On interlacing of zeros of certain family of modular forms

TL;DR

This paper proves that zeros of certain modular forms lie on a specific arc and that these zeros interlace with those of forms with weights differing by 12, extending previous results on zero distribution.

Contribution

It establishes the location of zeros on a specific arc and demonstrates their interlacing property for a family of modular forms, refining earlier results with new conditions.

Findings

Zeros of the modular forms lie on a specific arc in the fundamental domain.

Zeros of consecutive modular forms in the family interlace on the arc.

The results extend previous work by Nozaki and rectify earlier assumptions.

Abstract

Let $k=12 m(k)+s \ge 12$ for $s\in \{0,4,6,8,10,14\}$, be an even integer and $f$ be a normalised modular form of weight $k$ with real Fourier coefficients, written as $$ f=E_k+\sum_{j=1}^{m(k)}a_jE_{k-12j}\Delta^j. $$ Under suitable conditions on $a_j$ (rectifying an earlier result of Getz), we show that all the zeros of $f$, in the standard fundamental domain for the action of ${\bf SL}(2,\mathbb Z)$ on the upper half plane, lies on the arc $A:= \left\{ e^{i \theta} : \frac{\pi}{2} \le \theta \le \frac{2\pi}{3} \right\}$. Further, extending a result of Nozaki, we show that for certain family $\{f_k\}_k$ of normalised modular forms, the zeros of $f_k$ and $f_{k+12}$ interlace on $A^\circ:= \left\{ e^{i \theta} : \frac{\pi}{2} < \theta < \frac{2\pi}{3} \right\}$.

Equations

$$f=E_k+\sum _{j=1}^{m(k)}{a}_j{E}_{k-12j}{\Delta }^j.$$

$$z\mapsto \frac{az+b}{cz+d}$$

$$\mathbb{F}:=\left\{|z|\ge 1,-\frac{1}{2}\le \Re (z)\le 0\right\}\cup \left\{|z|>1,0<\Re (z)<\frac{1}{2}\right\}.$$

$$E_k(z):=\frac{1}{2}\sum _{\genfrac{}{}{0ex}{}{c,d\in \mathbb{Z}}{(c,d)=1}}\frac{1}{(cz+d{)}^k}.$$

$$\Delta :=\frac{E_4^3-E_6^2}{1728}.$$

$$A:=\left\{|z|=1,-\frac{1}{2}\le \Re (z)\le 0\right\}=\left\{e^{i\theta }:\frac{\pi }{2}\le \theta \le \frac{2\pi }{3}\right\}.$$

$${\delta }_t=\{\begin{array}{ll}-2& \text{if }t=0,\\ 1.009& \text{if }t=4,\\ 0.304& \text{if }t=6,\\ 0.122& \text{if }t=8,\\ 0.051& \text{if }t=10,\\ 0.022& \text{if }t\ge 12.\end{array}$$

$$f=E_k+\sum _{j=1}^{m(k)}{a}_j{E}_{k-12j}{\Delta }^j,$$

$$(3+{\delta }_{12})\sum _{j=1}^{m(k)-1}|a_j|{ϵ}^j+(3+{\delta }_s)|a_{m(k)}|{ϵ}^{m(k)}\le 1-{\delta }_{12}.$$

$$t_j^{\ast }<t_j<t_{j+1}^{\ast }\text{for}1\le j\le m.$$

$$F_k(\theta ):=e^{ik\theta /2}{E}_k(e^{i\theta }).$$

$$(3+{\delta }_{12})\sum _{j=1}^{m(k)-1}|a_j^{(k)}|{ϵ}^j+(3+{\delta }_s)|a_{m(k)}^{(k)}|{ϵ}^{m(k)}\le 20{\left(\frac{1}{2}\right)}^{k/2},$$

$$f_k:=E_k+\sum _{j=1}^{m(k)}{a}_j^{(k)}{E}_{k-12j}{\Delta }^j,$$

$$G_k(\theta ):=e^{ik\theta /2}{f}_k(e^{i\theta })$$

$$\beta <\alpha <\gamma .$$

$$\beta +\delta <\alpha -ϵ<\alpha +ϵ<\gamma -\mu ,$$

$${\beta }^{\ast }<{\alpha }^{\ast }<{\gamma }^{\ast }.$$

$$F_k(\theta )=2\cos (k\theta /2)+{\left(\frac{1}{2\cos (\theta /2)}\right)}^k+{\left(\frac{1}{2i\sin (\theta /2)}\right)}^k+P_k(\theta ),$$

$$|P_k(\theta )|<\{\begin{array}{ll}0.759& \text{if }k=4,\\ 0.179& \text{if }k=6,\\ 0.059& \text{if }k=8,\\ 0.019& \text{if }k=10,\\ 0.359(\frac{1}{2}{)}^{k/2}& \text{if }k\ge 12.\end{array}$$

