The ellipse law: Kirchhoff meets dislocations

In this paper we consider a nonlocal energy $I_\alpha$ whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter $\alpha\in \R$. The case $\alpha=0$ corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; $\alpha=1$ corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for $\alpha\in (0,1)$ the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an \emph{ellipse}. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are rotating vortex patches, called \emph{Kirchhoff ellipses}. Therefore we show a surprising connection between vortices and dislocations.


Introduction
The starting point of our analysis is the nonlocal energy defined on probability measures µ ∈ P(R 2 ), where the interaction potential W α is given by and α ∈ R.Here the parameter α has the role of tuning the strength of the anisotropic component of W α , making it more or less prominent.
In the particular case where the anisotropy is switched off, namely for α = 0, the minimiser is radial, and is given by the celebrated circle law µ 0 := 1 π χ B 1 (0) , the normalised characteristic function of the unit disc.This result is now classical and has been proved in a variety of contexts, from Fekete sets to orthogonal polynomials, from random matrices to Ginzburg-Landau vortices and Coulomb gases (see, e.g., [20,31], and the references therein).
In the case α = 1, the energy I 1 models interactions between edge dislocations of the same sign (see, e.g., [28,21]).The minimisers of I 1 were since long conjectured to be vertical walls of dislocations, and this has been confirmed only very recently, in [29], where the authors proved that the only minimiser of I 1 is the semi-circle law on the vertical axis.
In this paper we explicitly characterise the minimiser of I α for every α ∈ R. In particular, it turns out that the values α = ±1 correspond to maximal anisotropy.Increasing the value of the weight α above 1 has in fact no effect: a simple energy comparison argument shows that µ 1 is the only minimiser of I α for α ≥ 1.Moreover, the case α < 0 can be recovered from the knowledge of the case α > 0 by switching x 1 and x 2 , so we can limit our analysis to α ∈ (0, 1).
For α ∈ (0, 1) we prove that the unique minimiser of I α is the normalised characteristic function of the region surrounded by an ellipse of semi-axes √ 1 − α and √ 1 + α.This shows, in particular, that α = 1 is a critical value of the parameter, at which an abrupt change in the dimension of the support of the minimiser occurs.The main result of the paper is the following: Theorem 1.1.Let 0 ≤ α < 1.The measure where is the unique minimiser of the functional I α among probability measures P(R 2 ), and satisfies the Euler-Lagrange conditions with We emphasise that for 0 ≤ α ≤ 1 the Euler-Lagrange conditions (1.5)- (1.6) are a sufficient condition to minimality, since we will show that the energy I α is strictly convex for these values of α (see Proposition 2.1).Thus, proving that µ α satisfies (1.5)- (1.6) immediately entails the minimality of µ α .
1.1.Kirchhoff ellipses and dislocations.To prove that the ellipse law µ α satisfies the Euler-Lagrange conditions (1.5)-(1.6),we evaluate the convolution of the kernel W α with the characteristic function of the domain enclosed by a general ellipse.Let us define, for any a, b > 0, the domain , where z = x 1 + ix 2 is the complex variable in the plane, and of a second term containing the gradient of the anisotropic part of the potential.The convolution −(1/z) * χ Ω(a,b) has been computed before, for instance in [22], for rotating vortex patches in fluid dynamics.
Let us recall that a vortex patch is the solution of the vorticity form of the planar Euler equations in which the initial condition is the characteristic function of a bounded domain D 0 .Since vorticity is transported by the flow, the vorticity at time t is the characteristic function of a domain D t .In general the evolution of D t is an extremely complicated phenomenon, but Kirchhoff proved more than one century ago that if D 0 is the domain enclosed by an ellipse with semi-axes a and b, then D t is just a rotation of D 0 around its centre of mass with constant angular velocity ω = ab/(a + b), see [24,19,26].Domains with the simple evolution property described above are called V -states or rotating vortex patches.They can be viewed as stationary solutions in a reference system that rotates with the patch, and they can be described by means of an equation involving the stream function − log | • | * χ D 0 of the initial patch D 0 (see [7]), which is formally similar to the Euler-Lagrange equation (1.5).If one wants to verify that for the elliptical patch Ω(a, b) such equation is satisfied, one needs to compute explicitly , and this can be done by first computing its gradient −(1/z) * χ Ω(a,b) .
