Extremal determinants of Laplace–Beltrami operators for rectangular tori

Abstract

In this work we study the determinant of the Laplace–Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments. Refined logarithmic convexity and concavity results of the classical Jacobi theta functions of one real variable are deeply involved.

1 Introduction

The search for extremal geometries is a popular topic in many branches of mathematics and mathematical physics. In this work, we pick up a result by Osgood et al. [26] on extremals of determinants of Laplace–Beltrami operators on tori and restrict the assumptions, excluding their solution of the following problem.

Among all 2-dimensional tori of area 1, which torus maximizes the determinant of the Laplace–Beltrami operator?

The answer in [26] is that the torus identified with the plane modulo a hexagonal (sometimes called triangular or equilateral) lattice gives the unique solution. However, if we only consider rectangular lattices, this solution is not possible and the natural assumption is that the square lattice will lead to the optimal solution. We will prove that this is indeed the case. Both problems, the one for general and the one for rectangular lattices, are closely related to the study of extremal values of the heat kernel on the torus [3, 5], finding extremal bounds of Gaussian Gabor frames of given density [17,18,19], as well as the study of certain theta functions [25].

It is worth noting that in all cases the extremal solutions are the same as for the classical sphere packing and covering problem in the plane. This immediately raises the question about extremal solutions for the above problems in higher dimensions. An interesting aspect is that in higher dimensions the optimal arrangements for the sphere packing and covering problem differ from each other. Hence, in order to guess what the right solution could be, one first has to decide whether one deals with a packing or a covering problem. However, we will not discuss higher dimensions in this work.

Another common theme is that we deal with theta functions in one way or another, which take a prominent role in several branches of mathematics. They appear in the studies on energy minimization [6], the study of Riemann’s zeta and xi function [9, 12] or the theory of sphere packing and covering [11], including the recent breakthrough for sphere packings in dimension 8 by Viazovska [31] and in dimension 24 [10]. They also appear in the field of time–frequency analysis and the study of Gaussian Gabor frames [18, 19, 21] or the study of the heat kernel [30] to name just a few.

An open question concerns how the above problems can be linked to old, unsolved problems in geometric function theory, namely finding the exact values of Bloch’s constant [7] and Landau’s constant [24]. The correct solutions are conjectured to be given in the work of Ahlfors and Grunsky [1] and in the work of¹Rademacher [27], respectively.

Baernstein repeatedly suggested that a better understanding of the behavior of extremal values of the heat kernel on the torus might lead to new insights for the mentioned constants [2, 3, 5] and, after some research, the author shares this opinion. In fact, this work is a result of the author’s study on a conjecture of Baernstein et al. [4] related to Landau’s constant, which Eremenko posed again as an open problem in an unpublished preprint in 2011 [13]. However, besides the fact that in both cases varying metrics on rectangular tori are involved, it is not clear to the author how deep the connection between this work and the mentioned conjecture in [4, 13] truly is.

Let us return to the question posed at the beginning. In [26] we find the following result, which is²Corollary 1.3(b) in that work. For a lattice $\Lambda$, let $\Delta _\Lambda$ be the Laplace–Beltrami operator for the torus $\mathbb {T}_\Lambda = \mathbb {C}/\Lambda$ of unit area, then

$$\begin{aligned} {\det }^\prime \Delta _\Lambda \le \tfrac{\sqrt{3}}{2} \, | \eta ( \tfrac{1 + i\sqrt{3}}{2} ) |^4 \approx 0.35575 \dots \end{aligned}$$

with equality if and only if $\Lambda$ is hexagonal. The result was established by first showing that the hexagonal lattice gives a local maximum by exploiting general facts about modular forms. Then, a numerical check gave the result that this local maximum is indeed global.

In another work, Sarnak [29] mentions that³Karnaukh has shown that the square lattice gives the only other critical point and that it is a saddle point. In particular, this implies that there exists a one-parameter family of lattices, within which the square lattice yields a local maximum of the determinant. Actually, we will see that among all rectangular tori, the square torus gives the global maximum. Our main result is as follows.

Theorem 1.1

(Main result) For $\alpha \in \mathbb {R}_+$, we denote the rectangular torus of unit area by $\mathbb {T}_\alpha = \mathbb {C}/(\alpha ^{-1/2} \mathbb {Z}\times i \, \alpha ^{1/2} \mathbb {Z})$ and the Laplace–Beltrami operator by $\Delta _\alpha$. Then

$$\begin{aligned} {\det }^\prime \Delta _\alpha = \alpha \, | \eta (\alpha i) |^4 \le | \eta (i) |^4 \approx 0.34830 \dots \end{aligned}$$

with equality if and only if $\alpha = 1$.

Our proof of Theorem 1.1 will at first be parallel to the proof in [26], in particular, we will show that the problem about the determinant can be transferred to a problem of finding the maximum of the Dedekind eta function on a ray in the upper half plane. After that point, the proof will differ greatly from the methods in [26]. Not only will we show that the square lattice yields the global maximum, we will also give a precise behavior of the determinant as the lattice parameter varies. The key in our proof is to exploit the fact that the eta function can be decomposed into a product of Jacobi’s theta functions. The result will follow from certain logarithmic convexity and concavity results, partially established already in [19].

This work is structured as follows;

• In Sect. 2 we recall the definitions of Laplace–Beltrami operators on tori and their determinants as well as the results from the work of Osgood et al. [26]. Also, we will see how the determinant connects with the Dedekind eta function.

• In Sect. 3 we define Jacobi’s classical theta functions of one real variable and show how they can be used to express the Dedekind eta function. The proof of the main result will follow from refined logarithmic convexity and concavity statements related to Jacobi’s theta functions as described by Faulhuber and Steinerberger [19].

2 The Laplace–Beltrami operator on the torus

In this section we recall the results established in [26] and how the problem of finding extremal surfaces for determinants of Laplace–Beltrami operators connects with the Dedekind eta function. Osgood, Phillips and Sarnak studied the determinant of the Laplace–Beltrami operator on a surface with varying metric as a function of the metric. In the case of the torus with flat metric, the varying of the metric can be interpreted as varying the lattice associated to the torus (and keeping the standard metric). Furthermore, we will introduce the heat kernel of a Laplace–Beltrami operator and its determinant.

We denote the Laplace–Beltrami operator on the 2-dimensional torus $\mathbb {T}_\Lambda = \mathbb {C}/\Lambda$ by $\Delta _\Lambda$. The torus is identified with the fundamental domain of the lattice $\Lambda$ which is a discrete subgroup of $\mathbb {C}$. A lattice $\Lambda$ is generated by integer linear combinations of two nonzero complex numbers $z_1$ and $z_2$ with the property that $\frac{z_1}{z_2} \notin \mathbb {R}$;

$$\begin{aligned} \Lambda = \langle z_1, z_2 \rangle _\mathbb {Z}= \{m z_1 + n z_2 \mid m,n \in \mathbb {Z}, \, z_1, z_2 \in \mathbb {C}, \, \tfrac{z_1}{z_2} \notin \mathbb {R}\}. \end{aligned}$$

The area of the torus is then defined to be the area of a fundamental domain, i.e.,

$$\begin{aligned} \text {area}(\Lambda ) = | Im(\overline{z_1} z_2)| = |x_1 y_2 - x_2 y_1|, \qquad z_k = x_k + i \, y_k, \, k = 1,2. \end{aligned}$$

In this work we solely deal with rectangular lattices, i.e., we can choose a basis of the lattice $\langle z_1, z_2 \rangle _\mathbb {Z}$ with the property that the ratio of $z_1$ and $z_2$ is purely imaginary;

$$\begin{aligned} \Lambda \text { rectangular} \quad \Longleftrightarrow \quad \exists z_1, z_2 \in \mathbb {C}:\Lambda = \langle z_1, z_2 \rangle _\mathbb {Z}\; \wedge \; i \frac{z_1}{z_2} \in \mathbb {R}. \end{aligned}$$

We note that any rectangular lattice can be identified with $\alpha \mathbb {Z}\times i \, \beta \mathbb {Z}$ where $\alpha , \beta \in \mathbb {R}_+$. In this work, it will be no restriction to assume that the lattice has unit area, i.e., $\alpha \beta = 1$.

2.1 The Laplace–Beltrami operator on manifolds and its heat kernel

We will now introduce Laplace–Beltrami operators on connected Riemannian manifolds as well as the associated heat kernel and the determinant. For further reading on heat kernels we refer to [22]. Also, the procedure of introducing determinants of Laplace–Beltrami operators is described in [5, 26] or [28], and we will follow these references.

