On a Lattice Generalisation of the Logarithm and a Deformation of The

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1.1 Simple lattices and energies Before giving the precise deﬁnition of this deformation and generalisation of η, let us specify what are the spaces of lattices we are interested in. For any d ≥ 1 and any V > 0, we call Ld the set of all the simple lattices of the form

d d L = Zvi, (v1, ..., vd) basis of R Mi=1 and Ld◦(V ) those with covolume | det(v1, ..., vd)| = V , which is also the volume of their unit cell. For any f such that the following series is absolutely convergent, a natural question that can be asked is the optimization of 2 L 7→ Ef [L] := f(|p| ) (1.2) p XL 0 ∈ \{ } in Ld◦(V ) for any ﬁxed V > 0. This problem has been studied for example in [15, 18] for generalisation of the Jacobi theta function and the Riemann zeta function deﬁned by

πα p 2 1 θ (α) := e− | | ,α> 0, and ζ (s) := , s > d, (1.3) L L |p|s Xp L p XL 0 ∈ ∈ \{ } that are respectively called lattice theta function and Epstein zeta function, originally deﬁned in [9, 12]. Both functions are of interest because of their connection with other lattice “energies” (see e.g. [2, 3, 5] and references therein). In particular, since the energy Ef of any function f being the Laplace transform of a nonnegative Borel measure µ, and decaying fast enough at inﬁnity, can be written, as explained in [2, Prop 3.1], as

∞ Ef [L]= (θL(α/π) − 1) dµ(α), Z0 it follows that an optimum Ld for L 7→ θL(α) for all α> 0 is also an optimum for Ef . Such functions f are called completely monotone and play an important role in the optimal point conﬁgurations community (see [7, 8]). Only three global optimality results has been shown for the lattice theta function:

• Montgomery [15]: If d = 2, the triangular lattice Λ := 2V Z(1, 0) ⊕ Z 1 , √3 is the V √3 2 2 q h i unique minimizer of L 7→ θL(α) in L2◦(V ) for all V,α > 0.

• Cohn-Kumar-Miller-Radchenko-Viazovska [8]: If d ∈ {8, 24}, E8 and the Leech lattice Λ24 are the unique minimizers of L 7→ θL(α) in Ld◦(V ) for all V,α > 0. More importantly, they are also the unique minimizers of this theta function among all periodic conﬁgurations of points (i.e. where there are several points per unit cell).

The results we present in this paper are restricted to the simple lattice case and, in order to have the most general results in this case, we call D the set of dimensions d such that L 7→ θL(α) has the same unique minimizer Ld in Ld◦(V ) for all V,α > 0, in such a way that we already know that {2, 8, 24} ⊂ D 6= ∅. Furthermore, as explained in [20, p. 117], we remark that 3 6∈ D 6= N. From the above discussion, we also notice that, for all d ∈ D, the minimizer of the lattice theta function is also the same for all the energies Ef where f is completely monotone.

2 1.2 Gannon’s deformation of the Dedekind eta function The goal of this paper is to use these results in order to ﬁnd the maximal possible value of the deformed general eta function deﬁned by Gannon in [11, Eq (3.5a)] as follows. Let V > 0, d ≥ 1,

L, Λ ∈Ld◦(V ) be two simple lattices and PΛ be the set of all vectors v ∈ Λ such that the only scalar multiples λv ∈ Λ are integer multiples. We note that v ∈ PΛ ⇐⇒ −v ∈ PΛ and we write PΛ/± for a subset of PΛ with only one representative of each pair of vectors ±v ∈ Λ. Therefore, for any m,t > 0, we deﬁned

2 1 ∞ d 1 π m ∆ (L) := − s −2 e s (θ (s) − 1) ds, m 1/2 − L∗ 8π|L| Z0

and θL∗ is the lattice theta function deﬁned by (1.3) for the dual lattice L∗ of L deﬁned by

d L∗ := {y ∈ R : ∀p ∈ L,y · p ∈ Z}.

(m) For another formula which may look more familiar to the reader, see (3.1). This function EL,Λ can be viewed as a deformation of the Dedekind eta function η deﬁned by (1.1) in the following sense. We ﬁrst remark that, for any t> 0,

(m) (m) EZ,Z (it) =: η (it),

1/2 (m) (m) where (2πmt)− η (it) → η(it) as m → 0. As explained before, the function η (it) appears in a Perturbative String Theory [1]. We also notice that ∆m is the lattice generalisation of the Casimir eﬀect of a boson in a cylindrical worldsheet, the masslike parameter m measuring how the background space-time – where the D-branes are located – is curved.

1.3 Goal of our work In order to explore the properties of this deformation of η, we ask the following question leading to the knowledge of extremal values for possible associated physical systems:

(m) • Problem: What are the global maximizers of (L, Λ) 7→ EL,Λ (it) in Ld◦(V1) ×Ld◦(V2) where V1,V2 > 0 and m,t > 0 are ﬁxed?

