ON THE ZEROS OF CERTAIN LINEAR COMBINATIONS OF POINCARÉ CUSP FORMS

DIVYANSHU KALA; EKATA SAHA

doi:10.1017/S0004972726101245

ON THE ZEROS OF CERTAIN LINEAR COMBINATIONS OF POINCARÉ CUSP FORMS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press: 14 May 2026

DIVYANSHU KALA

and

EKATA SAHA

Show author details

DIVYANSHU KALA: Affiliation:
Institute of Mathematical Sciences , IV Cross Street, Taramani, Chennai 600113, India e-mail: divyanshuk@imsc.res.in
EKATA SAHA*: Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi , Hauz Khas, New Delhi 110016, India
*: e-mail: ekata@maths.iitd.ac.in

Article contents

Abstract
Introduction
Preliminary steps
Proof of Theorem
Proof of Lemma
Footnotes
References

Rights & Permissions

Abstract

Poincaré series are of fundamental importance in the theory of modular forms. For a fixed even integer $k \ge 4$, the space of modular forms of weight k can be spanned by certain Poincaré series. We provide a criterion for a modular form to have a specified number of zeros on the unit circle in the standard fundamental domain for the action of $\mathrm {SL}(2,{\mathbb {Z}})$ on the complex upper half-plane. The criterion is given in terms of the coefficients that are obtained by writing the modular form as a linear combination of the generating Poincaré series.

Keywords

linear combinations of Poincaré series location of zeros

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight

Secondary: 11F03: Modular and automorphic functions

Information

Type: Research Article
Information: Bulletin of the Australian Mathematical Society , Volume 114 , Issue 3 , December 2026 , pp. 447 - 454

DOI: https://doi.org/10.1017/S0004972726101245 [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

Let

$$ \begin{align*} {\mathbb{F}}:= \{|z| \ge 1, -\tfrac{1}{2} \le \mathrm{Re}(z) \le 0 \} \cup \{ |z|> 1, 0 < \mathrm{Re}(z) < \tfrac{1}{2} \} \end{align*} $$

denote the standard fundamental domain for the usual action of $\mathrm {SL}(2,{\mathbb {Z}})$ on the complex upper half-plane ${\mathbb {H}}$ . The location of the zeros of modular forms in the fundamental domain ${\mathbb {F}}$ has a long and rich history. One of the fundamental results is due to Rankin and Swinnerton-Dyer [Reference Rankin and Swinnerton-Dyer3]. For an even integer $k\ge 4$ and $z \in {\mathbb {H}}$ , let

$$ \begin{align*} E_k(z):=\dfrac{1}{2}\sum_{\substack{c,d\in{\mathbb{Z}} \\ \gcd(c,d)=1}} \frac{1}{(cz+d)^k} \end{align*} $$ upper E Subscript k Baseline left parenthesis z right parenthesis colon equals one half sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction 1 Over left parenthesis c z plus d right parenthesis Superscript k Baseline EndFraction

upper E Subscript k Baseline left parenthesis z right parenthesis colon equals one half sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction 1 Over left parenthesis c z plus d right parenthesis Superscript k Baseline EndFraction

denote the normalised Eisenstein series of weight k. They showed that the zeros of the Eisenstein series $E_k$ in ${\mathbb {F}}$ lie on the arc $A:=\{e^{i \theta } : \theta \in [\pi /2,2\pi /3]\}$ . This result was extended by Rankin [Reference Rankin4] to give partial information about the zeros of Poincaré cusp forms. For a positive integer m and $z\in {\mathbb {H}}$ , the Poincaré cusp form for $\mathrm {SL}(2,{\mathbb {Z}})$ of weight $k\ge 4$ and index m is defined by

$$ \begin{align*} G_k(z,m):=\dfrac{1}{2}\sum_{\substack{c,d\in{\mathbb{Z}} \\ \gcd(c,d)=1}} \frac{e^{2\pi i m\gamma(z)}}{(cz+d)^k}, \end{align*} $$

upper G Subscript k Baseline left parenthesis z comma m right parenthesis colon equals one half sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction e Superscript 2 pi i m gamma left parenthesis z right parenthesis Baseline Over left parenthesis c z plus d right parenthesis Superscript k Baseline EndFraction comma