Just for the moment, let us change to a more general notation. Let M be a connected Riemannian manifold and $\Delta _M$ the corresponding Laplace–Beltrami operator. The heat semigroup is defined as

$$\begin{aligned} P_t = \{ e^{t \Delta _M} \mid t \in \mathbb {R}_+ \}. \end{aligned}$$

The action of the group $P_t$ on a function $f \in L^2(M)$ is (by abuse of notation) given by the integral operator

$$\begin{aligned} P_t f(x) = \int _M p_t(x,y) f(y) \, d \mu (y), \end{aligned}$$

where $\mu (y)$ is the⁴Lebesgue measure on M. The integral kernel $p_t$ is called the heat kernel of the Laplace–Beltrami operator. For any $y \in M$, the heat kernel fulfills the heat equation

$$\begin{aligned} \Delta _M u - \partial _t u = 0 \end{aligned}$$

and for any $y \in M$

$$\begin{aligned} p_t(\, . \, , \, y) \rightarrow \delta _y, \qquad t \rightarrow 0. \end{aligned}$$

If the spectrum of $\Delta _M$ is discrete and consists of eigenvalues $\{\lambda _k\}_{k=1}^\infty$ with $\{\phi _k\}_{k=1}^\infty$ being the sequence of corresponding eigenfunctions, constituting an orthonormal basis for $L^2(M)$, then the heat kernel can be expanded as

$$\begin{aligned} p_t(x,y) = \sum _{k=1}^\infty e^{-\lambda _k t} \phi _k(x) \overline{\phi _k(y)}. \end{aligned}$$

Also, in this case the trace of the heat kernel is given by

$$\begin{aligned} \text {tr}(e^{t \Delta _M}) = \int _M p_t(x,x) \, d \mu (x) = \sum _{k=1}^\infty e^{-\lambda _k t}. \end{aligned}$$

The left-hand side of the above equation defines the trace of the heat kernel in general. To the Laplace–Beltrami operator one associates a zeta function in the following way

$$\begin{aligned} Z_M(s) = \sum _{k=1}^\infty \lambda _k^{-s}, \qquad Re(s) > 1. \end{aligned}$$

This function can be continued analytically in $\mathbb {C}\backslash \{ 1 \}$. Formally, the determinant of $\Delta _M$ is given by

$$\begin{aligned} \det \Delta _M = \prod _{\lambda _k \ne 0} \lambda _k. \end{aligned}$$

This product is not necessarily meaningful and the proper definition of the determinant is to use the zeta regularization

$$\begin{aligned} {\det }^\prime \Delta _M = e^{-\tfrac{d}{ds} Z_M \big |_{s=0}}. \end{aligned}$$

A closely related function is the height function of a Riemannian manifold [28],

$$\begin{aligned} h_M = - \log {\det }^\prime \Delta _M = \dfrac{d}{ds} Z_M \big |_{s=0}. \end{aligned}$$

This function is an isospectral invariant and it is clear that problems about ${\det }^\prime \Delta _M$ can be transferred to problems about $h_M$ and vice versa.

2.2 The determinant on the torus

We return to the case of the torus $\mathbb {C}/\Lambda$ where $\Lambda$ is a lattice of unit area. As we will see, it is no restriction to assume that $\Lambda = c \, \langle 1, z\rangle _\mathbb {Z}$, where c is chosen such that the area of the lattice is 1, i.e., $c = Im(z)^{-1/2}$. In other words, the problem under consideration is invariant under rotation and scaling, just like the classical sphere packing and covering problems.

In this case, the eigenvalues of the Laplace–Beltrami operator $\Delta _\Lambda$ are $(2 \pi |z_\lambda |)^2$, where $z_\lambda \in \Lambda$. At this point, we mention that the eigenvalues are actually given by $(2 \pi |z_\lambda ^\bot |)^2$, $z_\lambda ^\bot \in \Lambda ^\bot$, the dual lattice. However, for 2-dimensional lattices, the relation between a lattice and its dual lattice is simply given by

$$\begin{aligned} \Lambda ^\bot = \text {area}(\Lambda )^{-1} \, i \, \Lambda . \end{aligned}$$

This means that the dual lattice is a 90$^{\circ }$ rotated, scaled version of the original lattice. However, as mentioned we deal with lattices of unit area, hence the scaling factor is irrelevant, as is the rotation. Thus, in our concrete situation, there is no need to distinguish between the lattice $\Lambda$ and its dual $\Lambda ^\bot$ (this only results in a re-labeling of the eigenvalues). The zeta function is, consequently, given by

$$\begin{aligned} Z_\Lambda (s) = {\sum _{z_\lambda \in \Lambda }}^\prime (2 \pi | z_\lambda |)^{-2s}, \end{aligned}$$

where the prime indicates that the sum does not include the origin. Using the definition of the lattice $\Lambda = \langle z_1, z_2 \rangle _\mathbb {Z}$, we re-write the zeta function as

$$\begin{aligned} Z_\Lambda (s) = (2 \pi )^{-2s} {\sum _{k,l \in \mathbb {Z}}}^\prime \frac{1}{|k z_1 + l z_2|^{2s}} = (2 \pi )^{-2s} {\sum _{k,l \in \mathbb {Z}}}^\prime \frac{y^s}{|k + l z|^{2s}}. \end{aligned}$$

For the last equality we set $z = \tfrac{z_2}{z_1}$ and $y = Im(z) > 0$, where the second condition is imposed by the fact that the lattice has unit area. Osgood et al. [26] now use the fact that the last series is a multiple of the Eistenstein series

$$\begin{aligned} E_\Lambda (z,s) = {\sum _{k,l \in \mathbb {Z}}}^\prime \frac{y^s}{|k + l z|^{2s}}, \end{aligned}$$

hence,

$$\begin{aligned} Z_\Lambda (s) = (2 \pi )^{-2 s} E_\Lambda (z,s). \end{aligned}$$

The final step in order to compute ${\det }^\prime \Delta _\Lambda$ is now to differentiate $Z_\Lambda$ with respect to s and evaluate at 0. We compute that

$$\begin{aligned} \dfrac{d}{ds} Z_\Lambda (s) = (2 \pi )^{-2 s} \left( -2 \log (2 \pi ) \, E_\Lambda (z,s) + \dfrac{\partial }{\partial s} E_\Lambda (z,s) \right) . \end{aligned}$$

Now, the following result is a consequence of Kronecker’s limit formula.

$$\begin{aligned} E_\Lambda (z,0)&= -1,\\ \dfrac{\partial }{\partial s} E_\Lambda \big |_{s=0}&= - 2 \log \left( 2 \pi \, y^{1/2} | \eta (z) |^2 \right) . \end{aligned}$$

The Dedekind eta function $\eta (\tau )$ is defined in the upper half plane ${\mathbb {H}}$ by the infinite product

$$\begin{aligned} \eta (\tau ) = e^{\pi i \tau /12} \prod _{k=1}^\infty (1-e^{2 \pi i k \tau }). \end{aligned}$$

(2.1)

Hence,

$$\begin{aligned} \dfrac{d}{ds} Z_\Lambda \big |_{s=0} = - 2 \log \left( y^{1/2} | \eta (z) |^2 \right) . \end{aligned}$$

It follows that the (zeta regularized) determinant of the Laplace–Beltrami operator $\Delta _\Lambda$, with $\Lambda = y^{-1/2} \langle 1, z \rangle _\mathbb {Z}$, $z = x + i y$, $y>0$, is given by

$$\begin{aligned} {\det }^\prime \Delta _\Lambda = y |\eta (z)|^4. \end{aligned}$$

2.3 Rectangular tori

We have seen that maximizing the determinant ${\det }^\prime \Delta _\Lambda$, where the lattice is given by $\Lambda = y^{-1/2} \langle 1, z \rangle _\mathbb {Z}$, corresponds to maximizing $y |\eta (z)|^4$. In the case of a rectangular torus $\mathbb {C}/(y^{-1/2} \mathbb {Z}\times i \, y^{1/2} \mathbb {Z})$, the problem in focus is

$$\begin{aligned} \text {maximize} \quad y |\eta (i y)|^4, \qquad y \in \mathbb {R}_+. \end{aligned}$$

We could end this work now with the fact, already observed by Baernstein and Vinson [5], that maximizing ${\det }^\prime \Delta _\Lambda$ is implied by minimizing $\text {tr}(e^{t \Delta _\Lambda })$. Montgomery showed that the unique minimizer of $\text {tr}(e^{t \Delta _\Lambda })$ is the hexagonal lattice [25]. In a recent work on Gaussian Gabor frames, the results of Faulhuber and Steinerberger [19] imply that the square lattice is the unique minimizer of $\text {tr}(e^{t \Delta _\Lambda })$ within the class of rectangluar lattices. Hence, Theorem 1.1 is implied by the results in [19]. However, we will come up with a proof independent from the results in [19, 25], but, however, the techniques are similar. The rest of this work is meant to give a deeper insight into the mentioned problems and maybe these insights can be valuable for some of the related problems mentioned in the introduction.

