PROPERTIES OF LOGARITHMIC DERIVATIVES OF JACOBI’S THETA FUNCTIONS ON A LOGARITHMIC SCALE
arXiv:1709.06006v1 [math.NT] 18 Sep 2017
MARKUS FAULHUBER Analysis Group, Department of Mathematical Sciences, NTNU Trondheim, Sentralbygg 2, Gløshaugen, Trondheim, Norway Abstract. In this work we study and collect symmetry properties of the classical Jacobi theta functions. These properties concern the logarithmic derivatives of Jacobi’s theta functions on a logarithmic scale.
1. Introduction Theta functions take a prominent role in several branches of mathematics. They appear in geometric function theory [1], [2], [3] the studies on energy minimization [4], the study of Riemann’s zeta and xi function [5], [8] or the theory of sphere packing and covering [7], including the recent breakthrough for sphere packings in dimension 8 by Viazovska [19] and in dimension 24 [6]. They also appear in the field of time-frequency analysis and the study of Gaussian Gabor frames [11], [12], [14] or the study of the heat kernel [16] to name just a few. The aim of this work is to reveal properties of the classical Jacobi theta functions looked at on a logarithmic scale as well as setting known results into a new context. We will encounter ′ (x) , where f is at least continuously differentiable on R+ and expressions of the form x ff (x) d possesses no zeros. Also, we will frequently use the differential operator x dx and use the d n notation x dx for its iterated repetition, i.e., d n d d n−1 x =x . x dx dx dx
In particular, we are interested in properties of the logarithmic derivative of a certain function f on a logarithmic scale. This statement should be interpreted as follows. By using the variable transformation y = log(x) we extend the domain from R+ to R and consider the new function log (f (ey )). Therefore, we get an extra exponential factor each time we take a (logarithmic) derivative. We have f ′ (ey ) d log (f (ey )) = ey . dy f (ey ) E-mail address:
[email protected]. 2010 Mathematics Subject Classification. 26A51, 33E05. Key words and phrases. Convexity, Jacobi Theta Function, Logarithmic Derivative, Logarithmic Scale. The author was supported by the Austrian Science Fund (FWF): [J4100-N32] and [P27773-N23]. The computational results presented have been achieved (in part) using the Vienna Scientific Cluster (VSC). 1
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MARKUS FAULHUBER
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 [11], [12], [15] to establish uniqueness results about extremal theta functions on lattices. This work provides a new point of view on as well as new results related to the mentioned articles. Although we will mostly only consider real-valued functions in this work, we start by defining the theta function according to the textbook of Stein and Shakarchi [16]. Definition 1.1. For z ∈ C and τ ∈ H (the upper half plane) we define the theta function as X 2 (1.1) Θ(z, τ ) = eπik τ e2kπiz . k∈Z
This function is an entire function with respect to z and holomorphic with respect to τ . As stated in [16], 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 1.2 (Jacobi triple product). For z ∈ C and τ ∈ H we have Y Θ(z, τ ) = 1 − e2kπiτ 1 + e(2k−1)πiτ e2πiz 1 + e(2k−1)πiτ e−2πiz . k≥1
It also fulfills the following identity. Theorem 1.3. For z ∈ C and τ ∈ H we have 2 Θ z, − τ1 = (−iτ )1/2 eπiz τ Θ(τ z, τ ).
For a proof of Theorem 1.3 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 [16]. Whereas derivatives of Θ with respect to z are often studied, it seems that studies of its derivatives with respect to τ are less common. The following Lemma contains a symmetry result for the logarithmic derivative of Θ with respect to τ . Lemma 1.4. For z ∈ C and τ ∈ H we have
z ∂z Θ(τ z, τ ) + ∂τ Θ(τ z, τ ) 1 ∂τ Θ z, − τ1 1 =− . πiz + τ − 1 Θ(τ z, τ ) τ Θ z, − τ 2 2
In particular, for z = 0 we get
1 ∂τ Θ(0, τ ) 1 ∂τ Θ 0, − τ1 =− . τ − 1 Θ(0, τ ) τ Θ 0, − τ 2
Proof. We start by taking the logarithm on both sides of the identity in Theorem 1.3 1 log Θ z, − τ1 = log(−iτ ) + πiz 2 τ + log (Θ(τ z, τ )) . 2 Differentiating with respect to τ on both sides and a multiplication by τ yields, after rearranging the terms, the desired result.