where $\gamma \in \mathrm {SL}(2,{\mathbb {Z}})$ with the second row $(c\ \ d)$ . The above definition does not depend on the choice of $\gamma $ and the series is uniformly convergent on every compact subset of ${\mathbb {H}}$ . Rankin [Reference Rankin4, Theorem 1] proved that for $1\le m\le m(k)-1$ , at least $m(k)-m$ zeros of $G_k(z,m)$ in ${\mathbb {F}}$ lie on the arc $A^\circ :=\{e^{i \theta } : \theta \in (\pi /2,2\pi /3)\}$ , where $m(k)$ is the integer obtained by writing $k=12m(k)+s(k)$ with $s(k) \in \{0,4,6,8,10,14\}$ . In this context, it is relevant to note that the set $\{G_k(z,m):1\le m \le m(k)\}$ forms a basis of the ${\mathbb {C}}$ -vector space $S_k$ of weight k cusp forms for $\mathrm {SL}(2,{\mathbb {Z}})$ , whereas $\{G_k(z,m):0\le m \le m(k)\}$ forms a basis of the ${\mathbb {C}}$ -vector space $M_k$ of weight k modular forms for $\mathrm {SL}(2,{\mathbb {Z}})$ . Here, $G_k(z,0)$ denotes the Eisenstein series $E_k(z)$ .

Extending Rankin’s result, in [Reference Gun2, Theorem 1], Gun proved that any real linear combination of the form $\sum _{m=1}^j a_m G_k(z,m)$ has at least $m(k)-j$ zeros on the arc $A^{\circ }$ if $a_j>0$ and $a_m \ge 0$ for $1\le m\le j-1 \le m(k)-1$ .

Analogous results were also derived for linear combinations of another set of basis elements $\{E_{k-12j}\Delta ^j: 0 \le j \le m(k)\}$ of $M_k$ by Getz [Reference Getz1], where $\Delta =(E_4^3-E_6^2)/1728$ . In fact, Getz gave a criterion for a modular form to have all its zeros in ${\mathbb {F}}$ on the arc A. A minor error in [Reference Getz1, Theorem 1] was later corrected in [Reference Saha and Saradha5, Theorem 1].

In this article, we consider the basis $\{G_k(z,m):0\le m \le m(k)\}$ of $M_k$ and give a criterion for a modular form $f \in M_k$ to have at least a prescribed number of zeros in $A^{\circ }$ . More precisely, we prove the following theorem.

Theorem 1.1. For an even integer $k\ge 28$ and $0\le l\le m(k)-1$ , let

$$ \begin{align*} J_{k,l}(z):=\sum_{l\le m \le m(k)} a_m G_k(z,m), \end{align*} $$ StartLayout 1st Row upper J Subscript k comma l Baseline left parenthesis z right parenthesis colon equals sigma summation Underscript l less than or equals m less than or equals m left parenthesis k right parenthesis Endscripts a Subscript m Baseline upper G Subscript k Baseline left parenthesis z comma m right parenthesis comma EndLayout

where for $l\le m\le m(k)$ , $a_m\in {\mathbb {R}}$ and $ |a_l|= \max _{l\le m\le m(k)} |a_m|$ . Then, the modular form $J_{k,l}(z)$ has at least $m(k)-l$ zeros on the arc $A^{\circ }$ .

Clearly, our theorem extends Rankin’s result in [Reference Rankin4]. Our theorem also implies that the modular form $J_{k,0}(z)$ and the cusp form $J_{k,1}(z)$ have all of their zeros in ${\mathbb {F}}$ on the arc A, provided these functions are nonzero. We note that in the result of Gun [Reference Gun2, Theorem 1], the largest index present in the combination seems to determine the number of zeros on the arc $A^{\circ }$ , whereas in our theorem, the smallest index seems to determine the number of zeros on the arc $A^{\circ }$ . Also, in our theorem, we do not have any restriction on the nonnegativity of the coefficients of $J_{k,l}(z)$ . Moreover, if we review [Reference Getz1, Theorem 1] (or [Reference Saha and Saradha5, Theorem 1]), we see that for a normalised modular form, the coefficient of $E_{k-12j}\Delta ^j$ for the least value of j, namely $j=0$ , plays the key role in locating the zeros on the unit circle. Our theorem works in the same spirit, but it is applicable to cusp forms as well.

2 Preliminary steps

We now collect some preliminary estimates for our proof.

Lemma 2.1 [Reference Saha and Saradha6, Lemma 1]

Let $F_k({\theta ,m}):=e^{i k\theta /2}G_k(e^{i\theta },m)$ . For $\theta \in [\pi /2,2\pi /3]$ and an integer $m \ge 0$ , the function $F_k(\theta ,m)$ is real valued.