(m) Studying (L, Λ) 7→ EL,Λ (it) in Ld◦(V1) ×Ld◦(V2) is an interesting mathematical question by itself which encourages us to investigate the maximality properties of other lattice energies, expanding the use of known variational techniques to new objects, like the lattice-logarithm deﬁned in the next (m) subsection. Furthermore, even though EL,Λ (it) does not appear yet in any model of (Perturbative) String Theory, we hope that our results will help mathematical physicists to optimize quantities like an analog of the cylinder diagram in light-cone coordinates or the open-string one-loop diagram (see [1, Section 3]). We also believe that this type of lattice sums could be useful in Number Theory where such q-products appear frequently and are sometimes related to geometrical quantities, as η is related to the height of the ﬂat torus in dimension d = 2 (see [17]). This problem is solved in Theorem 3.2, showing that for any d ∈ D, the minimizers of the lattice theta functions are the (m) maximizers of (L, Λ) 7→ EL,Λ (it).

3 1.4 The lattice-logarithm As proposed by Gannon in the same paper (see [11, p. 64]), we deﬁne the lattice generalisation of (m) the logarithm as follows, which is connected to EL,Λ (it) by (3.1).

Deﬁnition 1.1 (Lattice-logarithm). For any d ≥ 1, any V > 0 and any simple lattice L ∈Ld◦(V ), we deﬁne the L-logarithm of any real number x ∈ (0, 1) by

1 (1 − x) p log (x) := − | | . L 2 |p| p XL 0 ∈ \{ }

Furthermore, if d = 1 and {tn}n Z is a N-periodic sequence of real numbers for some N ≥ 1, i.e. ∈ such that tn+N − tn = N for all n ∈ Z, then the {tn}-logarithm of x ∈ (0, 1) is deﬁned by

N tj ti 1 (1 − x)| − | log tn (x) := − . { } 4N |tj − ti| Xi=1 j XZ i ∈ \{ }

This L-logarithm is indeed a generalisation of the logarithm in the sense that logZ(x) = log(x) for any x ∈ (0, 1), and Gannon asked the question about the properties of this object.

Plan of the paper. In the next section, we study the maximization of L 7→ logL(x) for any given x ∈ (0, 1) as well as the problem of maximizing a lattice energy (see (2.2)) of type Ef deﬁned as in (1.2), where f depends on a lattice-logarithm. The fact that the L-logarithm is summed over another lattice Λ is the ﬁrst example of this kind, when L 6= Z. Finally, in Section 3, we 1 1 show our main result about the maximality of lattices V d L ,V d L for E(m)(it) when d ∈ D in 1 d 2 d L,Λ Ld◦(V1) ×Ld◦(V2).

2 Properties of the lattice-logarithm

We start by showing, in dimension d = 1, that the natural logarithm is characterized by a maximality problem. This result also allows to construct functions, in a canonical way, that are smaller and close to log(x).

Theorem 2.1 (Characterization of the logarithm). For any x ∈ (0, 1), we have, for any N,

Z log(x)= max log tn (x) : ∀n ∈ ,tn+N − tn = N , tn n Z { } { } ∈ n o and this maximum is achieved if and only if {tn} = Z + a for some a ∈ R.

(1 x)r Proof. Let x ∈ (0, 1) and, for any r > 0, φx(r) := −2ℓx(r) where ℓx(r) := − 2−r . We ﬁrst prove that φx is strictly convex, i.e. φx′′(r) > 0 for any r> 0. Indeed, we know that

r r log(1 x) (1 − x) e − ∞ tr = = e− dt, r r Z log(1 x) − − 2 tr and it follows that φx′′(r)= ∞log(1 x) t e− dt > 0 on (0, ∞). Thus, by Ventevogel’s optimalityR− result− [22, Theorem 1], we get, for any N ∈ N and any N-periodic sequence {tn}, N 1 ∞ φ (|t − t |) ≥ φ (k). 2N x j i x Xi=1 j XZ i Xk=1 ∈ \{ }

4 Thus, we get ∞ 1 ∞ (1 − x)k log tn (x) ≤ ℓx(k)= − = logZ(x) = log(x), { } 2 k Xk=1 Xk=1 and our result is proved because {tn} = Z + a are the only sequences satisfying the equality case in Jensen’s inequality.

Remark 2.2. One might be tempted to think that this result is true for any kind of lattice generalisation of a classical function, but one would be wrong. Indeed, we can deﬁne a {tn}- exponential of x> 0 by

N tj ti 1 x| − | exp tn (x) := , (2.1) { } 2N Γ(|tj − ti| + 1) Xi=1 j XZ i ∈ \{ } and ask the same question of the global optimality of {tn} = Z + a. It turns out that the function xr ψx(r) = Γ(r+1) is not convex for some values of the parameter, for example x = e, so our method cannot be used. Furthermore, for x = e, the function is not even decreasing on (0, ∞), which makes it a bad candidate for the optimality of the equidistant conﬁguration.