$$\underset{\theta \in [\pi /2,2\pi /3]}{\sup }|F_k(\theta )|=\underset{z\in A}{\sup }|E_k(z)|<\{\begin{array}{ll}4.009& \text{if }k=4,\\ 3.304& \text{if }k=6,\\ 3.122& \text{if }k=8,\\ 3.051& \text{if }k=10,\\ 3+1.359(\frac{1}{2}{)}^{k/2}& \text{if }k\ge 12.\end{array}$$

$$\underset{\theta \in [\pi /2,2\pi /3]}{\sup }|F_k(\theta )|\le 3+{\left|\frac{1}{2i\sin (\theta /2)}\right|}^k+|P_k(\theta )|<3+{\delta }_k,$$

$${\sigma }_N(\theta ):=\frac{1}{2}\sum _{\genfrac{}{}{0ex}{}{c,d\in \mathbb{Z}}{\genfrac{}{}{0ex}{}{c^2+d^2=N}{(c,d)=1}}}\frac{1}{(c{e}^{i\theta /2}+d{e}^{-i\theta /2}{)}^k}.$$

$$F_k(\theta )=\frac{1}{2}\sum _{\genfrac{}{}{0ex}{}{c,d\in \mathbb{Z}}{(c,d)=1}}\frac{1}{(c{e}^{i\theta /2}+d{e}^{-i\theta /2}{)}^k},$$

$$F_k(\theta )=\sum _{N\ge 1}{\sigma }_N(\theta ).$$

$${\sigma }_1(\theta )=2\cos (k\theta /2)\text{and}{\sigma }_2(\theta )={\left(\frac{1}{2\cos (\theta /2)}\right)}^k+{\left(\frac{1}{2i\sin (\theta /2)}\right)}^k.$$

$$F_k(\theta )=2\cos (k\theta /2)+{\left(\frac{1}{2\cos (\theta /2)}\right)}^k+{\left(\frac{1}{2i\sin (\theta /2)}\right)}^k+P_k(\theta ),$$

$$P_k(\theta ):=\sum _{N\ge 5}{\sigma }_N(\theta ).$$

$$P_k(\theta )=\sum _{5\le N\le A}{\sigma }_N(\theta )+\sum _{N\ge B}{\sigma }_N(\theta ),$$

$$|c{e}^{i\theta /2}+d{e}^{-i\theta /2}{|}^2=c^2+d^2+2cd\cos \theta \ge c^2+d^2-|cd|\ge \frac{c^2+d^2}{2}.$$

$$\left|\sum _{5\le N\le A}{\sigma }_N(\theta )\right|\le \frac{1}{2}\sum _{5\le N\le A}\sum _{\genfrac{}{}{0ex}{}{c,d\in \mathbb{Z}}{\genfrac{}{}{0ex}{}{c^2+d^2=N}{(c,d)=1}}}\frac{1}{(c^2+d^2-|cd|{)}^{k/2}}.$$

Full text

[TABLE]

Under suitable conditions on $a_{j}$ (rectifying an earlier result of Getz), we show that all the zeros of $f$, in the standard fundamental domain for the action of ${\bf SL}(2,{\mathbb{Z}})$ on the upper half plane, lies on the arc $A:=\left\{e^{i\theta}:\frac{\pi}{2}\leq\theta\leq\frac{2\pi}{3}\right\}$. Further, extending a result of Nozaki, we show that for certain family $\{f_{k}\}_{k}$ of normalised modular forms, the zeros of $f_{k}$ and $f_{k+12}$ interlace on $A^{\circ}:=\left\{e^{i\theta}:\frac{\pi}{2}<\theta<\frac{2\pi}{3}\right\}$.

1. Introduction

Let ${\mathbb{H}}$ denote the complex upper half plane. Then the full modular group ${\bf SL}(2,{\mathbb{Z}})$ acts on ${\mathbb{H}}$ by the transformation law

[TABLE]

for $\left({a\atop c}{b\atop d}\right)\in{\bf SL}(2,{\mathbb{Z}})$. The standard fundamental domain for this action of ${\bf SL}(2,{\mathbb{Z}})$ on ${\mathbb{H}}$ is the following subset of ${\mathbb{H}}$,

[TABLE]

Throughout this article we take $k\geq 4$ to be an even integer. For $z\in{\mathbb{H}}$, the Eisenstein series of weight $k$ for the full modular group ${\bf SL}(2,{\mathbb{Z}})$ is defined by the following absolutely convergent series,

[TABLE]

The Eisenstein series of weight [math] is defined by $E_{0}:=1$. The Eisenstein series $E_{k}$ is a modular form of weight $k$ for ${\bf SL}(2,{\mathbb{Z}})$. It is classical that the space of modular forms of weight $k$ is generated by the Eisenstein series $E_{k}$ and cusp forms of weight $k$. We write $k=12m(k)+s$ with $s\in\{0,4,6,8,10,14\}$. We will use this notation for $k$ throughout the article without any further mention. The unique normalised cusp form of weight $12$, denoted by $\Delta$, is defined as follows:

[TABLE]