To proceed further, for an integer $N\ge 1$ and $\theta \in I:=[\pi /2,2\pi /3]$ , we define

$$ \begin{align*} \sigma_N(\theta):=\dfrac{1}{2}\sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2=N \\ \gcd(c,d)=1}}\frac{e^{2\pi i m\gamma(e^{i\theta})}}{(ce^{i\theta/2}+de^{-i\theta/2})^k}, \end{align*} $$

sigma Subscript upper N Baseline left parenthesis theta right parenthesis colon equals one half sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row c squared plus d squared equals upper N 3rd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction e Superscript 2 pi i m gamma left parenthesis e Super Superscript i theta Superscript right parenthesis Baseline Over left parenthesis c e Superscript i theta divided by 2 Baseline plus d e Superscript minus i theta divided by 2 Baseline right parenthesis Superscript k Baseline EndFraction comma

where $m \ge 0$ is a fixed integer. In this definition, an empty sum is assumed to be 0. As the series $F_k(\theta ,m)$ is absolutely convergent for any $\theta \in {\mathbb {R}}$ , we have $F_k(\theta ,m)=\sum _{N\ge 1}\sigma _N(\theta )$ . We need the explicit expression of the first few terms. It is well documented that

$$ \begin{align*} \sigma_1(\theta)=2e^{-2 \pi m\sin\theta}\cos(k\theta/2+2 \pi m\cos\theta) \end{align*} $$

(see [Reference Saha and Saradha6, Section 3.1]). Moreover,

$$ \begin{align*} \sigma_2(\theta)=\frac{(-1)^me^{-\pi m\tan(\theta/2)}}{(2\cos\theta/2)^k}+\frac{(-1)^me^{-\pi m\cot(\theta/2)}}{(2i\sin\theta/2)^k} \end{align*} $$

(see [Reference Saha and Saradha6, Section 4.1]). So,

$$ \begin{align*} F_k(\theta,m) = \sigma_1(\theta) + \sigma_2(\theta) + R_{k,m}(\theta), \end{align*} $$

where

$$ \begin{align*} R_{k,m}(\theta)=\dfrac{1}{2}\sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 5 \\ \gcd(c,d)=1}}\frac{e^{2\pi i m\gamma(e^{i\theta})}}{(ce^{i\theta/2}+de^{- i\theta/2})^k}. \end{align*} $$

upper R Subscript k comma m Baseline left parenthesis theta right parenthesis equals one half sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row c squared plus d squared greater than or equals 5 3rd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction e Superscript 2 pi i m gamma left parenthesis e Super Superscript i theta Superscript right parenthesis Baseline Over left parenthesis c e Superscript i theta divided by 2 Baseline plus d e Superscript minus i theta divided by 2 Baseline right parenthesis Superscript k Baseline EndFraction period

2.1 Estimation of $R_{k,m}(\theta )$

We have

$$ \begin{align*} |R_{k,m}(\theta)|\le\dfrac{1}{2}\sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 5 \\ \gcd(c,d)=1}}\frac{1}{|ce^{i\theta/2}+de^{-i\theta/2}|^k}. \end{align*} $$

StartAbsoluteValue upper R Subscript k comma m Baseline left parenthesis theta right parenthesis EndAbsoluteValue less than or equals one half sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row c squared plus d squared greater than or equals 5 3rd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction 1 Over StartAbsoluteValue c e Superscript i theta divided by 2 Baseline plus d e Superscript minus i theta divided by 2 Baseline EndAbsoluteValue Superscript k Baseline EndFraction period

If $\gcd (c,d)=1$ , then $\gcd (-c,d)=1$ . So, the terms corresponding to these pairs give the contribution

$$ \begin{align*} (c^2+d^2+2cd\cos\theta)^{-k/2}+(c^2+d^2-2cd\cos\theta)^{-k/2}. \end{align*} $$

The derivative of this function is given by

$$ \begin{align*} kcd\sin\theta((c^2+d^2+2cd\cos\theta)^{-k/2-1}-(c^2+d^2-2cd\cos\theta)^{-k/2-1}). \end{align*} $$