3 Jacobi’s theta functions

In this section we study properties of Jacobi’s theta functions which will lead to a deeper understanding of Theorem 1.1. We start by defining the theta function in accordance with the textbook of Stein and Shakarchi [30].

Definition 3.1

For $z \in \mathbb {C}$ and $\tau \in {\mathbb {H}}$ (the upper half plane) we define the theta function as

$$\begin{aligned} \Theta (z, \tau ) = \sum _{k \in \mathbb {Z}} e^{\pi i k^2 \tau } e^{2 k \pi i z}. \end{aligned}$$

(3.1)

This function is an entire function with respect to z and holomorphic with respect to $\tau$. As stated in [30], the function arises in many different fields of mathematics, such as the theory of elliptic functions, the theory of modular functions, as a fundamental solution of the heat equation on the torus as well as in the study of Riemann’s zeta function. Also, it is used to prove results in combinatorics and number theory.

The function can also be expressed as an infinite product.

Proposition 3.2

(Jacobi triple product) For $z \in \mathbb {C}$ and $\tau \in {\mathbb {H}}$ we have

$$\begin{aligned} \Theta (z,\tau ) = \prod _{k \ge 1} \left( 1 - e^{2 k \pi i \tau }\right) \left( 1 + e^{(2k-1) \pi i \tau } e^{2\pi i z}\right) \left( 1 + e^{(2k-1) \pi i \tau } e^{-2\pi i z}\right) . \end{aligned}$$

It also fulfills the following identity.

Theorem 3.3

For $z \in \mathbb {C}$ and $\tau \in {\mathbb {H}}$ we have

$$\begin{aligned} \Theta \left( z, -\tfrac{1}{\tau }\right) = (-i \tau )^{1/2} e^{\pi i z^2 \tau } \Theta (\tau z, \tau ). \end{aligned}$$

For a proof of Theorem 3.3 and more details on theta functions of two complex variables as well as the product representation we refer to the textbook of Stein and Shakarchi [30].

Whereas derivatives of $\Theta$ with respect to z are often studied, it seems that studies of its derivatives with respect to $\tau$ are less common. The following Lemma contains a symmetry result for the logarithmic derivative of $\Theta$ with respect to $\tau$.

Lemma 3.4

For $z \in \mathbb {C}$ and $\tau \in {\mathbb {H}}$ we have

$$\begin{aligned} \pi i z^2 \tau + \tau \frac{z \, \partial _z \Theta (\tau z, \tau ) + \partial _\tau \Theta (\tau z, \tau )}{\Theta (\tau z, \tau )} - \frac{1}{\tau } \frac{\partial _\tau \Theta \left( z, -\tfrac{1}{\tau }\right) }{\Theta \left( z, -\tfrac{1}{\tau }\right) } = -\frac{1}{2}. \end{aligned}$$

In particular, for $z = 0$ we get

$$\begin{aligned} \tau \frac{\partial _\tau \Theta (0, \tau )}{\Theta (0, \tau )} - \frac{1}{\tau } \frac{\partial _\tau \Theta \left( 0, -\tfrac{1}{\tau }\right) }{\Theta \left( 0, -\tfrac{1}{\tau }\right) } = -\frac{1}{2}. \end{aligned}$$

Proof

We start by taking the logarithm on both sides of the identity in Theorem 3.3

$$\begin{aligned} \log \left( \Theta \left( z, -\tfrac{1}{\tau }\right) \right) = \frac{1}{2} \log (-i \tau ) + \pi i z^2 \tau +\log \left( \Theta (\tau z, \tau )\right) . \end{aligned}$$

Differentiating with respect to $\tau$ on both sides and a multiplication by $\tau$ yields, after rearranging the terms, the desired result. $\square$

As a next step, we define Jacobi’s theta functions of two variables which we will then restrict to certain domains. For $z \in \mathbb {C}$ and $\tau \in {\mathbb {H}}$ we define

$$\begin{aligned} \vartheta _1(z,\tau )&= -i \sum _{k \in \mathbb {Z}} (-1)^{k} e^{\pi i (k+1/2)^2 \tau } e^{(2k+1)\pi i z}\\&= 2 \sum _{k \in \mathbb {N}} (-1)^{k} e^{\pi i (k+1/2)^2 \tau } \sin ((2k+1)\pi z)\\ \vartheta _2(z,\tau )&= \sum _{k \in \mathbb {Z}} e^{\pi i (k-1/2)^2 \tau } e^{(2k+1)\pi i z}\\&= 2 \sum _{k \in \mathbb {N}} e^{\pi i (k-1/2)^2 \tau } \cos ((2k+1)\pi z)\\ \vartheta _3(z,\tau )&= \sum _{k \in \mathbb {Z}} e^{\pi i k^2 \tau } e^{2 k \pi i z}\\&= 1 + 2 \sum _{k \in \mathbb {N}} e^{\pi i k^2 \tau } \cos (2 k \pi z)\\ \vartheta _4(z,\tau )&= \sum _{k \in \mathbb {Z}} (-1)^k e^{\pi i k^2 \tau } e^{2 k \pi i z}\\&= 1 + 2 \sum _{k \in \mathbb {N}} (-1)^k e^{\pi i k^2 \tau } \cos (2 k \pi z). \end{aligned}$$

We note that $\vartheta _3(z,\tau ) = \Theta (z,\tau )$ and that any $\vartheta _j$ can be expressed via $\Theta$ from equation (3.1). The functions which we will study in the rest of this work are restrictions of the above functions with $(z,\tau ) = (0,i x)$, $x \in \mathbb {R}_+$. In particular, all these functions are real-valued.

Definition 3.5

For $x \in \mathbb {R}_+$ we define the real-valued theta functions in the following way.

$$\begin{aligned}&\theta _2(x) = \vartheta _2(0,i x) = \sum _{k \in \mathbb {Z}} e^{-\pi (k-1/2)^2 x} = 2 \sum _{k \in \mathbb {N}} e^{-\pi (k-1/2)^2 x}, \end{aligned}$$

(3.2)

$$\begin{aligned}&\theta _3(x) = \vartheta _3(0,i x) = \sum _{k \in \mathbb {Z}} e^{-\pi k^2 x} = 1 + 2 \sum _{k \in \mathbb {N}} e^{-\pi k^2 x}, \end{aligned}$$

(3.3)

$$\begin{aligned}&\theta _4(x) = \vartheta _4(0,i x) = \sum _{k \in \mathbb {Z}} (-1)^k e^{-\pi k^2 x} = 1 + 2\sum _{k \in \mathbb {N}} (-1)^k e^{-\pi k^2 x}. \end{aligned}$$

(3.4)

It does not make much sense to study properties of $\theta _1(x)$ if defined as above, because $\vartheta _1(0,i x) = 0$ for all $x \in \mathbb {R}_+$. However, we will see that $\theta _1$ is involved in the proof of our main result in some sense.

All of the above functions can also be expressed by infinite products.

$$\begin{aligned} \theta _2(x)&= 2 e^{-\tfrac{\pi }{4} x} \prod _{k \in \mathbb {N}} \left( 1 - e^{-2 k \pi x}\right) \left( 1 + e^{-2 k \pi x}\right) ^2\\ \theta _3(x)&= \prod _{k \in \mathbb {N}} \left( 1 - e^{-2 k \pi x}\right) \left( 1 + e^{-(2 k - 1) \pi x}\right) ^2\\ \theta _4(x)&= \prod _{k \in \mathbb {N}} \left( 1 - e^{-2 k \pi x}\right) \left( 1 - e^{-(2 k - 1) \pi x}\right) ^2 \end{aligned}$$

These representations can be quite useful when studying the logarithmic derivatives of these functions. We note that for $z \in \mathbb {R}$ and purely imaginary $\tau = i x$, $x \in \mathbb {R}_+$ the theta function $\Theta (z, ix)$ is maximal for $z \in \mathbb {Z}$ and minimal for $z \in \mathbb {Z}+ \frac{1}{2}$. These special cases correspond to Jacobi’s $\theta _3$ and $\theta _4$ function

$$\begin{aligned} \theta _3(x)&= \Theta \left( 0,ix \right) ,\\ \theta _4(x)&= \Theta \left( \tfrac{1}{2}, ix \right) . \end{aligned}$$