The challenge in our case is computing the gradient of the anisotropic part of W α * χ Ω(a,b) .The key observation is that it can be written in terms of suitable complex derivatives of the fundamental solution of the operator ∂ 2 , where ∂ = ∂/∂z.To compute such term explicitly we need the expression of −(1/z) * χ Ω(a,b) , which was known, as well as the expression of (z/z 2 ) * χ Ω(a,b) , which we obtain in Proposition 3.1.
What is surprising is that techniques developed in the context of fluid mechanics turn out to be crucial for the characterisation of the minimisers of the anisotropic energy I α , which arises, in the case α = 1, in the context of edge dislocations in metals.In particular the minimality of the semi-circle law for the dislocation energy I 1 can be deduced from Theorem 1.1 by a limiting argument based on Γ-convergence (see Corollary 3.3).That is, we obtain again the main result of [29], but with a different proof based on methods from fluid mechanics and complex analysis.
It is worth emphasising the special role that ellipses play in both contexts.On the one hand, in fluid mechanics they provide one of the few explicit solutions of the incompressible Euler equations.On the other hand, the characteristic function of the elliptical domains Ω(a, b) is one of the few measures µ for which the convolution potential W α * µ can be explicitly computed.
What is even more surprising is that, for 0 < α < 1, the normalised characteristic function of Ω( is actually the minimiser of the energy I α and that it is possible to prove it.In fact, in the literature there are very few explicit characterisations of minimisers for nonlocal energies and the only other example in the nonradial case is the result proved in [29] corresponding to α = 1.The reason why the minimality of µ α is somewhat unexpected is the following.Let us first consider the purely logarithmic case α = 0.By radial symmetry of the energy and uniqueness, the minimiser µ 0 must be radial.This case is well-known to be connected to the classical obstacle problem for the Laplace operator [6,9,8].Defining Ψ 0 = W 0 * µ 0 and assuming that µ 0 is supported on the closure of a smooth bounded open set Ω, the Euler-Lagrange equations (1.5)-(1.6)imply where Ω is the coincidence set, i.e., the points where Ψ 0 = C 0 −(|x| 2 /2).It is not surprising from (1.7) that µ 0 is the normalised characteristic function of the coincidence set Ω with constant density since 2πµ 0 = −∆Ψ 0 , and due to the radial symmetry the Euclidean ball is the clear candidate for Ω.
In the presence of the anisotropic term, that is, for α > 0, we write W α = W 0 + αF and define Ψ α = W α * µ α where µ α is the unique minimiser of I α .A corresponding obstacle problem as (1.7) can be formally written for the potential Ψ α , and the coincidence set is again determined by the condition Ψ α = C α − (|x| 2 /2).Assuming it is the closure of a smooth bounded open set Ω, one obtains If µ α is the normalised characteristic function of Ω, then ∆F * µ α should be constant on Ω as well.However, computing ∆F * χ Ω for a general domain Ω is a highly non-trivial task, and in principle ∆F * χ Ω could be a very complicated object.It is therefore surprising that, for elliptic domains Ω = Ω(a, b), ∆F * χ Ω(a,b) is constant in Ω(a, b).In fact, as we mentioned before, we are able to compute the convolution potential W α * χ Ω(a,b) in the whole of R 2 , and to show that in Ω(a, b) it is a homogeneous polynomial of degree 2 plus a constant.From this property, indeed, establishing the first Euler-Lagrange condition is a relatively easy task.The expression of the convolution potential (and of its gradient) outside Ω(a, b) is instead much more involved, so that establishing the second Euler-Lagrange condition is the challenge.
1.2.Dimension of the support of the equilibrium measure.We have seen that the values α = ±1 of the weight for the anisotropic term of the kernel W α determine a sharp transition in the dimension of the support of the minimising measure µ α from two (for α ∈ (−1, 1)) to one (for α ≤ −1 and α ≥ 1).
For general energies of the form where W : R d → R ∪ {+∞} is an interaction potential and V : R d → R ∪ {+∞} is a confining potential, understanding how the dimension of the support of the minimisers depends on W and V is a challenging question.