If $cd> 0$ , then

$$ \begin{align*} (c^2+d^2+2cd\cos\theta)^{-k/2-1}>(c^2+d^2-2cd\cos\theta)^{-k/2-1}, \end{align*} $$

since $ \cos \theta <0 \ \text { for }\theta \in I$ . If $cd< 0$ , then

$$ \begin{align*} (c^2+d^2+2cd\cos\theta)^{-k/2-1}<(c^2+d^2-2cd\cos\theta)^{-k/2-1}. \end{align*} $$

Hence, the derivative is positive for $\theta \in I$ and the maximum value, which is attained at $\theta =2\pi /3$ , is $ (c^2+d^2+cd)^{-k/2}+(c^2+d^2-cd)^{-k/2}. $ Accordingly,

$$ \begin{align*} |R_{k,m}(\theta)|\le \dfrac{1}{2}\sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 5 \\ \gcd(c,d)=1}}\frac{1}{(c^2+d^2+cd)^{k/2}}. \end{align*} $$

Note that

$$ \begin{align*} \sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 5 \\ \gcd(c,d)=1}}\frac{1}{(c^2+d^2+cd)^{k/2}}=\frac{2}{3^{k/2}}+\frac{2}{7^{k/2}}+\sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 8 \\ \gcd(c,d)=1}}\frac{1}{(c^2+d^2+cd)^{k/2}}. \end{align*} $$

sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row c squared plus d squared greater than or equals 5 3rd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction 1 Over left parenthesis c squared plus d squared plus c d right parenthesis Superscript k divided by 2 Baseline EndFraction equals StartFraction 2 Over 3 Superscript k divided by 2 Baseline EndFraction plus StartFraction 2 Over 7 Superscript k divided by 2 Baseline EndFraction plus sigma summation Underscript StartLayout 1st Row c comma d element of double struck upper Z 2nd Row c squared plus d squared greater than or equals 8 3rd Row gcd left parenthesis c comma d right parenthesis equals 1 EndLayout Endscripts StartFraction 1 Over left parenthesis c squared plus d squared plus c d right parenthesis Superscript k divided by 2 Baseline EndFraction period

Now, $ |c^2+d^2+cd|\ge |c^2+d^2|/2, $ which gives

$$ \begin{align*} \sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 8 \\ \gcd(c,d)=1}}\frac{1}{(c^2+d^2+cd)^{k/2}}\le \sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 8 \\ \gcd(c,d)=1}}\frac{2^{k/2}}{(c^2+d^2)^{k/2}}. \end{align*} $$

As the number of ways of writing an integer n as a sum of two squares is bounded by $4n$ ,

$$ \begin{align*} \sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 8 \\ \gcd(c,d)=1}}\frac{2^{k/2}}{(c^2+d^2)^{k/2}}\le 2^{{k}/{2}+2}\sum_{n\ge8}\frac{1}{n^{{k}/{2}-1}}. \end{align*} $$

Now,

$$ \begin{align*} \sum_{n\ge8}\frac{1}{n^{{k}/{2}-1}}\le \int_{7}^{\infty}\frac{1}{x^{{k}/{2}-1}}dx=\frac{7^{-{k}/{2}+2}}{{k}/{2}-2}. \end{align*} $$

Hence,

$$ \begin{align*} \sum_{\substack{c,d\in{\mathbb{Z}} \\ c^2+d^2\ge 8 \\ \gcd(c,d)=1}}\frac{2^{k/2}}{(c^2+d^2)^{k/2}}\le \frac{2^{{k}/{2}+2}}{7^{{k}/{2}-2}({k}/{2}-2)}. \end{align*} $$

So,

$$ \begin{align*} \sum_{\substack{c,d\in{\mathbb{Z}} \\c^2+d^2\ge 5 \\ \gcd(c,d)=1}}\frac{2^{k/2}}{(c^2+d^2)^{k/2}}\le \frac{2}{3^{k/2}}+\frac{2}{7^{k/2}}+ \frac{2^{{k}/{2}+2}}{7^{{k}/{2}-2}({k}/{2}-2)}. \end{align*} $$

Therefore,

$$ \begin{align*} |R_{k,m}(\theta)|\le \dfrac{1}{2}\bigg(\frac{2}{3^{k/2}}+\frac{2}{7^{k/2}}+\frac{2^{{k}/{2}+2}}{7^{{k}/{2}-2}({k}/{2}-2)}\bigg) = 3^{-k/2}\bigg(1+\frac{3^{k/2}}{7^{k/2}}+\frac{2\cdot6^{k/2}}{7^{{k}/{2}-2}({k}/{2}-2)}\bigg). \end{align*} $$