For $x \in \mathbb {R}_+$, the $\theta _2$-function can be expressed via $\Theta$ in the following way.

$$\begin{aligned} \theta _2(x) = e^{-\pi \frac{x}{4}} \, \Theta \left( \tfrac{i x}{2},ix \right) . \end{aligned}$$

A recurring theme will be the frequent use of the differential operator $x \, \frac{d}{dx}$. We will use the notation $\left( x \frac{d}{dx}\right) ^n$ for its iterated repetition, i.e.,

$$\begin{aligned} \left( x \frac{d}{dx}\right) ^n = x \frac{d}{dx} \left( x \frac{d}{dx}\right) ^{n-1}. \end{aligned}$$

In particular, we will study properties of the logarithmic derivative of a certain function $f: \mathbb {R}_+ \rightarrow \mathbb {R}_+$, on a logarithmic scale. Let us explain how this statement should be interpreted. By using the variable transformation $y = \log (x)$ we extend the domain from $\mathbb {R}_+$ to $\mathbb {R}$ and consider the new function $\log \left( f \left( e^y \right) \right)$. Therefore, we get an extra exponential factor each time we take a (logarithmic) derivative. We have

$$\begin{aligned} \frac{d}{dy} \log \left( f \left( e^y \right) \right) = e^y \frac{f'\left( e^y\right) }{f\left( e^y\right) }. \end{aligned}$$

By reversing the transformation of variables, we come back to the original scale, but the factor x stays. Without being explicitly mentioned, these methods were used in [18, 19, 25] to establish uniqueness results about extremal theta functions on lattices. We will now provide a new point of view on as well as new results related to the mentioned articles.

For what follows it is necessary to clarify the notation of the Fourier transform and Poisson’s summation formula which are given by

$$\begin{aligned} {\widehat{f}}(\omega ) = \int _{\mathbb {R}} f(x) e^{-2 \pi i \omega x} \, dx, \qquad x,\omega \in \mathbb {R}\end{aligned}$$

and

$$\begin{aligned} \sum _{k \in \mathbb {Z}} f(k+x) = \sum _{l \in \mathbb {Z}} {\widehat{f}}(l) e^{2 \pi i l x}, \qquad x \in \mathbb {R}. \end{aligned}$$

respectively. Both formulas certainly hold for Schwartz functions and since we will apply both only on Gaussians we do not have to worry about more general properties for the formulas to hold.

As a consequence of the Poisson summation formula we find the following, well-known identities

$$\begin{aligned}&\sqrt{x} \, \theta _3 \left( x \right) = \theta _3 \left( \tfrac{1}{x} \right) , \end{aligned}$$

(3.5)

$$\begin{aligned}&\sqrt{x} \, \theta _2 \left( x \right) = \theta _4 \left( \tfrac{1}{x} \right) \quad \text { and } \quad \sqrt{x} \, \theta _4 \left( x \right) = \theta _2 \left( \tfrac{1}{x} \right) . \end{aligned}$$

(3.6)

We note the common theme in the last identities which leads to the following lemma.

Lemma 3.6

Let $r \in \mathbb {R}$ and suppose that $f,g \in C^1\left( \mathbb {R}_+ , \mathbb {R}\right)$ do not possess zeros. If f and g satisfy the (generalized Jacobi) identity

$$\begin{aligned} x^r \, f\left( x \right) = g\left( \tfrac{1}{x} \right) , \end{aligned}$$

then

$$\begin{aligned} x \, \frac{f' \left( x \right) }{f \left( x \right) } + \tfrac{1}{x} \, \frac{g' \left( \tfrac{1}{x} \right) }{g \left( \tfrac{1}{x} \right) } = -r. \end{aligned}$$

Proof

Both, f and g are either positive or negative on $\mathbb {R}_+$ since they are real-valued, continuous and do not contain zeros. As they also fulfill the identity $x^r \, f(x) = g\left( \tfrac{1}{x}\right)$ both of them possess the same sign. Without loss of generality we may therefore assume that both functions are positive because otherwise we change the sign which does not affect the results. Therefore, we have

$$\begin{aligned} \log \left( x^r \, f(x) \right) = \log \left( g \left( \tfrac{1}{x} \right) \right) . \end{aligned}$$

Using the differential operator $x \frac{d}{dx}$ on both sides directly yields

$$\begin{aligned} r + x \, \frac{f'(x)}{f(x)} = -\tfrac{1}{x} \, \frac{g'\left( \tfrac{1}{x}\right) }{g\left( \tfrac{1}{x}\right) }. \end{aligned}$$

$\square$

We remark that an alternative proof of Lemma 3.6 is given in [19] for the special case $f = g = \theta _3$ and $r = \tfrac{1}{2}$ (a generalization of the proof in [19] is possible with the arguments given there). With this proof, the result

$$\begin{aligned} x \, \frac{\theta _3'(x)}{\theta _3(x)} + \tfrac{1}{x} \, \frac{\theta _3' \left( \tfrac{1}{x} \right) }{\theta _3 \left( \tfrac{1}{x} \right) } = -\frac{1}{2} \end{aligned}$$

(3.7)

was derived.

However, it seemed to have gone unnoticed that the result can be put into a more general context. With the previous lemma we also get the results

$$\begin{aligned} x \, \frac{\theta _2'(x)}{\theta _2(x)} + \tfrac{1}{x} \, \frac{\theta _4' \left( \tfrac{1}{x} \right) }{\theta _4 \left( \tfrac{1}{x} \right) } = -\frac{1}{2} \qquad \text { and } \qquad x \, \frac{\theta _4'(x)}{\theta _4(x)} + \tfrac{1}{x} \, \frac{\theta _2' \left( \tfrac{1}{x} \right) }{\theta _2 \left( \tfrac{1}{x} \right) } = -\frac{1}{2}, \end{aligned}$$

(3.8)

which were not given in [19]. Also, the author just learned that the above formulas can also be found in the monograph of Borwein and Borwein [8, chap. 2.3]. All of the above results actually contain symmetry statements about logarithmic derivatives of Jacobi’s theta functions on a logarithmic scale. We note that the identities (3.7) and (3.8) are special cases of Lemma 3.4 when $\Theta$ is restricted to certain rays in $\mathbb {C}\times {\mathbb {H}}$.

3.1 Properties of Theta-2 and Theta-4

We start with monotonicity properties of the logarithmic derivatives of Jacobi’s $\theta _2$ and $\theta _4$ functions on a logarithmic scale. The following result was already given in [19] and the proof can be established by using the product representation of $\theta _4$.

Proposition 3.7

The function $x \frac{\theta '_4(x)}{\theta _4(x)}$ is strictly decreasing on $\mathbb {R}_+$.

As a consequence of Proposition 3.7 we derive the following result.

Proposition 3.8

The Jacobi theta function $\theta _4$ is strictly logarithmically concave or, equivalently, $\frac{\theta _4'(x)}{\theta _4(x)}$ is strictly decreasing. Also, the expression $\frac{\theta _4'(x)}{\theta _4(x)}$ is positive.

Proof

The statement that $x \frac{\theta '_4(x)}{\theta _4(x)}$ is strictly decreasing on $\mathbb {R}_+$ was already proved in [19]. We observe that $\lim _{x \rightarrow \infty } x \frac{\theta _4'(x)}{\theta _4(x)} = 0$, hence, $x \frac{\theta _4'(x)}{\theta _4(x)} > 0$ and, therefore, $\frac{\theta _4'(x)}{\theta _4(x)} > 0$. Also,

$$\begin{aligned} \frac{d}{dx} \left( x \frac{\theta _4'(x)}{\theta _4(x)} \right) = \frac{\theta _4'(x)}{\theta _4(x)} + x \frac{d}{dx} \left( \frac{\theta _4'(x)}{\theta _4(x)} \right) < 0, \end{aligned}$$

and it follows that

$$\begin{aligned} 0< \tfrac{1}{x} \frac{\theta _4'(x)}{\theta _4(x)} < - \frac{d}{dx} \left( \frac{\theta _4'(x)}{\theta _4(x)} \right) , \end{aligned}$$

which proves that $\frac{\theta _4'(x)}{\theta _4(x)}$ is strictly decreasing, hence, $\theta _4$ is logarithmically concave. $\square$

As a consequence of Proposition 3.7 and Lemma 3.6 we obtain the following property of $\theta _2$, which was already claimed, but not proven, in [19].

Proposition 3.9

The function $x \frac{\theta _2'(x)}{\theta _2(x)}$ is strictly decreasing on $\mathbb {R}_+$.