In [3] the authors showed that the dimension of the support of a minimiser of E is directly related to the strength of the repulsion of the potential at the origin.What they showed is that the stronger the repulsion (up to Newtonian), the higher the dimension of the support.The case of mild repulsive potentials in which the minimisers are finite number of Dirac deltas has been recently studied in [13].
Our result shows that a change of the dimension of the support of the minimisers can also be obtained by tuning the asymmetry of the interaction potential.
Another challenging question arising from the results in this paper and in [3] is to give explicit examples in which a change of the dimension of the support of the minimisers is obtained by tuning the confining potential V , or the singularity of the interaction potential at zero.1.3.More general interactions and evolution.The problem of analysing the landscape of energies of the type (1.8) has triggered the attention of many analysts and applied mathematicians in the last 20 years.
One of the main reasons for this interest, from the analytic viewpoint, is that this question is directly linked to the stability properties of stationary solutions of its associated gradient flow in the Wasserstein sense [15,2], where µ : [0, ∞) → P(R d ) is a probability curve.Here, the variational derivative δE/δµ := W * µ + V is obtained by doing variations of the energy E(µ) preserving the unit mass of the density as originally introduced in [30]; see [36,2] for the general theory.Equations like (1.9) describe the macroscopic behaviour of agents interacting via a potential W , and are at the core of many applications ranging from mathematical biology to economics; see [34,27,23,5] and the references therein.
In most of the early works, interaction and confinement potentials were assumed to be smooth enough and convex in some sense, including interesting cases with applications in granular media modelling [15,35].In most of the applications however the potential W is singular, and in fact most of the rich structure of the minimisers happens when the potentials are singular at the origin; see [18,25,4,3,1,14] and [16] for a recent review in the subject.Typical interaction potentials in applications are repulsive at the origin and attractive at infinity (the latter guaranteeing confinement).
Euler-Lagrange necessary conditions for local minimisers of the energy E(µ) in a suitable topology were derived in [3], see also [31] for the particular case of the logarithmic potential.They were used to give necessary and sufficient conditions on repulsive-attractive potentials to have existence of global minimisers [10,32], and to analyse their regularity for potentials which are as repulsive as, or more singular than, the Newtonian potential [12].In both cases, global minimisers are solutions of some related obstacle problems for Laplacian or nonlocal Laplacian operators, implying that they are bounded and smooth in their support, or even continuous up to the boundary [6,9,12,16].Similar Euler-Lagrange equations were also used for nonlinear versions of the Keller-Segel model in order to characterise minimisers of related functionals [11].
The plan of the paper is as follows.The proof of the Euler Lagrange conditions in Theorem 1.1 will be done in Section 3. We start next section, Section 2, by showing the existence and uniqueness of global minimiser for I α .Section 4 contains some additional information.On the one hand, we discuss an alternative proof of the first Euler-Lagrange condition and compute the minimal energy.On the other hand, we study more general anisotropies.

Existence and uniqueness of the minimiser of I α
In this section we prove that for every α ∈ R the nonlocal energy I α defined in (1.1) has a unique minimiser µ α ∈ P(R 2 ), and that the minimiser has a compact support.
We observe that it is sufficient to consider the case α ∈ (0, 1).In fact, for α = 0, that is, for purely logarithmic interactions, it is well-known that there exists a unique minimiser of I 0 , which is given by the so-called circle law µ 0 := 1 π χ B 1 (0) (see, e.g., [20,31], and the references therein).The case α = 1, that is, the case of interacting edge dislocations, has been recently solved in [29], and it has been shown that I 1 has a unique minimiser, given by the semi-circle law (1.3).A simple comparison argument shows that µ 1 is indeed the unique minimiser of I α for any α ≥ 1.In fact, for any α ≥ 1 and any µ ∈ P(R 2 ) with µ = µ 1 , we have If α < 0, instead, we observe that hence all results in this case may be obtained from those with α > 0 just by swapping x 1 and In what follows we assume the kernel W α to be extended to the whole of R 2 by continuity, that is, we set W α (0) := +∞.Proposition 2.1.Let α ∈ [0, 1].Then the energy I α is well defined on P(R 2 ), is strictly convex on the class of measures with compact support and finite interaction energy, and has a unique minimiser in P(R 2 ).Moreover, the minimiser has compact support and finite energy.
