Mathematics authors titles recent submissions
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: markus.faulhuber@ntnu.no.
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
2
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.
4
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.
6
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–Hytönen 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–Hytönen 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–Hytönen [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–Hytönen
(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].
12
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
scale, i.e., applying the differential operator x dx
, yields
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.
14
MARKUS FAULHUBER
References
[1] Albert Baernstein II. Landau’s constant, and extremal problems involving discrete subsets of C. In Viktor P. Havin and Nikolai K. Nikolski, editors, Linear and Complex Analyis Problem Book 3, Part 2,
number 1574 in Lecture Notes in Mathematics, chapter 18, pages 404–407. Springer, 1994.
[2] Albert Baernstein II. A minimum problem for heat kernels of flat tori. In Extremal Riemann surfaces (San
Francisco, CA, 1995), volume 201 of Contemp. Math., pages 227–243. Amer. Math. Soc., Providence, RI,
1997.
[3] Albert Baernstein II and Jade P. Vinson. Local Minimality Results Related to the Bloch and Landau
Constants. In Peter Duren, Juha Heinonen, Brad Osgood, and Bruce Palka, editors, Quasiconformal
Mappings and Analysis: A Collection of Papers Honoring F.W. Gehring, pages 55–89. Springer New
York, New York, NY, 1998.
[4] Laurent Bétermin and Mircea Petrache. Dimension reduction techniques for the minimization of theta
functions on lattices. Journal of Mathematical Physics, 58(7):071902, 2017.
[5] Mark Coffey and George Csordas. On the log-concavity of a Jacobi theta function. Mathematics of Computation, 82(284):2265–2272, 2013.
[6] Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna S. Viazovska. The
sphere packing problem in dimension 24. Annals of Mathematics, 187(3):1035–1068, 2017.
[7] John H. Conway and Neil J. A. Sloane. Sphere packings, lattices and groups, volume 290 of Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag,
New York, third edition, 1998.
[8] George Csordas. Fourier Transforms of Positive Definite Kernels and the Riemann ξ -Function. Computational Methods and Function Theory, 15(3):373–391, Sep 2015.
[9] Anne-Maria Ernvall-Hytönen and Esa V. Vesalainen. On a conjecture of Faulhuber and Steinerberger on
the logarithmic derivative of ϑ4 . arXiv preprint arXiv:1709.05194, 2017.
[10] Markus Faulhuber. Extremal Bounds of Gaussian Gabor Frames and Properties of Jacobi’s Theta Functions. PhD thesis, University of Vienna, 2016.
[11] Markus Faulhuber. Minimal Frame Operator Norms Via Minimal Theta Functions. Journal of Fourier
Analysis and Applications, pages 1–15, 2017.
[12] Markus Faulhuber and Stefan Steinerberger. Optimal Gabor frame bounds for separable lattices and
estimates for Jacobi theta functions. Journal of Mathematical Analysis and Applications, 445(1):407–422,
January 2017.
[13] Beverly Hernandez. Results on the Secrecy Function Conjecture on the Theta Function of Lattices. Master’s thesis, California State University, Northridge, 2016.
[14] Augustus J. E. M. Janssen. Some Weyl-Heisenberg frame bound calculations. Indagationes Mathematicae,
7(2):165–183, 1996.
[15] Hugh L. Montgomery. Minimal theta functions. Glasgow Mathematical Journal, 30:75–85, January 1988.
[16] Elias Stein and Rami Shakarchi. Complex Analysis. Princeton University Press, Princeton, NJ, 2003.
[17] Esa V. Vesalainen and Anne-Maria Ernvall-Hytönen. Personal communication.
[18] Esa V. Vesalainen and Anne-Maria Ernvall-Hytönen. On the secrecy gain of ℓ-modular lattices. arXiv
preprint arXiv:1708.09239, 2017.
[19] Maryna S. Viazovska. The sphere packing problem in dimension 8. Annals of Mathematics, 187(3):991–
1015, 2017.
[20] Wolfram Research, Inc., Champaign, IL. Mathematica, Version 10.0, 2014.