Proof

Proposition 3.7 tells us that $x \frac{\theta _4'(x)}{\theta _4(x)}$ is strictly decreasing on $\mathbb {R}_+$. By using Lemma 3.6 or the first identity in (3.8) we obtain

$$\begin{aligned} \frac{d}{dx} \left( x \frac{\theta _2'(x)}{\theta _2(x)}\right) = \frac{d}{dx} \left( - \tfrac{1}{x} \frac{\theta _4' \left( \tfrac{1}{x} \right) }{\theta _4 \left( \tfrac{1}{x} \right) } -\frac{1}{2}\right) < 0. \end{aligned}$$

$\square$

In fact, Propositions 3.7 and 3.9 are equivalent by Lemma 3.6. In [19, Lemma 6.2] Proposition 3.9 was proved for $x > 1$ by directly estimating $x \frac{\theta _2'(x)}{\theta _2(x)}$, which is a hard task for small x. The use of Lemma 3.6 allows us to argue directly that $x \frac{\theta _2'(x)}{\theta _2(x)}$ and $x \frac{\theta _4'(x)}{\theta _4(x)}$ must possess the same monotonicity properties.

Proposition 3.10

The Jacobi theta function $\theta _2$ is strictly logarithmically convex or, equivalently, $\frac{\theta _2'(x)}{\theta _2(x)}$ is strictly increasing. Also, the expression $\frac{\theta _2'(x)}{\theta _2(x)}$ is negative.

Proof

The fact that $\frac{\theta _2'(x)}{\theta _2(x)} < 0$ is easily checked by using (3.2). From the definition we see that $\theta _2 > 0$ and due to its unconditional convergence, the derivatives can be computed by differentiating each term. We find out that $\theta _2'(x) < 0$ and the negativity is proven.

The logarithmic convexity statement basically follows from the Cauchy–Schwarz inequality for the Hilbert-space $\ell ^2(\mathbb {N})$. We compute

$$\begin{aligned} \frac{d^2}{dx^2} \Big (\log \big (\theta _2(x)\big )\Big ) = \frac{\theta _2''(x) \theta _2(x) - \theta _2'(x)^2}{\theta _2(x)^2}. \end{aligned}$$

Strict logarithmic convexity of $\theta _2$ is now equivalent to

$$\begin{aligned} \theta _2''(x) \theta _2(x) - \theta _2'(x)^2 > 0, \end{aligned}$$

which can be re-written as

$$\begin{aligned} \left( \sum _{k \ge 1} \pi ^2 \left( k-\tfrac{1}{2}\right) ^4 e^{-\pi \left( k-\tfrac{1}{2}\right) ^2 x}\right) \left( \sum _{k \ge 1} e^{-\pi \left( k-\tfrac{1}{2}\right) ^2 x}\right) > \left( \sum _{k \ge 1} \pi \left( k-\tfrac{1}{2}\right) ^2 e^{-\pi \left( k-\tfrac{1}{2}\right) ^2 x}\right) ^2. \end{aligned}$$

(3.9)

We set

$$\begin{aligned} \big (a_k\big )_{k = 1}^\infty = \left( \pi \left( k-\tfrac{1}{2}\right) ^2 e^{-\pi \left( k-\tfrac{1}{2}\right) ^2 x/2} \right) _{k = 1}^\infty \quad \text { and } \quad \big (b_k\big )_{k = 1}^\infty = \left( e^{-\pi \left( k-\tfrac{1}{2}\right) ^2 x/2} \right) _{k = 1}^\infty \, . \end{aligned}$$

For $x > 0$ fixed, $(a_k), (b_k) \in \ell ^2(\mathbb {N})$. Inequality (3.9) is now equivalent to

$$\begin{aligned} \big \Vert (a_k)\big \Vert _{\ell ^2(\mathbb {N})}^2 \big \Vert (b_k)\big \Vert _{\ell ^2(\mathbb {N})}^2 > \big \langle (a_k), (b_k) \big \rangle _{\ell ^2(\mathbb {N})}^2, \end{aligned}$$

where strict inequality follows since $(a_k)$ and $(b_k)$ are linearly independent in $\ell ^2(\mathbb {N})$. $\square$

Also, in [19] it was claimed that $x^2 \frac{\theta _4'(x)}{\theta _4(x)}$ is monotonically decreasing and convex. Both properties were recently proved by Ernvall–Hytönen and Vesalainen [15]. We have a similar statement for $\theta _2$.

Proposition 3.11

The functions $x^2 \frac{\theta _2'(x)}{\theta _2(x)}$ and $x^2 \frac{\theta _4'(x)}{\theta _4(x)}$ are strictly decreasing on $\mathbb {R}_+$.

Proof

The result involving $\theta _4$ was proved by Ernvall–Hytönen and Vesalainen [15].

We proceed with proving the result involving $\theta _2$. We already know that $\theta _4$ is logarithmically concave, which is equivalent to the fact that $\frac{\theta _4'(x)}{\theta _4(x)}$ is monotonically decreasing. By using identity (3.8) we get

$$\begin{aligned} x^2 \frac{\theta _2'(x)}{\theta _2(x)} = -\frac{x}{2} - \frac{\theta _4'(\tfrac{1}{x})}{\theta _4(\tfrac{1}{x})}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \frac{d}{dx} \left( x^2 \frac{\theta _2'(x)}{\theta _2(x)}\right) = -\frac{1}{2} - \, \underbrace{\frac{d}{dx} \left( \frac{\theta _4'(\tfrac{1}{x})}{\theta _4(\tfrac{1}{x})}\right) }_{> 0} < 0. \end{aligned}$$

$\square$

We note that a similar simple argument as used for the expression involving $\theta _2$ does not work for $\theta _4$. Due to identity (3.8) it also seems plausible to assume that for a monomial weight $x^r$, the second power, i.e. $x^2$, is the limit for the monotonicity properties of the logarithmic derivative of $\theta _2$ and $\theta _4$ as stated in Proposition 3.11 and numerical investigations point in that direction.

Using the fact that $x^2 \frac{\theta _4'(x)}{\theta _4(x)}$ is strictly decreasing and Proposition 3.10 we conclude that

$$\begin{aligned} 0< x^2 \frac{d^2}{dx^2} \log (\theta _2(x)) = x^2 \frac{d}{dx} \left( \frac{\theta _2'(x)}{\theta _2(x)}\right) < \frac{1}{2} \end{aligned}$$

Moreover, it seems to be true that $x^2 \frac{d^2}{dx^2} \log (\theta _2(x))$ is strictly decreasing, which would in return imply that $x^2 \frac{\theta _4'(x)}{\theta _4(x)}$ is strictly decreasing.

However, we leave this problem open and close this section with symmetry properties involving $\theta _2$, $\theta _4$ and repeated applications of $x \frac{d}{dx}$.

Proposition 3.12

For $x \in \mathbb {R}_+$ and $2 \le n \in \mathbb {N}$ the following holds.

$$\begin{aligned} \left( x \frac{d}{dx}\right) ^n \log \left( \theta _2(x) \right) = (-1)^n \left( x \frac{d}{dx}\right) ^n \log \left( \theta _4\left( \tfrac{1}{x} \right) \right) . \end{aligned}$$

Proof

The statement is an obvious consequence of identity (3.8) and follows by induction. $\square$

3.2 Properties of theta-3

We will now study properties of the logarithm of $\theta _3$ on a logarithmic scale.

Proposition 3.13

The function $x \frac{\theta _3'(x)}{\theta _3(x)}$ is strictly increasing on $\mathbb {R}_+$.

This result was proved in [19]. As a consequence of Lemma 3.6 we get the following results which were already proved in the author’s doctoral thesis [17].

Proposition 3.14

For $x \in \mathbb {R}_+$ and $2 \le n \in \mathbb {N}$ the following holds.

$$\begin{aligned} \left( x \frac{d}{dx}\right) ^n \log \left( \theta _3(x) \right) = (-1)^n \left( x \frac{d}{dx}\right) ^n \log \left( \theta _3\left( \tfrac{1}{x} \right) \right) . \end{aligned}$$

Proof

The statement is an obvious consequence of identity (3.7) and follows by induction. $\square$

The following statement and its proof were also already given in [17]. The techniques are similar to the techniques used by Montgomery in [25]. The computer algebra system Mathematica [32] was used at some points in the proof in order to compute explicit values or closed expressions for geometric series, but in principle all computations can also be checked by hand. The proof was adjusted to a level which should be quite accessible with no or only little help of computer algebra software.