For $k\ge 24$ ,

$$ \begin{align*} \frac{3^{k/2}}{7^{k/2}}+\frac{2\cdot6^{k/2}}{7^{{k}/{2}-2}({k}/{2}-2)} \le 2. \end{align*} $$ StartFraction 3 Superscript k divided by 2 Baseline Over 7 Superscript k divided by 2 Baseline EndFraction plus StartFraction 2 dot 6 Superscript k divided by 2 Baseline Over 7 Superscript k divided by 2 minus 2 Baseline left parenthesis k divided by 2 minus 2 right parenthesis EndFraction less than or equals 2 period

Hence, for $k\ge 24$ ,

(2.1) $$ \begin{align} |R_{k,m}(\theta)|\le \frac{3}{3^{k/2}}. \end{align} $$

3 Proof of Theorem 1.1

We define

$$ \begin{align*} I_{k,l}(\theta):= e^{2\pi l \sin\theta}e^{i k\theta/2}\sum_{m=l} ^{m(k)}a_mG_k(e^{i\theta},m). \end{align*} $$

The function $I_{k,l}(\theta )$ is real-valued, by Lemma 2.1. We write $I_{k,l}(\theta )$ as

$$ \begin{align*} I_{k,l}(\theta)=2a_l\cos(k\theta/2+2\pi l \cos\theta) +I_{k,l,1}(\theta)+I_{k,l,2}(\theta)+I_{k,l,3}(\theta)+I_{k,l,4}(\theta), \end{align*} $$

where

$$ \begin{align*} I_{k,l,1}(\theta)&:=\sum_{m=l+1} ^{m(k)}2a_me^{-2\pi (m-l)\sin\theta}\cos(k\theta/2+2\pi m\cos\theta),\\ I_{k,l,2}(\theta)&:=\sum_{m=l} ^{m(k)}a_m\frac{(-1)^me^{2\pi l\sin\theta-\pi m\tan(\theta/2)}}{(2\cos(\theta/2))^k},\\ I_{k,l,3}(\theta)&:=\sum_{m=l} ^{m(k)}a_m\frac{(-1)^me^{2\pi l\sin\theta-\pi m\cot(\theta/2)}}{(2i\sin(\theta/2))^k} \quad \text{ and }\\ I_{k,l,4}(\theta)&:=\sum_{m=l} ^{m(k)}a_me^{2\pi l\sin\theta}R_{k,m}(\theta). \end{align*} $$

StartLayout 1st Row 1st Column upper I Subscript k comma l comma 1 Baseline left parenthesis theta right parenthesis 2nd Column colon equals sigma summation Underscript m equals l plus 1 Overscript m left parenthesis k right parenthesis Endscripts 2 a Subscript m Baseline e Superscript minus 2 pi left parenthesis m minus l right parenthesis sine theta Baseline cosine left parenthesis k theta divided by 2 plus 2 pi m cosine theta right parenthesis comma 2nd Row 1st Column upper I Subscript k comma l comma 2 Baseline left parenthesis theta right parenthesis 2nd Column colon equals sigma summation Underscript m equals l Overscript m left parenthesis k right parenthesis Endscripts a Subscript m Baseline StartFraction left parenthesis negative 1 right parenthesis Superscript m Baseline e Superscript 2 pi l sine theta minus pi m tangent left parenthesis theta divided by 2 right parenthesis Baseline Over left parenthesis 2 cosine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction comma 3rd Row 1st Column upper I Subscript k comma l comma 3 Baseline left parenthesis theta right parenthesis 2nd Column colon equals sigma summation Underscript m equals l Overscript m left parenthesis k right parenthesis Endscripts a Subscript m Baseline StartFraction left parenthesis negative 1 right parenthesis Superscript m Baseline e Superscript 2 pi l sine theta minus pi m cotangent left parenthesis theta divided by 2 right parenthesis Baseline Over left parenthesis 2 i sine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction and 4th Row 1st Column upper I Subscript k comma l comma 4 Baseline left parenthesis theta right parenthesis 2nd Column colon equals sigma summation Underscript m equals l Overscript m left parenthesis k right parenthesis Endscripts a Subscript m Baseline e Superscript 2 pi l sine theta Baseline upper R Subscript k comma m Baseline left parenthesis theta right parenthesis period EndLayout

Our goal is to show that $|I_{k,l,1}(\theta )+I_{k,l,2}(\theta )+I_{k,l,3}(\theta )+I_{k,l,4}(\theta )|<2|a_l|$ , so that in between a pair of consecutive extreme values of $2a_l\cos (k\theta /2+2\pi l\cos \theta )$ , we get a zero of $I_{k,l}(\theta )$ . We record the following estimations, under the conditions in Theorem 1.1.