Proof.The case α = 0 is well-known.The proof for α ∈ (0, 1) follows the lines of the analogous result for α = 1; see [29,Section 2].For the convenience of the reader we recall the main steps of the proof.
Step 1: Existence of a compactly supported minimiser.We have that The lower bound (2.1) guarantees that I α is well defined and nonnegative on P(R 2 ) and, since I α (µ 0 ) < +∞, where µ 0 = 1 π χ B 1 (0) , it implies that inf P(R 2 ) I α < +∞.It also provides tightness and hence compactness with respect to narrow convergence for minimising sequences, that, together with the lower semicontinuity of I α , guarantees the existence of a minimiser.
As in [29, Section 2.2], one can show that any minimiser of I α has compact support, again by (2.1).
Step 2: Strict convexity of I α and uniqueness of the minimiser.We prove that for every ν 1 , ν 2 ∈ P(R 2 ), ν 1 = ν 2 , with compact support and finite interaction energy, namely such that R 2 (W α * ν i ) dν i < +∞ for i = 1, 2. Condition (2.2) implies strict convexity of I α on the set of probability measures with compact support and finite interaction energy and, consequently, uniqueness of the minimiser.
To prove (2.2), we argue again as in [29,Section 2.3].The heuristic idea is to rewrite the interaction energy of ν := ν 1 − ν 2 in Fourier space, as Since ν is a neutral measure, ν vanishes at ξ = 0. So, the claim (2.2) follows by showing the positivity of the Fourier transform of W α on positive test functions vanishing at zero.
) and has a logarithmic growth at infinity, it is a tempered distribution, namely W α ∈ S ′ , where S denotes the Schwartz space; hence Ŵα ∈ S ′ .We recall that Ŵα is defined by the formula Ŵα , ϕ := W α , φ for every ϕ ∈ S where, for Proceeding as in [29, Section 2.3], we have that the Fourier transform Ŵα of W α is given by for every ϕ ∈ S, where γ is the Euler constant.In particular, from (2.3), we have that for every ϕ ∈ S with ϕ(0) = 0. Thus, (2.4) implies that Ŵα , ϕ > 0 for every ϕ ∈ S with ϕ(0) = 0 and ϕ ≥ 0, ϕ ≡ 0. Finally, the approximation argument in the proof of [29, Theorem 1.1] allows one to pass from test functions in S to measures.Hence (2.2) is proved.
3. Characterisation of the minimiser of I α : The ellipse law.
It is a standard computation in potential theory (see [31,29]) to show that any minimiser µ of I α must satisfy the following Euler-Lagrange conditions: there exists C ∈ R such that where quasi everywhere (q.e.) means up to sets of zero capacity.The Euler-Lagrange conditions (3.1)-(3.2) are in fact equivalent to minimality for 0 ≤ α ≤ 1 due to Proposition 2.1.See [29, Section 3] for details.
In this section we show that, for every 0 ≤ α < 1, the measure µ α defined in (1.4) satisfies the Euler-Lagrange conditions (1.5)-(1.6),for some constant C α ∈ R. By the above discussion this immediately implies that µ α is the unique minimiser of I α , thus completing the proof of Theorem 1.1.The precise value of C α will be computed in Section 4.
We begin by studying W α * χ Ω(a,b) for every b ≥ a > 0. We note that the function [33]).As a first step, we compute the convolution ∇W α * χ Ω(a,b) (x) for every b ≥ a > 0 and at every point x ∈ R 2 .
In order to evaluate the convolution ∇W α * χ Ω(a,b) , it is convenient to work in complex variables.As usual, we identify z = x 1 + ix 2 ≡ x = (x 1 , x 2 ), and we write the standard differential operators as In complex variables the potential W α in (1.2) reads as and thus where x ⊥ = (x 2 , −x 1 ).The result is the following.
for every z ∈ Ω(a, b) and for every z ∈ Ω(a, b) c .Here and c 2 = b 2 − a 2 , where c is the eccentricity of the ellipse.
Proof of Proposition 3.1.We divide the proof into two steps.