LOGARITHMIC DERIVATIVES OF JACOBI’S THETA FUNCTIONS
3
2. Jacobi’s Theta Functions According to existing literature, we define the following functions which are usually denoted to Jacobi. We start with the following theta functions of two variables which we will then restrict to certain domains. For z ∈ C and τ ∈ H we define ϑ1 (z, τ ) =
X
2
(−1)(k−1/2) eπi(k−1/2) τ e(2k+1)πiz
k∈Z
=2
X
2τ
(−1)(k−1/2) eπi(k−1/2)
sin((2k + 1)πz)
k∈N
ϑ2 (z, τ ) =
X
2
eπi(k−1/2) τ e(2k+1)πiz
k∈Z
=2
X
2τ
eπi(k−1/2)
cos((2k + 1)πz)
k∈N
ϑ3 (z, τ ) =
X k∈Z
=1+2 ϑ4 (z, τ ) =
X
2
e−πik τ e2kπiz X
e−πik
X
(−1)k e−πik
2τ
cos(2kπz)
k∈N 2
(−1)k e−πik τ e2kπiz
k∈Z
=1+2
2τ
cos(2kπz).
k∈N
We note that ϑ3 (z, τ ) = Θ(z, τ ) and that any ϑj can be expressed via Θ from equation (1.1). The functions which we will study in the rest of this work are restrictions of the above functions with (z, τ ) = (0, ix), x ∈ R+ . In particular, all these functions are real–valued. Definition 2.1 (Jacobi’s Theta Functions). For x ∈ R+ we define Jacobi’s theta functions in the following way.
θ2 (x) = ϑ2 (0, ix) =
X
2x
e−π(k−1/2)
=
2
k∈Z
θ3 (x) = ϑ3 (0, ix) =
X
X
2x
e−π(k−1/2)
(2.1)
k∈N
e−πk
2x
=1+2
X
k∈Z
k∈N
k∈Z
k∈N
e−πk
2x
X X 2 2 θ4 (x) = ϑ4 (0, ix) = (−1)k e−πk x = 1 + 2 (−1)k e−πk x
(2.2) (2.3)
It does not make much sense to study properties of θ1 (x) if defined in the above sense as ϑ1 (0, ix) = 0 for all x ∈ R+ . However, we will see a result related to θ1 in Section 5.
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MARKUS FAULHUBER
All of the above functions can also be expressed by infinite products. 2 π Y 1 − e2kπs 1 + e2kπs θ2 (x) = 2e− 4 x θ3 (x) =
Y
k∈N
θ4 (x) =
Y
k∈N
k∈N
1 − e2kπs 1 − e2kπs
1 + e(2k−1)πs 1 − e(2k−1)πs
2
2
These representations can be quite useful when studying the logarithmic derivatives of these functions. We note that for z ∈ R and purely imaginary τ = ix, x ∈ R+ the theta function Θ(z, ix) is maximal for z ∈ Z and minimal for z ∈ Z + 12 . The first statement follows already from the definition of Θ whereas the latter argument can most easily be derived from the triple product representation. These special cases correspond to Jacobi’s θ3 and θ4 function θ3 (x) = Θ (0, ix) θ4 (x) = Θ 12 , ix .
For x ∈ R+ , Jacobi’s θ2 function can be expressed via Θ in the following way. x θ2 (x) = e−π 4 Θ ix 2 , ix .
For what follows it is necessary to introduce the Fourier transform and Poisson’s summation formula which are given by Z b f (x)e−2πiωx dx, x, ω ∈ R f (ω) = R
and
X
f (k + x) =
k∈Z
X l∈Z
fb(l)e2πilx ,
x ∈ R.
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 hold. As a consequence of the Poisson summation formula we find the following, well-known identities √ x θ3 (x) = θ3 x1 (2.4) (2.5)
√
x θ2 (x) = θ4
1 x
and
√
x θ4 (x) = θ2
1 x
.
We note the common theme in the last identities which leads to the following lemma. Lemma 2.2. Let r ∈ R and suppose that f, g ∈ C 1 (R+ ) do not possess zeros. If f and g satisfy the (generalized Jacobi) identity xr f (x) = g
1 x
,
LOGARITHMIC DERIVATIVES OF JACOBI’S THETA FUNCTIONS
5
then f ′ (x) + x f (x)
1 x
g′ g
1 x 1 x
= −r.