Proposition 3.15

For $x \in \mathbb {R}_+$, the function

$$\begin{aligned} \left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3(x) \right) \end{aligned}$$

(3.10)

is positive for $x \in (0,1)$ and negative for $x > 1$. Also, the function is anti-symmetric in the following sense

$$\begin{aligned} \left( x \frac{d}{dx} \right) ^3 \log \left( \theta _3(x) \right) = -\left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3\left( \tfrac{1}{x} \right) \right) . \end{aligned}$$

Proof

To simplify notation we set

$$\begin{aligned} \psi (x) = (\log \circ \, \theta _3)'(x) = \frac{\theta _3'(x)}{\theta _3(x)}. \end{aligned}$$

Proposition 3.14 settles the part concerning the anti-symmetry of expression (3.10). Therefore, it is also clear that we only need to prove the statement about the negativity for $x > 1$, the result for $x \in (0,1)$ then follows immediately. We start with the following calculation;

$$\begin{aligned} \begin{aligned} \left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3(x) \right)&= \left( x \, \frac{d}{dx}\right) ^2 \left( x \, \psi (x) \right) \\&= x \, \psi (x) + 3 x^2 \, \psi '(x) + x^3 \, \psi ''(x). \end{aligned} \end{aligned}$$

(3.11)

In particular, Proposition 3.14 implies that $\psi (1) + 3 \psi '(1) + \psi ''(1) = 0$. We proceed in two steps:

• (i) We will show that $\left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3(x) \right)$ is negative for $x > 1.1$.

• (ii) We will show that $\left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3(x) \right)$ is decreasing on an interval containing $\left( 1, 1.1 \right)$.

First step: We will use a combination of the asymptotic and the local behavior to establish the result claimed in (i).

Since $\psi$ is the logarithmic derivative of $\theta _3$, we use Jacobi’s triple product formula for $\theta _3$ to obtain a series representation for $\psi$. In order to control the expression in Eq. (3.11) we compute the derivatives of $\psi$ up to order 2.

$$\begin{aligned} \psi (x)&= \sum _{k \ge 1} \left( \frac{2 k \pi e^{-2 k \pi x}}{1-e^{-2 k \pi x}} - 2 \frac{(2 k -1) \pi e^{-(2 k -1) \pi x}}{1+e^{-(2 k -1) \pi x}} \right) = \frac{\theta _3'(x)}{\theta _3(x)}\\ \psi '(x)&= \sum _{k \ge 1} \left( -\frac{(2 k \pi )^2 e^{-2 k \pi x}}{\left( 1-e^{-2 k \pi x}\right) ^2} + 2 \frac{((2 k -1) \pi )^2 e^{-(2 k -1) \pi x}}{\left( 1+e^{-(2 k -1) \pi x}\right) ^2} \right) \\ \psi ''(x)&= \sum _{k \ge 1} \left( \frac{(2 k \pi )^3 e^{-2 k \pi x}\left( 1+e^{-2 k \pi x}\right) }{\left( 1-e^{-2 k \pi x}\right) ^3} - 2 \frac{((2 k -1) \pi )^3 e^{-(2 k -1) \pi x} \left( 1 - e^{-(2k-1)\pi x}\right) }{\left( 1+e^{-(2 k -1) \pi x}\right) ^3} \right) \end{aligned}$$

It is easy to verify that $\psi ^{(n)}(x) = {\mathcal {O}}(e^{-\pi x})$, $n = 0,1,2$ (this actually holds for any $n \in \mathbb {N}$). However, due to the monomial term in k, which is of the order of the derivative plus 1, we will also consider the contributions of terms involving $e^{- 2 \pi x}$ to gain more control on the local behavior close to 1.

The techniques are standard and the proof is inspired by the proofs in Montgomery’s article [25]. In particular, we will estimate parts of the series by the leading term(s), by appropriate geometric series or by their values at $x = 1$ (or use combinations of the mentioned methods). We will try to indicate which kind of estimates are used at which point in the proof.

We start with an estimate for $\psi$ for $x > 1$. We will first estimate the denominators and after that estimate the positive part by a geometric series and the negative part by the leading term. This will also be the common theme throughout the proof.

$$\begin{aligned} \psi (x)&= \sum _{k \ge 1} \left( \frac{2 k \pi e^{-2 k \pi x}}{\underbrace{1-e^{-2 k \pi x}}_{\tfrac{1}{1.002}<}} - 2 \frac{(2 k -1) \pi e^{-(2 k -1) \pi x}}{\underbrace{1+e^{-(2 k -1) \pi x}}_{<\tfrac{1}{0.958}}} \right) \\&< 1.002 \cdot 2 \pi \sum _{k\ge 1} \left( k \, e^{-2 k \pi x} \right) - 0.958 \cdot 2 \pi \, e^{-\pi x}\\&= 1.002 \cdot 2 \pi \underbrace{\left( 1-e^{-2 \pi x}\right) ^{-2}}_{<1.004} \, e^{-2 \pi x} - 0.958 \cdot 2 \pi \, e^{-\pi x}\\&< 6.35 \, e^{-2\pi x} - 6 \, e^{- \pi x} \end{aligned}$$

We proceed with an upper bound for $\psi '$ for $x > 1$.

$$\begin{aligned} \psi '(x)&= \sum _{k \ge 1} \left( -\frac{(2 k \pi )^2 e^{-2 k \pi x}}{\underbrace{\left( 1-e^{-2 k \pi x}\right) ^2}_{<1}} + 2 \frac{((2 k -1) \pi )^2 e^{-(2 k -1) \pi x}}{\underbrace{\left( 1+e^{-(2 k -1) \pi x}\right) ^2}_{1<}} \right) \\&< -4 \pi ^2 e^{-2 \pi x} + 2 \pi ^2 \sum _{k \ge 1} \left( (2 k -1)^2 e^{-(2 k -1) \pi x} \right) \\&= -4 \pi ^2 e^{-2 \pi x} + 2 \pi ^2 \underbrace{\left( 1 + 6 e^{-2 \pi x} + e^{-4 \pi x} \right) \left( 1 - e^{-2 \pi x}\right) ^{-3}}_{<1.017} \, e^{- \pi x} \\&< -39.4 \, e^{-2 \pi x} + 20.1 \, e^{-\pi x}. \end{aligned}$$

In the second to last line we used the fact that (for $|q|<1$)

$$\begin{aligned} \sum _{k \ge 1}(2k-1)^2 q^{2k-1}&= \sum _{k \ge 1} k^2 q^k - \sum _{k \ge 1} (2k)^2 q^{2k}\\&= q \frac{1+q}{(1-q)^3} - 4 q^2 \frac{1+q^2}{(1-q^2)^3} \\&= q \frac{1 + 6 q^2 + q^4}{(1-q^2)^3}. \end{aligned}$$

The closed expressions for the according geometric series are obtained by taking derivatives of the classical geometric series and for the series involving only even term, q is substituted by $q^2$.

With the same techniques we bound $\psi ''$ from above for $x > 1$.

$$\begin{aligned} \psi ''(x)&= \sum _{k \ge 1} \left( (2 k \pi )^3 e^{-2 k \pi x} \, \frac{\left( 1+e^{-2 k \pi x}\right) }{\left( 1-e^{-2 k \pi x}\right) ^3} \right. \\&\qquad \qquad - \left. 2 \, ((2 k -1) \pi )^3 e^{-(2 k -1) \pi x} \, \frac{\left( 1 - e^{-(2k-1)\pi x}\right) }{\left( 1+e^{-(2 k -1) \pi x}\right) ^3} \right) \\&< \sum _{k \ge 1} \left( \left( 2 k \pi \right) ^3 e^{-2 k \pi x}\underbrace{\left( 1 + e^{-2\pi }\right) }_{< 1.002} \underbrace{\left( 1 - e^{-2\pi }\right) ^{-3}}_{< 1.006}\right) \\&\qquad \qquad - 2 \pi ^3 e^{-\pi x} \left( \underbrace{\left( 1 - e^{-\pi x} \right) \left( 1 + e^{-\pi x}\right) ^{-3} + 27 \, e^{-2 \pi x} \left( 1 - e^{-3\pi x} \right) \left( 1 + e^{-3\pi x}\right) ^{-3}}_{>0.89}\right) \\&< 8.07 \, \pi ^3 \, e^{-2 \pi x} \underbrace{\left( e^{- 4 \pi x}+4 e^{-2 \pi x} + 1 \right) \left( 1-e^{-2 \pi x}\right) ^{-4}}_{< 1.02} - 1.78 \, \pi ^3 e^{-\pi x}\\&< 255.5 \, e^{-2 \pi x} - 55.1 \, e^{-\pi x}. \end{aligned}$$

This time we truncated the negative series over the odd integers after 2 terms, as the monomial already has degree 3 and, therefore, the second term contributes some weight to the series evaluated near $x = 1$.