Rankin and Swinnerton-Dyer [8] proved that for $k\geq 4$, all the zeros of the Eisenstein series $E_{k}$ lie in the arc

[TABLE]

In 2004, extending the arguments of Rankin and Swinnerton-Dyer, Getz [4] gave a criterion for a normalised modular form of weight $k$ for ${\bf SL}(2,{\mathbb{Z}})$, written as $f=E_{k}+\sum_{j=1}^{m(k)}a_{j}E_{k-12j}\Delta^{j}$, to have all its zeros on the arc $A$, in terms of $a_{j}$’s. However, there is a rectifiable error in his proof. While estimating $H(\theta)$ in [4, p. 2225, eq. (2.5)], he used an upper bound for $R_{k-12j},1\leq j\leq m(k)$ from [4, p. 2224, eq. (2.3)] which is valid if $k-12j\geq 12$. But $k-12m(k)$ is always less than $12$, unless it is $14$. We present below a corrected version of his theorem.

Let us define

[TABLE]

Theorem 1**.**

Let $k\geq 12$ and $f$ be a normalised modular form of weight $k$, written as

[TABLE]

with $a_{j}\in{\mathbb{R}}$ for $1\leq j\leq m(k)$. Let $\epsilon:=\sup_{z\in A}|\Delta(z)|$. Suppose that

[TABLE]

Then $f$ has $m(k)$ zeros (other than possible zeros at $i,\rho:=e^{2\pi i/3}$) in the fundamental domain ${\mathbb{F}}$ and they all lie on the arc $A$.

Remark 1**.**

Note that $\delta_{12}$ above is smaller than the $\delta$ of [4, Theorem 1]. This better value is due to a more accurate estimation of a finite sum using computation. See §2, Lemma 1. Getz [4] computed $\epsilon\sim 0.004809\ldots$.

Apart from this kind of normalised modular forms there are other examples of families of modular forms, which have been shown to have their zeros on the arc $A$ (see [9, 1, 2]). Now for the zeros of these families of modular forms, one interesting question is to ask about their possible interlacing property.

Definition 1**.**

Let $\alpha<\beta$. Suppose that $f,g$ are two complex valued functions with simple zeros in the open interval $(\alpha,\beta)$. Let $\alpha<t_{1}<\cdots<t_{m}<\beta$ and $\alpha<t_{1}^{*}<\cdots<t_{m+1}^{*}<\beta$ be the zeros of $f$ and $g$, respectively. We say that zeros of $f$ and $g$ interlace in $(\alpha,\beta)$ if

[TABLE]

For example, the zeros of $\cos(n\theta)$ and $\cos((n+1)\theta)$ interlace in $(0,\pi)$ for an integer $n\geq 1$. Rankin and Swinnerton-Dyer [8] proved that the Eisenstein series $E_{k}$ has $m(k)$ simple zeros in the open arc $A^{\circ}:=\left\{e^{i\theta}:\frac{\pi}{2}<\theta<\frac{2\pi}{3}\right\}.$ For this, they considered the function

[TABLE]

This is a real valued function for $\theta$ real and it has $m(k)$ zeros in the interval $(\pi/2,2\pi/3)$ and so does $E_{k}(e^{i\theta})$ in $A^{\circ}$. Note that $m(k+12)=m(k)+1$. Hence one may look for the interlacing property for the zeros of $F_{k}(\theta)$ and $F_{k+12}(\theta)$ for $\theta\in(\pi/2,2\pi/3)$. In this instance, we say that the zeros of $E_{k}(e^{i\theta})$ and $E_{k+12}(e^{i\theta})$ interlace in $A^{\circ}$. This was predicted by Gekeler [3] and proved by Nozaki [7].

For the zeros of certain families of weakly holomorphic modular forms considered by Asai, Kaneko and Ninomiya [1], their interlacing property was established by Jermann [6]. Similar properties for the zeros of the weakly holomorphic modular forms, studied by Duke and Jenkins [2], were proved by Jenkins and Pratt [5]. Here we establish the interlacing property of the zeros of certain family of normalised modular forms that were considered in Theorem 1.

Theorem 2**.**

For each $k\geq 12$, let $(a_{j}^{(k)})_{1\leq j\leq m(k)}$ be real numbers such that

[TABLE]

where $\delta_{s}$ is as in (1). Then for the family of normalised modular forms $(f_{k})_{k}$ for ${\bf SL}(2,{\mathbb{Z}})$ defined by

[TABLE]

the zeros of $f_{k}$ in the fundamental domain ${\mathbb{F}}$ lie on the arc $A$. Further, the zeros of $f_{k}$ and those of $f_{k+12}$ interlace in $A^{\circ}$ for each $k\geq 12$.

Remark 2**.**

By (4), we see that (2) is satisfied. Hence by Theorem 1, all the zeros of $(f_{k})_{k}$ lie on the arc $A$, thus giving the first assertion of Theorem 2.

Remark 3**.**

Nozaki’s result is a special case of Theorem 2 when $a_{j}^{(k)}=0$ for all $1\leq j\leq m(k)$.