Step 1: Computation of 1 z * χ Ω(a,b) and 1 z * χ Ω(a,b) .We observe that 1 z * χ Ω(a,b) is the Cauchy transform of the (characteristic function of the) ellipse Ω(a, b), up to a multiplicative constant.Indeed, the Cauchy transform of a C 1 domain Ω ⊂ C is defined as Clearly C(χ Ω ) is a continuous function in C, holomorphic in C \ Ω and vanishes at infinity.In the special case of an ellipse, namely for Ω = Ω(a, b), the expression (3.9) can be computed explicitly (see [22, page 1408]), and is given by where λ and h are as in (3.8).By taking the conjugate of (3.10) we obtain directly and hence the first two terms of ∇W α * χ Ω(a,b) are now computed.
Step 2: Computation of z z2 * χ Ω(a,b) .We start by observing that The previous expression implies that with h i 1 , h i 2 , h o 1 and h o 2 holomorphic functions in their respective domains (and the indices i and o stand for "inner" and "outer").
It remains to determine the functions h i 1 , h i 2 , h o 1 and h o 2 explicitly.By applying the operator ∂ to both sides of (3.13), we deduce which, together with (3.11), leads to the identification of h i 1 and h o 1 , as . Substituting these expressions into (3.13)we then have with h i 2 and h o 2 holomorphic functions in their respective domains, still to be determined.By (3.12), however, it is sufficient to determine their derivatives, since, by applying the operator ∂ to both sides of (3.14), we have Now we observe that the function 1 π z z2 * χ Ω(a,b) on the left-hand side of (3.15) is continuous in C and decays to zero as z → ∞; see e.g.[33].Therefore, also the right-hand side of (3.15) is continuous in C, which implies in particular that for every z ∈ ∂Ω(a, b).By using the expression of the boundary of the ellipse in complex variables, namely ∂Ω(a, b) = {z ∈ C : z = λz + 2abh(z)} where λ and h are defined as in (3.8), and by rearranging the terms in (3.16), we obtain that on ∂Ω(a, b).Consider now the auxiliary function Because of the continuity condition (3.17), R(z) is an anti-holomorphic function in C.Moreover, it easy to see that R has zero limit at ∞.This is clear for all the terms in the expression of R in Ω(a, b) c involving h and h ′ , by (3.8); for the term (h o 2 ) ′ it follows by (3.15).The Liouville Theorem then implies that R(z) ≡ 0. As a consequence, both expressions on the right-hand side of (3.18) are zero, which gives in Ω(a, b) c , and hence the identification of (h i 2 ) ′ and (h o 2 ) ′ in their respective domains.Plugging these formulas into (3.15),we finally conclude that Finally, using (3.10), (3.11) and (3.19) we have that (3.6) and (3.7) immediately follow.
We are now in the position to prove our main result.
Step 2: The measure µ α satisfies (3.4).By (3.7) we have that To simplify the expression (3.22) we also observe that where we have used the fact that c 2 = b 2 − a 2 = 2α.Substituting (3.23) and (3.24) into (3.22), and performing some simple algebraic manipulations, we deduce that Proving that (3.21) holds with a = √ 1 − α and b = √ 1 + α is then equivalent to showing that Now, we recall that √ z 2 + 2α denotes the branch of the complex square root preserving the quadrants, that is, for every . This is true since |z 2 + 2α| + |z| 2 − 2 is a level-set function for the ellipse Ω( √ 1 − α, √ 1 + α).This is a general statement for ellipses Ω(a, b) with b ≥ a > 0, that we prove in Lemma 3.2 below.
The proof of Theorem 1.1 is thus complete, up to the computation of the constant C α , that we postpone to Section 4. ) where Proof.By dilating z by a factor of 2 a+b we can further assume that a + b = 2, and write a = 1 − β and b = 1 + β, for some 0 ≤ β < 1.Thus , and the claim becomes ) which completes the proof of (3.28) (and then of (3.26)).The statement (3.27) can be proved in the same way.
The limiting case α = 1 studied in [29] can be obtained from our analysis, valid for 0 ≤ α < 1, by means of a simple argument based on Γ-convergence.As a first step, we note that (I α ) α∈(0,1) is an increasing family of lower semicontinuous functionals (with respect to the narrow convergence of measures).Hence I 1 is not only the pointwise limit of I α as α → 1 − , but also the Γ-limit, namely Γ-lim see, e.g., [17,Proposition 5.4].Let now µ α and µ 1 be the measures defined in (1.4) and in (1.3), respectively.Since µ α is a minimiser of I α for every α ∈ (0, 1), and since µ α ⇀ µ 1 narrowly as α → 1 − , the Fundamental Theorem of Γ-convergence implies that µ 1 is a minimiser of I 1 .It is in fact the unique minimiser, by the strict convexity of I 1 .