Now, for $x > 1$ we have the following estimate;

$$\begin{aligned}&\left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3(x) \right) = x \, \psi (x) + 3 x^2 \, \psi '(x) + x^3 \, \psi ''(x)\\&\quad < \left( 6.35 \, x - 118.2 \, x^2 + 255.5 \, x^3 \right) e^{-2 \pi x} + \left( -6 \, x + 60.3 \, x^2 - 55.1 \, x^3 \right) e^{-\pi x}. \end{aligned}$$

It is obvious that the last expression asymptotically tends to zero from below, hence, there exists some $x_0 > 1$ such that

$$\begin{aligned} \left( x \dfrac{d}{dx} \right) ^3 \log (\theta _3(x)) < 0, \qquad \forall x > x_0. \end{aligned}$$

We will now determine $x_0$. By the above estimates, it is enough to show that the following inequality holds for $x> x_0 > 1$;

$$\begin{aligned} \left( 6.35 - 118.2 \, x + 255.5 \, x^2 \right) e^{-\pi x} + \left( -6 + 60.3 \, x - 55.1 \, x^2 \right) < 0 \end{aligned}$$

(3.12)

It is not hard to check that $-6 + 60.3 \, x - 55.1 \, x^2$ is strictly decreasing for $x > 1$. Also, we note that expressions of the form $\left( a + b \, x + c \, x^2\right) e^{-\pi x}$ are strictly decreasing for

$$\begin{aligned} x > \frac{2c - \pi b + \sqrt{\pi ^2 b^2 - 4 \pi ^2 a c + 4 c^2}}{2 \pi c}. \end{aligned}$$

For $a = 6.35$, $b = -118.2$ and $c = 255.5$, we find out that for $x>1$ the expression is indeed strictly decreasing. Therefore, it suffices to find a value $x_0$ such that (3.12) is true, as it then holds for all $x > x_0$. For $x = 1.1$ the left-hand side of (3.12) is smaller than $-0.4$ and we found the desired value $x_0 = 1.1$.

Second step: We will now show that $\left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3(x) \right)$ is strictly decreasing on an interval containing $\left( 1, x_0 \right)$. To do so, we apply the differential operator $x \frac{d}{dx}$ on (3.10), i.e.,

$$\begin{aligned} \begin{aligned} \left( x \frac{d}{dx}\right) ^4 \log \left( \theta _3(x) \right)&= x \frac{d}{dx} \left( x \, \psi (x) + 3 x^2 \, \psi '(x) + x^3 \, \psi ''(x)\right) \\&= x \, \psi (x) + 7 x^2 \, \psi '(x) + 6 x^3 \, \psi ''(x) + x^4 \, \psi '''(x). \end{aligned} \end{aligned}$$

(3.13)

In order to control the last expression for $x \in (1,x_0)$, we need good estimates on $\psi '''$ on this interval. We start in the same manner as for the lower order derivatives.

$$\begin{aligned} \psi '''(x)&= \sum _{k \ge 1} \left( -(2 k \pi )^4 e^{-2 k \pi x} \frac{\left( e^{-4 k \pi x} + 4 e^{-2 \pi k x} + 1 \right) }{\left( 1 - e^{-2 k \pi x}\right) ^4} \right. \\&\qquad \qquad \left. + 2 \, ((2k-1) \pi )^4 \, e^{-(2k-1)\pi x} \frac{\left( e^{-2 (2k-1)\pi x} - 4 e^{-(2k-1)\pi x} + 1\right) }{\left( 1 + e^{-(2k-1)\pi x}\right) ^4} \right) \\&< -16 \, \pi ^4 e^{-2 \pi x} \underbrace{(e^{-4 \pi x} + 4 e^{-2 \pi x} + 1) \left( 1- e^{-2 \pi x}\right) ^{-4}}_{> 1}\\&\qquad \qquad + 2 \pi ^4 \sum _{k \ge 1} \left( (2k-1)^4 \, e^{-(2k-1)\pi x} \left( 1 - e^{-(2k-1)\pi x}\right) ^2 \left( 1 + e^{-(2k-1)\pi x}\right) ^{-4}\right) . \end{aligned}$$

Now, we make use of the fact that we only need to establish that the expression in (3.13) is negative on the interval $(1,x_0)$. Due to the continuity of the function, this estimate then also holds on a slightly larger interval, which we do not have to specify. We estimate that, on $(1, x_0)$,

$$\begin{aligned} \left( 1 - e^{-(2k-1)\pi x}\right) ^2 \left( 1 + e^{-(2k-1)\pi x}\right) ^{-4}< \left( 1 - e^{-1.1 \cdot \pi }\right) ^2 \left( 1 + e^{-1.1 \cdot \pi }\right) ^{-4} < 0.83, \quad k = 1. \end{aligned}$$

As $k \rightarrow \infty$, the value of the left-hand side of the above inequality tends to 1 from below, which gives the uniform estimate

$$\begin{aligned} \left( 1 - e^{-(2k-1)\pi x}\right) ^2 \left( 1 + e^{-(2k-1)\pi x}\right) ^{-4} < 1, \quad \forall k \ge 2. \end{aligned}$$

Therefore, on the interval $(1,x_0)$ we get the estimate

$$\begin{aligned} \psi '''(x)&< - 16 \, \pi ^4 \, e^{-2 \pi x} + 1.66 \, \pi ^4 \, e^{-\pi x} + 2 \, \pi ^4 \sum _{k \ge 2} (2k-1)^4 \, e^{-(2k-1) \pi x}. \end{aligned}$$

We now use the closed expression for the following geometric series, established in the same manner as in the estimates for $\psi ''$;

$$\begin{aligned} \sum _{k \ge 2} (2k-1)^4 q^{2k-1} = q \, \frac{81 q^2 +220 q^4 + 86 q^6 - 4 q^8 + q^{10}}{\left( 1 - q^ 2 \right) ^5}, \qquad |q|<1. \end{aligned}$$

For $q = e^{- \pi x}$ and $x \in (1, x_0)$ we have that

$$\begin{aligned} \frac{81 q^2 +220 q^4 + 86 q^6 - 4 q^8 + q^{10}}{\left( 1 - q^ 2 \right) ^5}< \frac{81 q^2 +220 q^4 + 86 q^6 + q^{10}}{\left( 1 - q^ 2 \right) ^5} \big |_{q = e^{-\pi }} < 0.16. \end{aligned}$$

Therefore, we get that, on the interval $(1,x_0)$,

$$\begin{aligned} \psi '''(x)&< -16 \, \pi ^4 \, e^{-2 \pi x} + 1.98 \, \pi ^4 e^{-\pi x} \\&< -1558 \, e^{-2 \pi x} + 193 \, e^{- \pi x}. \end{aligned}$$

In total, it follows that for $x \in (1,x_0)$ the following inequality holds;

$$\begin{aligned} \begin{aligned} \left( x \frac{d}{dx}\right) ^4 \log \left( \theta _3(x) \right)&= x \, \psi (x) + 7 x^2 \, \psi '(x) + 6 x^3 \, \psi ''(x) + x^4 \psi '''(x)\\&< \left( 6.35 \, x - 275.8 \, x^2 + 1533 \, x^3 - 1558 \, x^4 \right) e^{- 2 \pi x}\\&\quad + \left( -6 \, x + 140.7 \, x^2 - 330.6 \, x^3 + 193 \, x^4 \right) e^{-\pi x}. \end{aligned} \end{aligned}$$

We need to verify that

$$\begin{aligned}&\left( 6.35 \, x - 275.8 \, x^2 + 1533 \, x^3 - 1558 \, x^4 \right) e^{- 2 \pi x} \\&\quad + \left( -6 \, x + 140.7 \, x^2 - 330.6 \, x^3 + 193 \, x^4 \right) e^{-\pi x} < 0 \end{aligned}$$

for $x \in (1,x_0)$. This is equivalent to showing that

$$\begin{aligned} \left( 6.35 - 275.8 \, x + 1533 \, x^2 - 1558 x^3 \right) e^{- \pi x} + \left( -6 + 140.7 \, x - 330.6 x^2 + 193 x^3 \right) < 0 \end{aligned}$$

In order to establish this result, we note that

$$\begin{aligned} \left( 6.35 - 275.8 \, x + 1533 \, x^2 - 1558 x^3 \right) < 0, \qquad x > 1. \end{aligned}$$

Hence, we get the estimate

$$\begin{aligned} \left( 6.35 - 275.8 \, x + 1533 \, x^2 - 1558 \, x^3 \right) e^{- \pi x}&< \left( 6.35 - 275.8 \, x + 1533 \, x^2 - 1558 \, x^3 \right) e^{- 1.1 \cdot \pi }\\&< 0.3 - 8.7 \, x + 48.6 \, x^ 2 - 49 \, x^3, \end{aligned}$$

for $x \in (1,x_0)$. Therefore, by showing the truth of the (stronger) inequality

$$\begin{aligned} -5.7 + 132 \, x - 282 \, x^2 + 144 \, x^3 < 0, \quad x \in (1,x_0), \end{aligned}$$

(3.14)

we can finish the proof. It is not hard to show that the left-hand side of the above inequality has critical points at $\frac{11}{36} \notin (1, x_0)$ (local maximum) and 1 (local minimum). Finally, by checking the values on the boundary of the interval $(1, x_0)$ we see that (3.14) indeed holds.