For proving the interlacing of the zeros of the Eisenstein series $E_{k}(z)$, Nozaki showed that $F_{k}(\theta)$ as defined in (3), is very well approximated by $2\cos(k\theta/2)$ for $\theta\in(\pi/2,2\pi/3)$. This is an important step in his method. We are able to show that

[TABLE]

is also well approximated by $2\cos(k\theta/2)$ for $\theta\in(\pi/2,2\pi/3)$. See §3.1 for details. Both these functions have $m(k)$ zeros in $(\pi/2,2\pi/3)$. If $\alpha$ is a zero of $\cos(k\theta/2)$ for $\theta\in(\pi/2,2\pi/3)$, then there is a neighbourhood of $\alpha$, say $(\alpha-\epsilon,\alpha+\epsilon)$, containing exactly one zero $\alpha^{*}$ of $G_{k}(\theta)$. It can be easily seen that, the zeros of $\cos(k\theta/2)$ and $\cos((k+12)\theta/2)$ interlace in $(\pi/2,2\pi/3)$ (see §3.2). Thus there exist successive zeros $\beta,\gamma$ of $\cos((k+12)\theta/2)$ with

[TABLE]

Again, there exist intervals of the form $(\beta-\delta,\beta+\delta)$ and $(\gamma-\mu,\gamma+\mu)$, each containing exactly one zero of $\cos((k+12)\theta/2)$, say $\beta^{*},\gamma^{*}$ respectively. Thus if

[TABLE]

then we obtain that

[TABLE]

This argument is used to show that the zeros of $G_{k}(\theta)$ and $G_{k+12}(\theta)$ interlace in $(\pi/2,23\pi/36)$ (see §3.2, 4.1). This method does not work as we approach $2\pi/3$. For proving the interlacing property in the remaining interval we consider the interval $(19\pi/32,2\pi/3)$, which overlaps with $(\pi/2,23\pi/36)$. Here the method depends on analysing different cases according to the increasing or decreasing property of the cosine function at their respective zeros (see §4.2).

2. A lemma and Proof of Theorem 1

We begin this section with the following lemma. This will be used in the proof of both the Theorems 1 and 2.

Lemma 1**.**

Let $k\geq 4$ and $F_{k}(\theta):=e^{ik\theta/2}E_{k}(e^{i\theta})$ for $\theta\in[\pi/2,2\pi/3]$. Then

[TABLE]

where

[TABLE]

Remark 4**.**

From Lemma 1 we obtain that

[TABLE]

In particular, we shall use the following bounds for the proof of Theorem 1. For $k\geq 4$,

[TABLE]

where $\delta_{k}$ is as in (1).

2.1. Proof of Lemma 1

For $N\geq 1$, let us define

[TABLE]

In the above definition, whenever empty sum appears, it is assumed to be [math]. Now

[TABLE]

for $\theta\in[\pi/2,2\pi/3]$. Since the series defining the Eisenstein series of weight $k\geq 4$ is absolutely convergent, we can write

[TABLE]

One can easily see that

[TABLE]

Hence

[TABLE]

where

[TABLE]

Now we split the above sum and write

[TABLE]

for some integer $A\geq 5$ and $B$, the least integer larger than $A$ which can be written as sum of squares of two co-prime integers.

Note that for any two real numbers $c,d$ one has $|cd|\leq\frac{c^{2}+d^{2}}{2}$. Since $-1\leq 2\cos\theta\leq 0$ for $\theta\in[\pi/2,2\pi/3]$, we obtain

[TABLE]

Thus we get

[TABLE]

For any natural number $N$, there are at most $2(2N^{1/2}+1)$ pairs $(c,d)$ of integers such that $c^{2}+d^{2}=N$. Hence for the second sum in (7), using (8) we get

[TABLE]

So we have

[TABLE]

Let us denote the right hand side of (9) by $p_{k}$. Using a C++ programming we obtain optimal values of $p_{k}$ as given in Table 1. The values of $p_{k}$ is best possible up to second decimal place. Our choice of $A$ is also indicated in the table.

[TABLE]

Table 1

The computation of the values of $p_{k}$, when $k\geq 6$ has taken only a few seconds, whereas for $p_{4}$, the programme ran for about two hours. Our proof is now complete. ∎

2.2. Proof of Theorem 1

It is easy to see that $F_{k}(\theta)$ is real valued for $\theta\in[\pi/2,2\pi/3]$. Further,

[TABLE]

Now,

[TABLE]

Since $a_{j}\in{\mathbb{R}}$ for $1\leq j\leq m(k)$, we see that $G(\theta)$ is real valued for $\theta\in[\pi/2,2\pi/3]$. Also by Lemma 1 we have,

[TABLE]

We show that

[TABLE]

By Remark 4,

[TABLE]

