain as wanted the equality ψ b pIJq " ψ b pIq ψ b pJq.21) may be written as an Eulerian product By Equations ( 22) and ( 23), and by Lemma 4.4, we have This equation writes c m,k as a series N pbq 2 where f : I K Ñ R is a multiplicative function, which vanishes on the nontrivial powers of prime ideals.By Eulerian product, we have therefore proved Equation ( 16).
Let us now prove that 0 ď c m,k ď 1.Note that for every prime ideal p, we have In particular all the factors of the two products over p in Equation ( 16) belong to r0, 1s, hence 0 ď c m,k ď 1 Npmq ď 1.Let us finally prove that c 1 m ą 0. For every prime ideal p, let w p " κ m,h ppq κ 1 h ppq Nppp,mqq Nppq 2 . By Formula (17), if Nppq " 2, we have In particular 1 ´wp ‰ 0 if Nppq " 2. From the inequalities (24) and by Equation ( 16), we have Since there are only finitely many primes ideals p dividing m, the term on the right hand side is bounded from below by a positive constant c 1 m " min kPOK c m,k ą 0. This concludes the proof of Lemma 4.
Recall that a system of two congruences " n " α 0 mod α n " β 0 mod β with unknown n P O K , where α, β, α 0 , β 0 P O K and α, β ‰ 0, has a solution if and only if α 0 ´β0 " 0 mod pα, βq.Furthermore, if this congruence condition is satisfied, that is, if there exists n 0 , m 0 P O K such that α 0 ´β0 " βm 0 ´αn 0 , then n is a solution if and only if n ´α0 ´αn 0 P αO K X βO K " rα, βsO K .
This is equivalent to asking n to belong to the translate Λ α,β,α0,β0 " α 0 `αn 0 ` Λ α,β of the Z-lattice Λ α,β " rα, βsO K .Applying this with α " m pb,mq , β " c pb,cq , α 0 " 0 and β 0 " ´k Thus Equation ( 27) becomes, using Equation ( 12), Let b, c be as in the index of the first sum above.Using again the standard Gauss counting argument, using Formula (11) for the second equality and the equation Nprα, βsq " Npαq Npβq Nppα,βqq for the last equality, we have, uniformly in b, c, m P O K ´t0u, k P O K , z P C ˆ, θ P s0, 2πs and y ě 1, CardpΛ α,β,α0,β0 Using this with y " x |b| , which is at least 1 since |b| ď x, we have N `pmpb, cq, cpb, mqq ˘1{2 By Equation (19) (replacing therein x by x 2 ), completing the first sum of the above equation with the indices b P O K ´t0u such that |b| ą x introduces an error of the form Op 1 x q (uniformly in m P O K ´t0u, k P O K and x ě 1).A computation similar to the one done for Equation (19) gives that the second sum in Equation ( 30) is actually bounded by 1 |O K | 2 ζ K p 3 2 q 2 ζ K p2q, which is uniform since there are only finitely many such fields K.
By the definition of the constant c m,k in Equation ( 15), this proves Equation ( 26 Replacing f `by f ´gives the same minoration to S z,θ,k,m pxq, hence Theorem 4.1 follows.l Finally, the multiplicativity property χ b pIJq " χ b pIqχ b pJq of the function χ b is a consequence of the multiplicativity of the map ψ b and of the fact that ψ b pIq and ψ b pJq are coprime.l Lemma 4.4 For every prime ideal p and every b P I K , we have ψ b ppq " " p if p | b, pp, mq otherwise, and χ b ppq χ b ppq " 1 ô $ & % p | pb, hq or p ∤ b and pp, mq | h .Proof.The first formula follows from the definition of ψ b ppq (see Formula (20)) by considering the three cases ‚ p | b, ‚ p ∤ b and p | m, and ‚ p ∤ b and p ∤ m.The second formula follows from the first one, from the definitions of χ b ppq and χ b ppq, and from the fact that χ b ppq χ b ppq " 1 if and only if χ b ppq " χ b ppq " 1, by considering the two cases ‚ p | b and ‚ p ∤ b. l The arithmetic function c Þ Ñ µ K pcq χ b pcq χ b pcq Npψ b pcqq being multiplicative by Lemma 4.3 and the complete multiplicativity of the norm, and vanishing on the nontrivial powers of primes, the series defining C ˚pbq in Formula ( 2. lNow that we understand the constant c m,k , we continue towards the proof of Theorem 4.1 by giving an asymptotic formula for the sum Uniformly in m P I K , k P O K , z P C ˆ, θ P s0, 2πs and x ě 1, we haver Spxq " θ c m,k a |D K x 2 `Opxq .(26)Proof.For all nonzero elements a and b in the factorial ring O K , we denote by pa, bq any fixed choice of gcd of a and b, and by ra, bs any fixed choice of lcm of a and b.By Equation (4), for every a P O K ´t0u, we haveϕ K paq Npaq " 1 |O K | ÿ bPOK ´t0u : b | a µ K pbq Npbq .Let x ě 1. Applying twice this equality, since Npbq ď Npaq when b | a, we have by Fubini'c P O K ´t0u.The system of three congruences $ & % a " 0 mod m a " 0 mod b a " ´k mod c has a solution a P O K ´t0u such that |a| ď x if and only if there exists an element n P O K ´t0u such that a " bn, |n| ď x |b| and " bn " 0 mod m bn " ´k mod c .(28) When pb, cq ∤ k, no solution exists.Assume that pb, cq | k.Since b pb,cq is invertible modulo c pb,cq , we denote by b pb,cq a multiplicative inverse of b pb,cq modulo c pb,cq .Then the system of congruences (28) is equivalent to # b pb,mq n " 0 mod m pb,mq b pb,cq n " ´k pb,cq mod c pb,cq ô # n " 0 mod m pb,mq n " ´k pb,cq b pb,cq mod c pb,cq .