By combining the two steps we see that the function $\left( x \frac{d}{dx}\right) ^3 \log \left( \theta _3(x) \right)$ is strictly decreasing at least on the interval (1, 1.1) and negative for $x > 1.1$. As the value at $x = 1$ is zero, we can finally conclude that the expression given in Eq. (3.10) is negative for $x > 1$. Due to the already mentioned anti-symmetry with respect to the point (1, 0) the function has to be positive for $0< x <1$. $\square$

Vesalainen and Ernvall–Hytönen [14, 16] proved Proposition 3.15 independently from this work. They used the result to prove a conjecture by Hernandez and Sethuraman⁵ [20] (see also the references in [16]). The following result by Vesalainen and Ernvall–Hytönen is an immediate consequence of Proposition 3.15.

Theorem 3.16

(Ernvall–Hytönen, Vesalainen) For $\alpha , \beta \in \mathbb {R}_+$ with $1 \le \alpha < \beta$ and $x \in \mathbb {R}_+$ we define the function

$$\begin{aligned} g(x) = \frac{\theta _3(\beta x) \theta _3(\tfrac{x}{\beta })}{\theta _3(\alpha x) \theta _3(\tfrac{x}{\alpha })}. \end{aligned}$$

Then

$$\begin{aligned} g(x) = g(\tfrac{1}{x}) \end{aligned}$$

and g assumes its global maximum only for $x = 1$. Also, the function is strictly increasing for $x \in (0,1)$ and strictly decreasing for $x \in (1,\infty )$.

3.3 A result involving theta-1 and a proof of Theorem 1.1

In contrast to the fact that $\vartheta _1(0,\tau ) = 0$ for all $\tau \in {\mathbb {H}}$, its derivative with respect to z evaluated at (0, ix), $x \in \mathbb {R}_+$ is not the zero function, in particular

$$\begin{aligned} \begin{aligned} \partial _z \vartheta _1(0,i x)&= \sum _{k \in \mathbb {Z}} (-1)^k (2k+1) e^{-\pi (k+1/2)^2 x}\\&= 2 \, e^{-\tfrac{\pi }{4} x} \prod _{k \in \mathbb {N}} \left( 1 - e^{-2 k \pi x}\right) ^3, \quad x \in \mathbb {R}_+. \end{aligned} \end{aligned}$$

(3.15)

The above function is often denoted by $\theta _1'(x)$ (see e.g., [11]), where the prime indicates differentiation with respect to z. The representation as an infinite product is also well-known and follows from the Jacobi triple product representation of $\Theta$ (Proposition 3.2). Since the notation $\theta _1'$ could be misleading in this work, we rather use the notation from [8] and set

$$\begin{aligned} \theta _1^+(x) = \partial _z \vartheta _1(0, i x), \quad x \in \mathbb {R}_+. \end{aligned}$$

Now we have the following identity (see e.g., [8, chap. 3] or [11, Chap. 4])

$$\begin{aligned} \theta _2(x) \theta _3(x) \theta _4(x) = \theta _1^+(x). \end{aligned}$$

(3.16)

From this identity we derive the following result.

Theorem 3.17

For $x \in \mathbb {R}_+$, we define the function

$$\begin{aligned} \psi _1(x) = x^{3/4} \theta _1^+(x). \end{aligned}$$

This function also has the following representations

$$\begin{aligned} \psi _1(x) = x^{3/4} \prod _{j=2}^4 \theta _j(x) = 2 \, x^{3/4} e^{-\tfrac{\pi }{4} x} \prod _{k \in \mathbb {N}} \left( 1 - e^{-2 k \pi x}\right) ^3. \end{aligned}$$

The function is positive and assumes its global maximum only for $x = 1$ and $x \frac{\psi _1'(x)}{\psi _1(x)}$ is strictly decreasing. Furthermore,

$$\begin{aligned} \psi _1(x) = \psi _1(\tfrac{1}{x}) \end{aligned}$$

and, hence,

$$\begin{aligned} \left( x \frac{d}{dx} \right) ^n \psi _1(x) = (-1)^n \left( x \frac{d}{dx} \right) ^n \psi _1(\tfrac{1}{x}), \quad n \in \mathbb {N}. \end{aligned}$$

Proof

The different representations of $\psi _1$ follow immediately from (3.15) and (3.16). It is obvious from the product representation that $\psi _1$ is positive. The symmetry of $\psi _1$ follows easily from (3.5) and (3.6). The (anti-)symmetry for the repeated application of the differential operator $x \frac{d}{dx}$ immediately follows by induction. Differentiating $\log (\psi _1)$ on a logarithmic scale, i.e., applying the differential operator $x \frac{d}{dx}$, yields

$$\begin{aligned} x \frac{d}{dx} \log (\psi _1(x)) = \frac{3}{4} + \sum _{j=2}^4x \frac{\theta _j'(x)}{\theta _j(x)}. \end{aligned}$$

In particular, this means that $\psi _1$ has a critical point at $x = 1$ since

$$\begin{aligned} \sum _{j=2}^4 \frac{\theta _j'(1)}{\theta _j(1)} = -\frac{3}{4}, \end{aligned}$$

which follows from (3.7) and (3.8). The fact that $\psi _1$ assumes its global maximum only for $x = 1$ will follow from the fact that $x \frac{\psi _1'(x)}{\psi _1(x)}$ is strictly decreasing because, since $x > 0$ and $\psi _1(x) > 0$ we have that

$$\begin{aligned} \psi _1'(x)< 0 \Leftrightarrow x \frac{\psi _1'(x)}{\psi (x)} < 0. \end{aligned}$$

Since $\frac{\psi _1'(1)}{\psi _1(1)} = 0$, the negativity of $\psi _1'$ for $x > 1$ already follows if we can show that $x \frac{\psi _1'(x)}{\psi _1(x)}$ is decreasing. We use the infinite product representation to establish this result.

$$\begin{aligned} \begin{aligned} x \frac{d}{dx} \log (\psi _1(x))&= x \frac{d}{dx} \log \left( 2 \, x^{3/4} e^{-\tfrac{\pi }{4} x} \prod _{k \in \mathbb {N}} \left( 1 - e^{-2 k \pi x}\right) ^3 \right) \\&= x \frac{d}{dx} \left( \log (2) +\tfrac{3}{4} \log (x) - \tfrac{\pi }{4}x + 3 \sum _{k \in \mathbb {N}} \log \left( 1 - e^{-2 k \pi x}\right) \right) \\&= \tfrac{3}{4} - \tfrac{\pi }{4} x + 3 \sum _{k \in \mathbb {N}} \frac{2 k \pi x}{e^{2 k \pi x} - 1}. \end{aligned} \end{aligned}$$

It is quickly verified that, except for the constant term, all terms in this expression are decreasing and the proof is finished. $\square$

Fig. 1

The function $\psi _1(x)$ and its derivative $x \frac{d}{dx} \psi _1(x)$ plotted on a logarithmic scale

Fig. 2

The logarithmic derivative $x \frac{d}{dx} \log (\psi _1(x))$ plotted on a logarithmic scale and split up in its components $\phi _j(x) = x \frac{\theta _j'(x)}{\theta _j(x)}+\tfrac{1}{4}$, $j=2,3,4$

Figure 1 shows the function $\psi _1(x)$ and its derivative with logarithmic scaling on the x-axis, revealing the symmetry properties. In Fig. 2 we see the logarithmic derivative of $\psi _1$ on a logarithmic scale, i.e., $x \tfrac{d}{dx} \log (\psi _1(x))$. The behavior of this function is strongly influenced by the above established properties of Jacobi’s theta functions.

Finally, we observe that, for $x \in \mathbb {R}_+$,

$$\begin{aligned} \psi _1(x) = 2 x^{3/4} |\eta (i x)|^3, \end{aligned}$$

which follows from Eq. (2.1). Theorem 1.1 now follows immediately from Theorem 3.17.