Thus, $|G(\theta)-2\cos(k\theta/2)|<2$ by our hypothesis. Now we argue as in [8]. By (10), we get that between two consecutive extremum points of $\cos(k\theta/2)$, there is a zero of $G(\theta)$. Now the extremum points of $\cos(k\theta/2)$ for $\theta\in[\pi/2,2\pi/3]$ are given by $\frac{2\pi n}{k}$, where $\frac{k}{4}\leq n\leq\frac{k}{3}$ i.e. $3m(k)+\frac{s}{4}\leq n\leq 4m(k)+\frac{s}{3}$. So we have $m(k)+1$ such $n$’s. Therefore $G(\theta)$ has $m(k)$ zeros on $(\pi/2,2\pi/3)$. This, together with the valence formula, completes the proof. ∎

3. Properties of $G_{k}(\theta)$

3.1. Approximation of $G_{k}(\theta)$ by $2\cos(k\theta/2)$

We write $G_{k}(\theta)=e^{ik\theta/2}f_{k}(e^{i\theta})$ as

[TABLE]

where by Lemma 1,

[TABLE]

with $|P_{k}(\theta)|<0.359\left(\frac{1}{2}\right)^{k/2}$. Let

[TABLE]

Then by Lemma 1, Remark 4 and our hypothesis we get

[TABLE]

Thus

[TABLE]

with $|Q_{k}(\theta)|<21.359\left(\frac{1}{2}\right)^{k/2}$. By following the proof of [7, Lemma 4.1], we observe that

[TABLE]

Further, we argue as in the proof of [7, Lemma 4.4] to obtain that for $\theta\in[19\pi/32,\alpha_{m(k)}]$, the function

[TABLE]

is minimised at $\theta=19\pi/32$. Hence for $k\geq 24$

[TABLE]

Thus for $k\geq 24$ and $\theta\in[19\pi/32,\alpha_{m(k)}]$,

[TABLE]

3.2. Zeros of $G_{k}(\theta)$ and $G_{k+12}(\theta)$

Let $\alpha_{1}<\cdots<\alpha_{m(k)}$ and $\beta_{1}<\cdots<\beta_{m(k)+1}$ denote the zeros of $\cos(k\theta/2)$ and $\cos((k+12)\theta/2)$ in $(\pi/2,2\pi/3)$, respectively. Then

[TABLE]

for $1\leq j\leq m(k)$ and

[TABLE]

for $1\leq j\leq m(k)+1$. We observe the following properties: for $1\leq j\leq m(k)$,

[TABLE]

[TABLE]

and

[TABLE]

Hence we have,

[TABLE]

and

[TABLE]

Remark 5**.**

From (15) and (16) it follows that $k\geq 24$, whenever $\alpha_{j}+\frac{6\pi}{k(k+12)}>\frac{23\pi}{36}$ or $\alpha_{j}\geq\frac{19\pi}{32}+\frac{\pi}{3k}$ or $\beta_{j}+\frac{6\pi}{(k+12)(k+24)}>\frac{23\pi}{36}$.

We now associate the zeros of $G_{k}(\theta)$ with the zeros of $\cos(k\theta/2)$.

Lemma 2**.**

Let $k\geq 12$ and $\alpha_{1}^{*}<\cdots<\alpha_{m(k)}^{*}$ be the zeros of $G_{k}(\theta)$ in $(\pi/2,2\pi/3)$. Then

[TABLE]

for all $j$ such that $\alpha_{j}+\frac{6\pi}{k(k+12)}\leq\frac{23\pi}{36}$ and

[TABLE]

for all $j$ such that $\alpha_{j}\geq 19\pi/32+\pi/3k$. In particular, $\alpha_{j}^{*}\in\left(\alpha_{j}-\frac{\pi}{3k},\alpha_{j}+\frac{\pi}{3k}\right)$ for $1\leq j\leq m(k)$.

Proof of Lemma 2.

Observe that

[TABLE]

We first claim that $2\sin\left(\frac{3\pi}{k+12}\right)>|S_{k}(\theta)|$ for $\theta\in(\pi/2,23\pi/36]$. Note that

[TABLE]

Hence it is enough to show that

[TABLE]

Now this is true if

[TABLE]

In particular if

[TABLE]

The quantity $\sqrt[k]{\frac{k+12}{10.355}}$ is maximum for $k=12$, and therefore $\sqrt[k]{\frac{k+12}{10.355}}<1.073$. Note that for $\theta\in(\pi/2,23\pi/36]$, $2\cos(\theta/2)>1.074$. This proves our first claim.

Since

[TABLE]

for $\theta\in(\pi/2,23\pi/36]$, we get that

[TABLE]

have same sign as $2\cos\left(\frac{k}{2}\left(\alpha_{j}\pm\frac{6\pi}{k(k+12)}\right)\right)$. Then from (20) we get that

[TABLE]

This implies that there exists a zero of $G_{k}(\theta)$ in the interval $\left(\alpha_{j}-\frac{6\pi}{k(k+12)},\alpha_{j}+\frac{6\pi}{k(k+12)}\right)$ for all $j$ such that $\alpha_{j}+\frac{6\pi}{k(k+12)}\leq\frac{23\pi}{36}$.