Lemma 3.1. For $\theta \in I$ ,

$$ \begin{align*} |I_{k,l,1}(\theta)|\le 0.1|a_l|, \quad |I_{k,l,2}(\theta)|\le 1.2|a_1|,\quad |I_{k,l,3}(\theta)|<0.2|a_l|, \quad |I_{k,l,4}(\theta)|<0.1|a_l|. \end{align*} $$

3.1 Concluding the proof

For $k\ge 28$ , using Lemma 3.1,

$$ \begin{align*} |I_{k,l,1}(\theta)|+|I_{k,l,2}(\theta)|+|I_{k,l,3}(\theta)|+ |I_{k,l,4}(\theta)|<1.4|a_l|<2|a_l|. \end{align*} $$

Therefore, the function $I_{k,l}(\theta )$ changes sign between two consecutive extreme values of $\cos (k\theta /2+2\pi l\cos \theta )$ . We enumerate the number of points of extreme values of $\cos (k\theta /2+2 \pi l\cos \theta )$ . For $\theta \in I$ , we have $k\theta /2+2\pi l\cos \theta \in [k\pi /4,k\pi /3-m\pi ]$ . Note that $\cos (k\theta /2+2\pi l\cos \theta ) =\pm 1$ if and only if $(k\theta /2+2\pi l\cos \theta )$ is an integer multiple of $\pi $ . Now, the interval $[k/4,k/3-l]$ contains exactly $m(k)-l+1$ integers. Hence, there are at least $m(k)-l$ zeros of $J_{k,l}(z)$ on the arc $A^{\circ }$ .

In the next section, we give the proof of Lemma 3.1.

4 Proof of Lemma 3.1

4.1 Estimation of $|I_{k,l,1}(\theta )|$

We see that

$$ \begin{align*} |I_{k,l,1}(\theta)|\le 2|a_l| \sum_{m=l+1} ^{m(k)}|e^{-2\pi (m-l)\sin\theta}\cos(k\theta/2+2\pi m\cos\theta)| \le 2|a_l| \sum_{m=l+1} ^{m(k)}e^{-2\pi (m-l)\sin\theta}. \end{align*} $$

Since $\sin \theta \ge \sqrt 3/2$ for $\theta \in I$ ,

$$ \begin{align*} |I_{k,l,1}(\theta)| \le 2 |a_l| \sum_{m=1} ^{\infty}e^{-\sqrt3\pi m} \le 2|a_l| \frac{e^{-\sqrt3\pi}}{1-e^{-\sqrt3\pi}} < 0.1|a_l|. \end{align*} $$

4.2 Estimation of $|I_{k,l,2}(\theta )|$

Here,

$$ \begin{align*} |I_{k,l,2}(\theta)|=\bigg|\!\sum_{m=l} ^{m(k)}a_m\frac{(-1)^me^{2\pi l\sin\theta-\pi m\tan(\theta/2)}}{(2\cos(\theta/2))^k}\bigg| &\le \frac{e^{\pi l(2\sin\theta-\tan(\theta/2))}}{(2\cos(\theta/2))^k}\sum_{m=l} ^{m(k)}|a_m|e^{-\pi(m-l)\tan(\theta/2)}\\ &\le \frac{e^{m(k)\pi(2\sin\theta-\tan(\theta/2))}}{(2\cos(\theta/2))^k}|a_l|\sum_{m=0}^{\infty}e^{-m\pi}\\ &\le \frac{e^{\pi k(2\sin\theta-\tan(\theta/2))/12}}{(2\cos(\theta/2))^k}|a_l|\sum_{m=0}^{\infty}e^{-m\pi}. \end{align*} $$