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: markus.faulhuber@ntnu.no.
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
2
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.
4
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.
6
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–Hytönen 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–Hytönen 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–Hytönen [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–Hytönen
(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].
12
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
scale, i.e., applying the differential operator x dx
, yields
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.
14
MARKUS FAULHUBER
References
[1] Albert Baernstein II. Landau’s constant, and extremal problems involving discrete subsets of C. In Viktor P. Havin and Nikolai K. Nikolski, editors, Linear and Complex Analyis Problem Book 3, Part 2,
number 1574 in Lecture Notes in Mathematics, chapter 18, pages 404–407. Springer, 1994.
[2] Albert Baernstein II. A minimum problem for heat kernels of flat tori. In Extremal Riemann surfaces (San
Francisco, CA, 1995), volume 201 of Contemp. Math., pages 227–243. Amer. Math. Soc., Providence, RI,
1997.
[3] Albert Baernstein II and Jade P. Vinson. Local Minimality Results Related to the Bloch and Landau
Constants. In Peter Duren, Juha Heinonen, Brad Osgood, and Bruce Palka, editors, Quasiconformal
Mappings and Analysis: A Collection of Papers Honoring F.W. Gehring, pages 55–89. Springer New
York, New York, NY, 1998.
[4] Laurent Bétermin and Mircea Petrache. Dimension reduction techniques for the minimization of theta
functions on lattices. Journal of Mathematical Physics, 58(7):071902, 2017.
[5] Mark Coffey and George Csordas. On the log-concavity of a Jacobi theta function. Mathematics of Computation, 82(284):2265–2272, 2013.
[6] Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna S. Viazovska. The
sphere packing problem in dimension 24. Annals of Mathematics, 187(3):1035–1068, 2017.
[7] John H. Conway and Neil J. A. Sloane. Sphere packings, lattices and groups, volume 290 of Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag,
New York, third edition, 1998.
[8] George Csordas. Fourier Transforms of Positive Definite Kernels and the Riemann ξ -Function. Computational Methods and Function Theory, 15(3):373–391, Sep 2015.
[9] Anne-Maria Ernvall-Hytönen and Esa V. Vesalainen. On a conjecture of Faulhuber and Steinerberger on
the logarithmic derivative of ϑ4 . arXiv preprint arXiv:1709.05194, 2017.
[10] Markus Faulhuber. Extremal Bounds of Gaussian Gabor Frames and Properties of Jacobi’s Theta Functions. PhD thesis, University of Vienna, 2016.
[11] Markus Faulhuber. Minimal Frame Operator Norms Via Minimal Theta Functions. Journal of Fourier
Analysis and Applications, pages 1–15, 2017.
[12] Markus Faulhuber and Stefan Steinerberger. Optimal Gabor frame bounds for separable lattices and
estimates for Jacobi theta functions. Journal of Mathematical Analysis and Applications, 445(1):407–422,
January 2017.
[13] Beverly Hernandez. Results on the Secrecy Function Conjecture on the Theta Function of Lattices. Master’s thesis, California State University, Northridge, 2016.
[14] Augustus J. E. M. Janssen. Some Weyl-Heisenberg frame bound calculations. Indagationes Mathematicae,
7(2):165–183, 1996.
[15] Hugh L. Montgomery. Minimal theta functions. Glasgow Mathematical Journal, 30:75–85, January 1988.
[16] Elias Stein and Rami Shakarchi. Complex Analysis. Princeton University Press, Princeton, NJ, 2003.
[17] Esa V. Vesalainen and Anne-Maria Ernvall-Hytönen. Personal communication.
[18] Esa V. Vesalainen and Anne-Maria Ernvall-Hytönen. On the secrecy gain of ℓ-modular lattices. arXiv
preprint arXiv:1708.09239, 2017.
[19] Maryna S. Viazovska. The sphere packing problem in dimension 8. Annals of Mathematics, 187(3):991–
1015, 2017.
[20] Wolfram Research, Inc., Champaign, IL. Mathematica, Version 10.0, 2014.