It is easy to see from (15) that $\alpha_{m(k)}\leq\frac{2\pi}{3}-\frac{\pi}{k}$. Hence we get $\alpha_{m(k)}+\frac{\pi}{3k}\leq\frac{2\pi}{3}-\frac{2\pi}{3k}$. Since $\cos(k\alpha_{j}/2)=0$, we have

[TABLE]

As $G_{k}(\theta)=2\cos(k\theta/2)+S_{k}(\theta)$, by (13) we get that the sign of $G_{k}\left(\alpha_{j}\pm\frac{\pi}{3k}\right)$ is same as that of $2\cos\left(\frac{k}{2}\left(\alpha_{j}\pm\frac{\pi}{3k}\right)\right)$, whenever $\alpha_{j}\geq\frac{19\pi}{32}+\frac{\pi}{3k}$. Hence it follows from (21) that

[TABLE]

This implies that there exists a zero of $G_{k}(\theta)$ in the interval $\left(\alpha_{j}-\frac{\pi}{3k},\alpha_{j}+\frac{\pi}{3k}\right)$ for all $j$ such that $\alpha_{j}\geq\frac{19\pi}{32}+\frac{\pi}{3k}$.

From Remark 5 we deduce that a zero $\alpha_{j}$ of $\cos(k\theta/2)$ in $(\pi/2,2\pi/3)$ satisfies $\alpha_{j}+\frac{6\pi}{k(k+12)}\leq\frac{23\pi}{36}$ or $\alpha_{j}\geq\frac{19\pi}{32}+\frac{\pi}{3k}$. Moreover,

[TABLE]

By (19) all the intervals of the form $\left(\alpha_{j}-\frac{\pi}{3k},\alpha_{j}+\frac{\pi}{3k}\right)$ are disjoint for $1\leq j\leq m(k)$. We have shown above that each of them contains at least one zero of $G_{k}(\theta)$. We also know that $G_{k}(\theta)$ has exactly $m(k)$ zeros in $(\pi/2,2\pi/3)$. Since the zeros of $\cos(k\theta/2)$ and $G_{k}(\theta)$ have been labelled as per the increasing order of magnitude, the assertion of the lemma follows. ∎

4. Proof of Theorem 2

We closely follow the arguments of Nozaki [7]. However at several places our arguments are simpler. We know that a zero $\alpha_{j}$ of $\cos(k\theta/2)$ in $(\pi/2,2\pi/3)$ satisfies $\alpha_{j}+\frac{6\pi}{k(k+12)}\leq\frac{23\pi}{36}$ or $\alpha_{j}\geq\frac{19\pi}{32}+\frac{\pi}{3k}$. Hence we prove Theorem 2 for these two cases.

4.1. Case I

Let $\alpha_{j}\in\left(\frac{\pi}{2},\frac{23\pi}{36}-\frac{6\pi}{k(k+12)}\right]$. Then we prove the following:

• (i)

$\beta_{j}^{*}<\alpha_{j}^{*}$.

• (ii)

$\alpha_{j}^{*}<\beta_{j+1}^{*}$ if $\beta_{j+1}+\frac{6\pi}{(k+12)(k+24)}\leq\frac{23\pi}{36}$.

4.1.1. Proof of (i)

Since $\alpha_{j}+\frac{6\pi}{k(k+12)}\leq\frac{23\pi}{36}$, we have $\beta_{j}+\frac{6\pi}{(k+12)(k+24)}\leq\frac{23\pi}{36}$. Applying Lemma 2 for $G_{k}(\theta)$ and $G_{k+12}(\theta)$, we get

[TABLE]

Therefore by (18), we have

[TABLE]

4.1.2. Proof of (ii)

When $\beta_{j+1}+\frac{6\pi}{(k+12)(k+24)}\leq\frac{23\pi}{36}$, by Lemma 2

[TABLE]

Therefore by (18), we have

[TABLE]

4.2. Case II

Let $\alpha_{j}\in\left[\frac{19\pi}{32}+\frac{\pi}{3k},\frac{2\pi}{3}\right)$. Then we prove the following:

• (iii)

$\beta_{j}^{*}<\alpha_{j}^{*}$ if $\beta_{j}\geq\frac{19\pi}{32}+\frac{\pi}{3(k+12)}$.

• (iv)

$\alpha_{j}^{*}<\beta_{j+1}^{*}$.

For an integer $n$,

[TABLE]

Further $\cos(n\theta)$ is decreasing at $\pi/2$ if $n\equiv 1\bmod 4$ and increasing at $\pi/2$ if $n\equiv 3\bmod 4$. Thus if $\alpha$ denotes the first zero of $\cos(n\theta)$ in $(\pi/2,\infty)$, then $\cos(n\theta)$ is decreasing at $\alpha$ if $n\equiv 0,3\bmod 4$ and increasing at $\alpha$ if $n\equiv 1,2\bmod 4$. Hence if $\beta$ denotes the first zero of $\cos((n+6)\theta)$ in $(\pi/2,\infty)$, then $\cos((n+6)\theta)$ is increasing at $\beta$ if $n\equiv 0,3\bmod 4$ and decreasing at $\beta$ if $n\equiv 1,2\bmod 4$.