In higher dimension d ≥ 2, we restrict our study to the simple lattice case and we have the following result connecting the minimizers of the lattice theta function and the one of L 7→ logL(x) for any given real number x ∈ (0, 1).

Theorem 2.3 (Maximizer of the L-logarithm). Let d ∈ D, x ∈ (0, 1) and V > 0. Then, the unique 1 d minimizer V Ld of the lattice theta function is the unique maximizer of L 7→ logL(x) in Ld◦(V ).

(1 x)√r Proof. − It is suﬃcient to show that φx(r) := √r is a completely monotone function and to use ea√r [2, Prop 3.1] already mentioned in the introduction in order to conclude. Writing φx(r)= √r and remarking that a := log(1−x) < 0, fx is therefore completely monotone as a product of completely monotone functions (see [14]).

Since logL is a function by itself on (0, 1), we can use it to define a new lattice sum where the L-logarithm is an interacting potential. The problem of finding optimal point configurations for logarithmic interaction is indeed of great interest in dimension d = 2 [21] (where − log is the Coulomb potential) and on the (d − 1)-dimensional sphere [6]. We therefore obtain an energy depending on two simple lattices that we can maximize as follows.

Theorem 2.4 (The L-logarithm viewed as interacting potential). Let d ∈ D and V1,V2 > 0. For any simple lattices L ∈ Ld◦(V1) and Λ ∈ Ld◦(V2), and for any function f such that f ′ is completely monotone, we deﬁne f( q 2) Ef [L, Λ] := logL(1 − e− | | ). (2.2) Xq Λ ∈ 1 d Then, for any ﬁxed L and f, V2 Ld is the unique maximizer of Λ 7→ Ef [L, Λ] in Ld◦(V2). Further- 1 1 more, V d L ,V d L is the unique pair of maximizers of E in L (V ) ×L (V ). 1 d 2 d f d◦ 1 d◦ 2 Proof. We have, by absolute convergence,

p f( q 2) 1 e−| | | | 1 1 p f( q 2) Ef [L, Λ] = − = −  e−| | | |  , (2.3) 2 |p| 2 |p| Xq Λ p XL 0 p XL 0 Xq Λ ∈ ∈ \{ } ∈ \{ }  ∈ 

5 and it is clear that the energy is well-defined. Since f is such that f ′ is a completely monotone p r function and r 7→ e−| | is also completely monotone for any p ∈ L\{0}, it follows from [14, Thm 1 p f(r) d 2] that r 7→ e−| | is completely monotone. Therefore, by [2, Prop 3.1], V2 Ld is the unique p f( q 2) minimizer of Λ 7→ q Λ e−| | | | in Ld◦(V2) and the first part of the theorem is proved. The second part followsP from∈ the first part by using (2.3) and Theorem 2.3. Remark 2.5. As already mentioned in the introduction, it is, as far as we know, the first time that such energy with an interaction potential depending itself on a lattice is studied. It might be interesting to studied a lattice generalisation of the lattice theta function, using the L-exponential defined by (2.1) and see if one can again derive optimality results of this kind. The fact that the associated potential ψx defined in Remark 2.2 is not convex might be a problem but also an interesting direction to explore.

E(m) it 3 Maximization of L,Λ ( ) among simple lattices

(m) It is straightforward to obtain the following formula that connects EL,Λ (it) to the lattice-logarithm.

Lemma 3.1 ([11]). For any t,m > 0 and any simples lattices L, Λ ∈Ld, we have

d+1 2 d 1 m2 d 1 2 2 (m) t ∞ − π − m + p log E (it)= − s 2 e− s (θL (s) − 1) ds + t 2 log (1 − q | | ). (3.1) L,Λ 8|L|1/2 Z ∗ Λ 0 Xp L ∈ (m) Therefore, we can easily derive the maximum of (L, Λ) 7→ EL,Λ (it) among simple lattices.

1 1 Theorem 3.2 (Maximizer of E(m)(it)). Let d ∈ D, t,m > 0 and V ,V > 0. Then V d L ,V d L L,Λ 1 2 1 d 2 d (m) is the unique maximizer of (L, Λ) 7→ EL,Λ (it) in Ld◦(V1) ×Ld◦(V2).

d 1 m2 D −2 π s Proof. We use the previous formula (3.1). Since d ∈ , s e− > 0 for any s> 0 and Ld = Ld∗ as a simple consequence of the Poisson Summation Formula, we know that

2 ∞ d 1 m −2 π s L 7→ s e− (θL∗ (s) − 1) ds Z0

1 1 d d is minimized by V1 Ld in Ld◦(V1). Therefore the ﬁrst term is maximized by V1 Ld. For the second 2 term, we use Theorem 2.4 with f(r)= πt(m + r) and we remark that f ′(r)= πt is a completely 1 1 monotone function, which implies that the second term is maximized by V d L ,V d L , which 1 d 2 d concludes the proof.

Acknowledgement: I acknowledge support from VILLUM FONDEN via the QMATH Centre of Excellence (grant No. 10059).

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