Proof. Both, f and g are either positive or negative on R+ since they are real-valued, con tinuous and do not contain zeros. As they also fulfill the identity xr f (x) = g x1 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 log (xr f (x)) = log g x1 .
d on both sides directly yields Using the differential operator x dx ′ 1 f ′ (x) 1 g x . r+x = −x f (x) g x1
We remark that an alternative proof of Lemma 2.2 is given in [12] for the special case f = g = θ3 and r = 21 (a generalization of the proof in [12] is possible with the arguments given there). With this proof, the result 1 θ3′ (x) 1 θ3′ x1 =− + (2.6) x θ3 (x) x θ3 x1 2
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 θ2′ (x) 1 θ4′ x1 θ4′ (x) 1 θ2′ x1 1 1 =− =− (2.7) x +x and x +x 1 1 θ2 (x) 2 θ4 (x) 2 θ4 x θ2 x
which were not given in [12]. 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 (2.6) and (2.7) are special cases of Lemma 1.4 when Θ is restricted to certain rays in C × H. 3. Properties of Theta-2 and Theta-4 We start with monotonicity properties of the logarithmic derivatives of Jacobi’s θ2 and θ4 functions on a logarithmic scale. The following result was already given in [12] and the proof can be established by using the product representation of θ4 . θ ′ (x)
Proposition 3.1. The function x θ44 (x) is strictly decreasing on R+ . As a consequence of Proposition 3.1 we derive the following result. Proposition 3.2. The Jacobi theta function θ4 is strictly logarithmically concave or, equivθ ′ (x) θ ′ (x) alently, θ44 (x) is strictly decreasing. Also, the expression θ44 (x) is positive.
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MARKUS FAULHUBER θ ′ (x)
Proof. The statement that x θ44 (x) is strictly decreasing on R+ was already proved in [12]. We θ ′ (x)
θ ′ (x)
θ ′ (x)
observe that limx→∞ x θ44 (x) = 0, hence, x θ44 (x) > 0 and, therefore, θ44 (x) > 0. Also, ′ d θ4′ (x) θ4 (x) θ4′ (x) d +x x = < 0, dx θ4 (x) θ4 (x) dx θ4 (x)
and it follows that
0< which proves that
θ4′ (x) θ4 (x)
′ 1 θ4 (x) x θ (x) 4
d 1 by directly estimating x θ22 (x) , which is a hard task θ ′ (x)
θ ′ (x)
for small x. The use of Lemma 2.2 allows us to argue directly that x θ22 (x) and x θ44 (x) must possess the same monotonicity properties. Proposition 3.4. The Jacobi theta function θ2 is strictly logarithmically convex or, equivaθ ′ (x) θ ′ (x) lently, θ22 (x) is strictly increasing. Also, the expression θ22 (x) is negative. θ ′ (x)
Proof. The fact that θ22 (x) < 0 is easily checked by using (2.1). From the definition we see that θ2 > 0 and due to its unconditional convergence, the derivatives can be computed by differentiating each term. We find out that θ2′ (x) < 0 and the negativity is proven. The logarithmic convexity statement basically follows from the Cauchy-Schwarz inequality for the Hilbert–space ℓ2 (N). We compute θ2′′ (x)θ2 (x) − θ2′ (x)2 d2 = . log θ (x) 2 dx2 θ2 (x)2
Strict logarithmic convexity of θ2 is now equivalent to
θ2′′ (x)θ2 (x) − θ2′ (x)2 > 0,
which can be re-written as 2 1 2 1 2 1 2 X X X 4 −π k− 2 x 2 −π k− 2 x −π k− 2 x > . π k − 12 e e (3.1) π 2 k − 12 e k≥1
k≥1
k≥1
LOGARITHMIC DERIVATIVES OF JACOBI’S THETA FUNCTIONS
We set ak
∞
k=1
=
For x > 0 fixed,
∞ ∞ 1 2 1 2 ∞ −π k− 2 x/2 1 2 −π k− 2 x/2 π k− 2 e and bk k=1 = e k=1 k=1 2 (ak ), (bk ) ∈ ℓ (N). Inequality (3.1) is now equivalent to
2
(ak ) 22 (bk ) 22 , > (a ), (b ) k k ℓ2 (N) ℓ (N) ℓ (N)
and strict inequality follows since (ak ) and (bk ) are linearly independent in ℓ2 (N).
7
.