Therefore, for all $1\leq j\leq m(k)$, if $\cos(k\theta/2)$ is increasing (resp. decreasing) at $\alpha_{j}$, then $\cos((k+12)\theta/2)$ is decreasing (resp. increasing) at $\beta_{j}$. We consider two subcases.

• (a)

The function $\cos(k\theta/2)$ is increasing at $\alpha_{j}$.

• (b)

The function $\cos(k\theta/2)$ is decreasing at $\alpha_{j}$.

Note that by Remark 5, we have $k\geq 24$. Now for $k\geq 24$ and $\theta\in[19\pi/32,2\pi/3-2\pi/3k]$, $S_{k}(\theta)>0$ (see (13)). Hence $\alpha_{j}^{*}\lessgtr\alpha_{j}$ according as $\cos(k\theta/2)$ is increasing or decreasing at $\alpha_{j}$.

4.2.1. Proof of (iii) and (iv) when (a) occurs

It follows by our earlier analysis that $\cos((k+12)\theta/2)$ is increasing at $\beta_{j+1}$. Also $\beta_{j+1}>\alpha_{j}>\frac{19\pi}{32}+\frac{\pi}{3(k+12)}$. Hence

[TABLE]

Further

[TABLE]

The Case (a) is described pictorially below.

In the above one and other picture below, the respective arrows indicate the direction in which $\beta_{j}^{*},\alpha_{j}^{*}$ and $\beta_{j+1}^{*}$ lie. We first prove (iii). In fact, we show that the intersection point of the two cosine curves between $\beta_{j}$ and $\alpha_{j}$ separates $\beta_{j}^{*}$ and $\alpha_{j}^{*}$. The function

[TABLE]

has a zero between $\beta_{j}$ and $\alpha_{j}$, say $\gamma_{j}$. For $1\leq j\leq m(k)$,

[TABLE]

We prove that $\beta_{j}^{*}\in(\beta_{j},\gamma_{j})$ and $\alpha_{j}^{*}\in(\gamma_{j},\alpha_{j})$. By (13), $G_{k}(\alpha_{j})>0$ and $G_{k+12}(\beta_{j})>0$. Thus, we are led to show that $G_{k}(\gamma_{j}),G_{k+12}(\gamma_{j})<0$. As in the proof of [7, Lemma 4.2], it follows that

[TABLE]

for any $j$ such that $\gamma_{j}\geq\frac{19\pi}{32}$ and $\theta\in[19\pi/32,2\pi/3)$.

Since $\cos(k\theta/2)$ is increasing at $\alpha_{j}$ and $\cos(k\alpha_{j}/2)=0$, we get $\cos(k\gamma_{j}/2)<0$. Hence $G_{k}(\gamma_{j})<0$. Similarly one obtains that $G_{k+12}(\gamma_{j})<0$. This completes the proof of (iii) when (a) occurs.

Next we show (iv) i.e. $\alpha_{j}^{*}<\beta_{j+1}^{*}$. Now if $\alpha_{j}\leq\beta_{j+1}-\frac{\pi}{3(k+12)}$, then by (22) and Lemma 2,

[TABLE]

which proves (iv). Hence we need to consider only

[TABLE]

From (14) we have, $S_{k}(\theta)>S_{k+12}(\theta)$ for $\theta\in[\beta_{j+1}-\pi/3(k+12),\alpha_{j}]$. Further, the function $\cos(k\theta/2)-\cos((k+12)\theta/2)$ takes positive value in the interval $(\gamma_{j},\gamma_{j+1})$, where $\gamma_{j}$’s are as in (23). It is easy to check that $[\beta_{j+1}-\pi/3(k+12),\alpha_{j}]\subset(\gamma_{j},\gamma_{j+1})$. Therefore, we obtain that

[TABLE]

for $\theta\in[\beta_{j+1}-\pi/3(k+12),\alpha_{j}]$. Taking $\theta=\alpha_{j}^{*}$, we find that $G_{k+12}(\alpha_{j}^{*})<0$. On the other hand, $G_{k+12}(\beta_{j+1})>0$. Hence $\alpha_{j}^{*}<\beta_{j+1}^{*}$. This completes the proof of Case (a).