StartLayout 1st Row 1st Column StartAbsoluteValue upper I Subscript k comma l comma 2 Baseline left parenthesis theta right parenthesis EndAbsoluteValue equals StartAbsoluteValue sigma summation Underscript m equals l Overscript m left parenthesis k right parenthesis Endscripts a Subscript m Baseline StartFraction left parenthesis negative 1 right parenthesis Superscript m Baseline e Superscript 2 pi l sine theta minus pi m tangent left parenthesis theta divided by 2 right parenthesis Baseline Over left parenthesis 2 cosine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction EndAbsoluteValue 2nd Column less than or equals StartFraction e Superscript pi l left parenthesis 2 sine theta minus tangent left parenthesis theta divided by 2 right parenthesis right parenthesis Baseline Over left parenthesis 2 cosine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction sigma summation Underscript m equals l Overscript m left parenthesis k right parenthesis Endscripts StartAbsoluteValue a Subscript m Baseline EndAbsoluteValue e Superscript minus pi left parenthesis m minus l right parenthesis tangent left parenthesis theta divided by 2 right parenthesis Baseline 2nd Row 1st Column Blank 2nd Column less than or equals StartFraction e Superscript m left parenthesis k right parenthesis pi left parenthesis 2 sine theta minus tangent left parenthesis theta divided by 2 right parenthesis right parenthesis Baseline Over left parenthesis 2 cosine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction StartAbsoluteValue a Subscript l Baseline EndAbsoluteValue sigma summation Underscript m equals 0 Overscript infinity Endscripts e Superscript minus m pi Baseline 3rd Row 1st Column Blank 2nd Column less than or equals StartFraction e Superscript pi k left parenthesis 2 sine theta minus tangent left parenthesis theta divided by 2 right parenthesis right parenthesis divided by 12 Baseline Over left parenthesis 2 cosine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction StartAbsoluteValue a Subscript l Baseline EndAbsoluteValue sigma summation Underscript m equals 0 Overscript infinity Endscripts e Superscript minus m pi Baseline period EndLayout

We now show that the function

$$ \begin{align*} a_k(\theta):=\frac{e^{\pi k(2\sin\theta-\tan(\theta/2))/12}}{(2\cos(\theta/2))^k} \end{align*} $$ a Subscript k Baseline left parenthesis theta right parenthesis colon equals StartFraction e Superscript pi k left parenthesis 2 sine theta minus tangent left parenthesis theta divided by 2 right parenthesis right parenthesis divided by 12 Baseline Over left parenthesis 2 cosine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction

is increasing. Note that the derivative of the function $a_k(\theta )$ is

$$ \begin{align*} \tfrac{1}{12}\, k (2\cos(\theta/2))^{-k-1} \, e^{{\pi k (2 \sin \theta - \tan(\theta/2))}/{12}} (4\pi \cos(\theta/2) \cos\theta + 12 \sin(\theta/2) - \pi \sec(\theta/2)). \end{align*} $$

For $\theta \in I$ , we have $4\pi \cos (\theta /2) \cos \theta + 12 \sin (\theta /2) - \pi \sec (\theta /2)>0.$ To see this inequality, we first write

$$ \begin{align*} 4\pi \cos(\theta/2) \cos \theta + 12 \sin(\theta/2) - \pi \sec(\theta/2)=\frac{4\pi \cos^2(\theta/2) \cos\theta+6\sin\theta-\pi}{\cos(\theta/2)}. \end{align*} $$

So, we need to prove $4\pi \cos ^2(\theta /2) \cos \theta +6\sin \theta -\pi>0$ . For this, we break I into two parts, namely, $[\pi /2,1.95]$ and $[1.95,2\pi /3]$ . On $[\pi /2,1.95]$ ,

$$ \begin{align*} 4\pi \cos^2(\theta/2) \cos\theta+6\sin\theta\ge4\pi \cos^2(\pi/4) \cos(1.95)+6\sin1.95>\pi. \end{align*} $$

On $[1.95,2\pi /3]$ ,

$$ \begin{align*} 4\pi \cos^2(\theta/2) \cos\theta+6\sin\theta\ge4\pi \cos^2(1.95/2) \cos(2\pi/3)+6\sin(2\pi/3)>\pi. \end{align*} $$

Hence, $a_k(\theta )$ is an increasing function for $\theta $ in I. Since $a_k(2\pi /3)=1$ , we have $0<a_k(\theta )\le 1$ for $\theta \in I$ . Thus,

$$ \begin{align*} |I_{k,l,2}(\theta)|\le |a_l|\sum_{m=0}^{\infty}e^{-\pi m} \le |a_l|\bigg(\frac{1}{1-e^{-\pi}}\bigg)< 1.2|a_l|. \end{align*} $$