′ 2 θ4 (x) θ4 (x)
is monotonically decreasing and convex. Both Also, in [12] it was claimed that x properties were recently proved by Ernvall–Hyt¨ onen and Vesalainen [9], [17]. We have a similar statement for θ2 . θ ′ (x)
θ ′ (x)
Proposition 3.5. The functions x2 θ22 (x) and x2 θ44 (x) are strictly decreasing on R+ . Proof. The result involving θ4 was proved by Ernvall–Hyt¨ onen and Vesalainen [9]. We proceed with proving the result involving θ2 . We already know that θ4 is logarithmically θ ′ (x) concave, which is equivalent to the fact that θ44 (x) is monotonically decreasing. By using identity (2.7) we get x2
x θ′ ( 1 ) θ2′ (x) = − − 4 x1 . θ2 (x) 2 θ4 ( x )
Therefore, we have d dx
θ ′ (x) x2 2 θ2 (x)
d 1 =− − 2 dx |
! θ4′ ( x1 ) < 0. θ4 ( x1 ) {z }
>0
We note that a similar simple argument as used for the expression involving θ2 does not work for θ4 . Due to identity (2.7) it also seems plausible to assume that for a monomial weight xr , the second power, i.e. x2 , is the limit for the monotonicity properties of the logarithmic derivative of θ2 and θ4 as stated in Proposition 3.5 and numerical investigations point in that direction. θ ′ (x) Using the fact that x2 θ44 (x) is strictly decreasing and Proposition 3.4 we conclude that ′ 2 θ2 (x) 1 2 d 2 d 0 1. Also, the function is anti–symmetric in the following sense d 3 d 3 log (θ3 (x)) = − x log θ3 x1 . x dx dx Proof. To simplify notation we set
ψ(x) = (log ◦ θ3 )′ (x) =
θ3′ (x) . θ3 (x)
Proposition 4.3 settles the part concerning the anti–symmetry of expression (4.1). Therefore, it is also clear that we only need to prove the statement about the negativity for x > 1, the result for x ∈ (0, 1) follows immediately. We start with the following calculation d 2 d 3 log (θ3 (x)) = x (x ψ(x)) x dx dx (4.2) = x ψ(x) + 3x2 ψ ′ (x) + x3 ψ ′′ (x).
In particular, Proposition 4.3 implies that ψ(1) + 3ψ ′ (1) + ψ ′′ (1) = 0. We proceed in two steps: d 3 (i) We will show that x dx log (θ3 (x)) is negative for x ≥ 56 . 3 d log (θ3 (x)) is decreasing on an interval including 1, 65 . (ii) We will show that x dx
LOGARITHMIC DERIVATIVES OF JACOBI’S THETA FUNCTIONS
9
First step: Since ψ is the logarithmic derivative of θ3 , we use Jacobi’s triple product formula for θ3 to obtain a series representation for ψ. In order to control the expression in equation (4.2) we compute the derivatives of ψ up to order 2. ! X 2kπe−2kπx θ3′ (x) (2k − 1)πe−(2k−1)πx = − 2 < 0, for x > 0 ψ(x) = −(2k−1)πx 1 − e−2kπx θ3 (x) 1 + e k≥1 ! 2 e−(2k−1)πx 2 e−2kπx X ((2k − 1)π) (2kπ) ψ ′ (x) = − 2 2 +2 (1 − e−2kπx ) 1 + e−(2k−1)πx k≥1 ! 3 e−(2k−1)πx 1 − e−(2k−1)πx X (2kπ)3 e−2kπx 1 + e−2kπx ((2k − 1)π) ψ ′′ (x) = −2 3 3 (1 − e−2kπx ) 1 + e−(2k−1)πx k≥1 θ ′ (x)
The observation that ψ(x) = θ33 (x) < 0, for x > 0 follows easily from (2.2). In what follows we will have to work a bit more and estimate parts of the series by the leading term, by appropriate geometric series or by their values at x = 1 (or use combinations of the mentioned methods). We proceed with an upper bound for ψ ′ for x > 1. ! 2 e−(2k−1)πx 2 e−2kπx X ((2k − 1)π) (2kπ) ψ ′ (x) = − 2 2 +2 (1 − e−2kπx ) 1 + e−(2k−1)πx k≥1 ! X ((2k − 1)π)2 e−(2k−1)πx < 2 2 1 + e−(2k−1)πx k≥1 X < 2π 2 k2 e−kπx k≥1
−3 = 2π 2 e−πx e−πx + 1 1 − e−πx | {z } | {z } 0.04
−2 X +2 ((2k − 1)π)4 e−(2k−1)πx 1 + e−(2k−1)πx {z } | k≥1 1.2. As the value at x = 1 is zero, we can finally conclude that the expression given in equation(4.1) 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. The estimates in the proof are quite rough, but at the same time fine enough for the proof to work which is due to the rapid decay of the involved terms. Numerical inspections beforehand suggested that d 3 log (θ3 (x)) < 0 for x > 1.05. x dx and that
d x dx
4
log (θ3 (x)) < 0
for 1 < x < 1.5.