4.2.2. Proof of (iii) and (iv) when (b) occurs

In this case $\cos((k+12)\theta/2)$ is decreasing at $\beta_{j+1}$. Hence

[TABLE]

Further

[TABLE]

From (24) and (25) we easily get

[TABLE]

which proves (iii). Thus we proceed to prove (iv) i.e. $\alpha_{j}^{*}<\beta_{j+1}^{*}$. If $\alpha_{j}+\frac{\pi}{3k}\leq\beta_{j+1}$, then by Lemma 2 and (24), we have

[TABLE]

Hence we can assume that

[TABLE]

Note that $\beta_{j+1}<\alpha_{j}+\frac{\pi}{3k}$ implies

[TABLE]

Here the aim is to show that

[TABLE]

If (27) holds, then $G_{k}(\beta_{j+1}^{*})<0$. Since $G_{k}(\alpha_{j})>0$, we get $\alpha_{j}<\alpha_{j}^{*}<\beta_{j+1}^{*}$ as required. Thus it remains to prove (27). Note that

[TABLE]

Let $r(\theta):=2\cos((k+12)\theta/2)-2\cos(k\theta/2)$. We first find a lower bound for $r(\theta)$ in $[\beta_{j+1},\alpha_{j}+\pi/3k]$. For this we show that $r(\theta)$ is decreasing in this interval. Note that $r(\theta)=-4\sin((k+6)\theta/2)\sin(3\theta)$. Now $-\sin(3\theta)$ is decreasing from $1$ to [math] in $(\pi/2,2\pi/3)$. Also $r(\theta)>0$ for $\theta\in(\gamma_{j},\gamma_{j+1})$. Thus $\sin((k+6)\theta/2)>0$ for $\theta\in(\gamma_{j},\gamma_{j+1})$. As $\gamma_{j}$ and $\gamma_{j+1}$ are two consecutive zeros of $\sin((k+6)\theta/2)$, we therefore obtain that $\sin((k+6)\theta/2)$ is decreasing in $\left(\frac{\gamma_{j}+\gamma_{j+1}}{2},\gamma_{j+1}\right)$. From (15), (16) and (23), we deduce that

[TABLE]

Hence $\sin((k+6)\theta/2)$ is decreasing in $[\beta_{j+1},\alpha_{j}+\pi/3k]$. As both the functions $\sin((k+6)\theta/2)$ and $-\sin(3\theta)$ are positive and decreasing, we get that $r(\theta)$ is decreasing in $[\beta_{j+1},\alpha_{j}+\pi/3k]$. Therefore $r(\theta)$ is minimised at $\alpha_{j}+\pi/3k$ for $\theta\in[\beta_{j+1},\alpha_{j}+\pi/3k]$. Note that $2\cos\left(\frac{k}{2}\left(\alpha_{j}+\frac{\pi}{3k}\right)\right)=-1$. Now

[TABLE]

Hence

[TABLE]

From (26) and the condition $j\leq(k-s)/12$, we get that if $k\equiv 0\bmod 4$,

[TABLE]

and if $k\equiv 2\bmod 4$,

[TABLE]

Therefore,

[TABLE]

The equation of the tangent line of the sine curve at $\pi/6$ is

[TABLE]

Hence $2\sin\left(\frac{\pi}{6}-\frac{4\pi}{k}\right)<1-\frac{4\sqrt{3}\pi}{k}$.

Next let $t(\theta):=(2\cos(\theta/2))^{-(k+12)}-(2\cos(\theta/2))^{-k}$. Then

[TABLE]

So $t(\theta)$ is minimised when $(2\cos(\theta/2))^{-12}=\frac{k}{k+12}$. We thus get,

[TABLE]

This completes the proof of Case (b).

4.3. Remaining cases

By Cases I and II, in order to complete the proof of Theorem 2, we need to prove the following two cases.

• (v)

$\alpha_{j}^{*}<\beta_{j+1}^{*}$ when $\alpha_{j}\in\left(\frac{\pi}{2},\frac{23\pi}{36}-\frac{6\pi}{k(k+12)}\right]$ and $\beta_{j+1}+\frac{6\pi}{(k+12)(k+24)}>\frac{23\pi}{36}$.

• (vi)

$\beta_{j}^{*}<\alpha_{j}^{*}$ when $\alpha_{j}\in\left[\frac{19\pi}{32}+\frac{\pi}{3k},\frac{2\pi}{3}\right)$ and $\beta_{j}<\frac{19\pi}{32}+\frac{\pi}{3(k+12)}$.

4.3.1. Proof of (v)

We show that $\alpha_{j}\geq\frac{19\pi}{32}+\frac{\pi}{3k}$. Hence by §4.2, $\alpha_{j}^{*}<\beta_{j+1}^{*}$. Suppose that $\alpha_{j}<\frac{19\pi}{32}+\frac{\pi}{3k}$. Then using (15) we get that

[TABLE]

Since $\beta_{j+1}+\frac{6\pi}{(k+12)(k+24)}>\frac{23\pi}{36}$, it follows from Remark 5 and (16) that $k\geq 24$ and

[TABLE]

respectively. Using (28) in (29) we get that

[TABLE]

This is a contradiction since $k\geq 24$.

4.3.2. Proof of (vi)

As in (v), we can deduce that $\alpha_{j}+\frac{6\pi}{k(k+12)}\leq\frac{23\pi}{36}$. Hence $\beta_{j}^{*}<\alpha_{j}^{*}$ by §4.1. ∎

Acknowledgement: We would like to thank Biswajyoti Saha and Jhansi Bhavani V. for helping us with the computations in Lemma 1.