Corollary 3.3.The unique minimiser of I 1 is given by the semi-circle law

Further Comments
4.1.Stationarity of µ α : an alternative proof.Here, we provide an alternative proof of the fact that, for every α ∈ (0, 1), the ellipse-law µ α in (1.4) is a stationary solution of the gradient flow (1.9) associated to (1.5), namely it satisfies the Euler-Lagrange condition (1.5) inside its support, for some constant C α ∈ R. In doing so, we also compute the exact value of C α and the minimum value of I α .Finally, we explicitly compute the function W α * µ a,b in the whole of R 2 for a general ellipse (see Remark 4.1).We recall that µ a,b = 1 πab χ Ω(a,b) , with 0 < a ≤ b, is the (normalised) characteristic function of the ellipse of semi-axes a and b.
The proof we propose in this section uses the explicit expression of the logarithmic potential of µ a,b , namely − log |•| * µ a,b , which is well-known in the literature, in the context of fluid mechanics.This potential represents the stream function associated to the vorticity corresponding to an elliptic vortex patch (the Kirchhoff ellipse) rotating with constant angular velocity about its centre, and it was computed in order to prove that the Kirchhoff ellipses are V-states of the Euler equations in two dimensions [24,19,26,22].
The explicit expression of the logarithmic potential for any ellipse is well-known and is given by where the function H is defined as We note that H is real-valued, H(z) = H(z), and that for a < b.Note that Φ a,b is only one part (the radial component) of the convolution potential W α * µ a,b .We now show that the anisotropic part of W α * µ a,b can be obtained from Φ a,b by means of an ingenious differentiation.We first write (4.1) explicitly, for x ∈ Ω(a, b) and 0 < a < b: We perform a change of variables in order to write the integral in the expression above as an integral on the fixed domain B 1 (0), the unit disc.In terms of the new variables u = (u 1 , u 2 ) := , and the aspect ratio k := a/b, k ∈ (0, 1), the expression in (4.4) becomes By differentiating the previous expression (4.5) with respect to the aspect ratio k we obtain the identity 1 which, expressed in the original variables x and y, and a, b, becomes Note that the left-hand side of (4.6) is exactly the convolution of the anisotropic term of the potential W α with the measure µ a,b .This allows us to compute the whole convolution potential W α * µ a,b on Ω(a, b): Then we can evaluate the value of the energy I α on ellipses µ a,b , namely In particular, by (4.8) and (4.9) we obtain the minimum value I α (µ α ) of the energy I α , that is, + β − b γ 2 a 2 + γ 2 , so that, up to this change of variables, the study of the minimality of I α,β,γ reduces again to the original case.In particular, for b < 1 the minimiser is an ellipse with major axis along the line y 1 = 0, that is, −ax 1 + γx 2 = 0, while for b ≥ 1 the minimiser is the semi-circle law on that line.
The two orthogonal lines y 1 = 0 and y 2 = 0 are the zero set of the anisotropic force F α,β,γ , given by where x ⊥ = (x 2 , −x 1 ).The force F α,β,γ is perpendicular to the radial direction, and it is indeed zero only when which correspond to y 1 = 0 and y 2 = 0. Looking at the sign of the force in (4.11) it is clear that F α,β,γ points towards the line y 1 = 0.
which is the region surrounded by an ellipse centred at the origin with horizontal semi-axis a and vertical semi-axis b.As a first step, we compute explicitly the gradient of W α * χ Ω(a,b) , both inside and outside Ω(a, b); see equations (3.6)-(3.7) in Proposition 3.1.As we shall see, this is enough to conclude the proof of Theorem 1.1, but for completeness we shall also explicitly compute W α * χ Ω(a,b) in the whole plane (see Remark 4.1).The gradient of W α * χ Ω(a,b) is the sum of −(1/z) * χ Ω(a,b)