We remark that the proof is quite sensible to the estimated constants and does for example not work if in the last step ψ ′′′ (x) is estimated by 184 e−πx instead of 183 e−πx . Most recently, Vesalainen and Ernvall–Hyt¨ onen [17], [18] proved Proposition 4.3 independently from this work. They used the result to prove a conjecture by Hernandez and Sethuraman1 [13] (see also the references in [18]). The result by Vesalainen and Ernvall–Hyt¨ onen (which the author was aware of) is an immediate consequence of Proposition 4.3 and reads as follows. Theorem 4.4. For α, β ∈ R+ with 1 ≤ α ≤ β and x ∈ R+ we define the function g(x) =
θ3 (βx)θ3 ( βx )
θ3 (αx)θ3 ( αx )
.
Then g(x) = g( x1 ) and g assumes its global maximum only for s = 1. Also, the function is strictly increasing for s ∈ (0, 1) and strictly decreasing for s ∈ (1, ∞). 5. A Result Involving Theta-1 In contrast to the fact that ϑ1 (0, τ ) = 0 for all τ ∈ H, its derivative with respect to z evaluated at (0, ix), x ∈ R+ is not the zero function, in particular X 2 ∂z ϑ1 (0, ix) = (−1)k (2k + 1)e−π(k+1/2) x
(5.1)
k∈Z
π
= 2 e− 4 x
Y
k∈N
1 − e−2kπx
3
,
x ∈ R+ .
1The author does not claim any credit for solving the conjecture by Hernandez and Sethuraman since the author was not aware of the conjecture prior to the appearance of [18].
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MARKUS FAULHUBER
The above function is often denoted by θ1′ (x), where the prime indicates differentiation with respect to z. The representation as infinite product is also well–known and follows from the Jacobi triple product representation of Θ (Proposition 1.2). Since the notation θ1′ would be very misleading in this work we set z θ1
= ∂z ϑ1 (0, ix),
x ∈ R+ .
Now we have the following identity (see e.g. [7, Chap. 4]) (5.2)
θ2 (x)θ3 (x)θ4 (x) = z θ1 (x).
From this identity we derive the following result. Proposition 5.1. For x ∈ R+ , we define the function ψ1 (x) = x3/4 z θ1 (x). This function also has the following representations ψ1 (x) = x3/4
4 Y
π
θj (x) = 2 x3/4 e− 4 x
j=2
Y
k∈N
1 − e−2kπx
3
. ψ′ (x)
The function is positive and assumes its global maximum only for x = 1 and x ψ11 (x) is strictly decreasing. Furthermore, ψ1 (x) = ψ1 ( x1 ) and, hence,
d x dx
n
n
ψ1 (x) = (−1)
d x dx
n
ψ1 ( x1 ),
n ∈ N.
Proof. The different representations of ψ1 follow immediately from (5.1) and (5.2). It is obvious from the product representation that ψ1 is positive. The symmetry of ψ1 follows easily from (2.4) and (2.5). The (anti–)symmetry for the repeated application of the differential d operator x dx immediately follows by induction. Differentiating log(ψ1 ) on a logarithmic d , yields scale, i.e., applying the differential operator x dx 4 ′ 3 X θj (x) d x . x log(ψ1 (x)) = + dx 4 θj (x) j=2
In particular, this means that ψ1 has a critical point at x = 1 since 4 X θj′ (1) j=2
3 =− . θj (1) 4 ψ′ (x)
The fact that ψ1 assumes its global maximum will follow from the fact that x ψ11 (x) is strictly decreasing because, since x > 0 and ψ1 (x) > 0 we have that ψ1′ (x) < 0 ⇔ x
ψ1′ (x) < 0. ψ(x)
LOGARITHMIC DERIVATIVES OF JACOBI’S THETA FUNCTIONS
13 ψ′ (x)
ψ′ (1)
Since ψ11 (1) = 0 the negativity of ψ1′ for x > 1 already follows if we can show that x ψ11 (x) is decreasing. We use the infinite product representation to establish this result. ! 3 π Y d d 3/4 − 4 x −2kπx x log(ψ1 (x)) = x log 2 x e 1−e dx dx k∈N ! X d log(2) + 43 log(x) − π4 x + 3 =x log 1 − e−2kπx dx k∈N X 2kπx . = 34 − π4 x + 3 e2kπx − 1 k∈N
It is quickly verified that, except for the constant term, all terms in this expression are decreasing and the proof is finished.
d Figure 1. The function ψ1 (x) and its derivative x dx ψ1 (x) plotted on a logarithmic scale.
d log(ψ1 (x)) plotted on a Figure 2. The logarithmic derivative x dx θ ′ (x)
logarithmic scale and split up in its components φj (x) = x θjj (x) + 14 , j = 2, 3, 4.
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MARKUS FAULHUBER
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