4.3 Estimation of $|I_{k,l,3}(\theta )|$

Note that

$$ \begin{align*} \bigg|\!\sum_{m=l} ^{m(k)}a_m\frac{e^{2\pi l\sin\theta-\pi m\cot(\theta/2)}}{(2i\sin(\theta/2))^k}\bigg| &\le |a_l|\frac{e^{\pi l(2\sin\theta-\cot(\theta/2))}}{(2\sin(\theta/2))^k}\sum_{m=l} ^{m(k)}e^{-\pi(m-l)\cot(\theta/2)}. \end{align*} $$

To estimate the right-hand side, we consider the function $c_k(\theta )$ for $\theta \in I$ and $k\ge 28$ , where

$$ \begin{align*} c_k(\theta)= \frac{e^{\pi k(2\sin\theta-\cot(\theta/2))/12}}{(2\sin(\theta/2))^k}. \end{align*} $$ c Subscript k Baseline left parenthesis theta right parenthesis equals StartFraction e Superscript pi k left parenthesis 2 sine theta minus cotangent left parenthesis theta divided by 2 right parenthesis right parenthesis divided by 12 Baseline Over left parenthesis 2 sine left parenthesis theta divided by 2 right parenthesis right parenthesis Superscript k Baseline EndFraction period

To obtain an upper bound of $c_k(\theta )$ , we prove that it is decreasing on I. We note that its derivative is

$$ \begin{align*} \tfrac{1}{12}\, k (2\sin(\theta/2))^{-k-1} \, e^{{\pi k(2 \sin\theta - \cot(\theta/2))}/{12}} (4 \pi\sin(\theta/2) \cos\theta + \pi \csc(\theta/2) - 12 \cos(\theta/2)), \end{align*} $$

which is negative for $\theta \in I$ . Hence, an upper bound of $c_k(\theta )$ comes at $\theta =\pi /2$ and, for $k\ge 28$ ,

$$ \begin{align*} \frac{e^{\pi k(2\sin\theta-\cot(\theta/2))/12}}{(2\sin(\theta/2))^k}\le \frac{e^{k\pi/12}}{2^{k/2}}<0.1. \end{align*} $$

Hence,

$$ \begin{align*} \bigg|\!\sum_{m=l} ^{m(k)}a_m\frac{e^{2\pi l\sin\theta-\pi m\cot(\theta/2)}}{(2i\sin(\theta/2))^k}\bigg| &< 0.1 |a_l|\sum_{m=l} ^{m(k)}e^{-\pi(m-l)\cot(\theta/2)} \le 0.1 |a_l|\sum_{m=0} ^{\infty}e^{-m\pi/\sqrt 3} \le 0.2|a_l|. \end{align*} $$

4.4 Estimation of $|I_{k,l,4}(\theta )|$

From the expression of $I_{k,l,4}(\theta )$ ,

$$ \begin{align*} |I_{k,l,4}(\theta)|\le |a_l|e^{2\pi l\sin\theta}\sum_{m=l} ^{m(k)}|R_{k,m}(\theta)|. \end{align*} $$

For $k\ge 28$ , using (2.1),

$$ \begin{align*} |I_{k,l,4}(\theta)|\le |a_l| m(k)e^{2\pi (m(k)-1)}3^{1-{k}/{2}}. \end{align*} $$

As $m(k)-1\le k/12-1$ ,

$$ \begin{align*} |I_{k,l,4}(\theta)|\le \frac{k}{4}|a_l| e^{-2\pi}\bigg(\frac{e^{\pi/3}}{3}\bigg)^{k/2}. \end{align*} $$

This gives the desired upper bound $|I_{k,l,4}(\theta )|\le 0.1|a_l|.$

Footnotes

The research of the second author is partially supported by the SERB (ANRF) Core Research Grant (CRG/2023/004746).

References

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Article contents

ON THE ZEROS OF CERTAIN LINEAR COMBINATIONS OF POINCARÉ CUSP FORMS

Abstract

Keywords

MSC classification

Information

1 Introduction

2 Preliminary steps

Lemma 2.1 [Reference Saha and Saradha6, Lemma 1]

2.1 Estimation of $R_{k,m}(\theta )$

3 Proof of Theorem 1.1

3.1 Concluding the proof

4 Proof of Lemma 3.1

4.1 Estimation of $|I_{k,l,1}(\theta )|$

4.2 Estimation of $|I_{k,l,2}(\theta )|$

4.3 Estimation of $|I_{k,l,3}(\theta )|$

4.4 Estimation of $|I_{k,l,4}(\theta )|$

Footnotes

References

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