Interlacing of zeros of weakly holomorphic modular forms
By Paul Jenkins and Kyle Pratt
Abstract
We prove that the zeros of a family of extremal modular forms interlace, settling a question of Nozaki. Additionally, we show that the zeros of almost all forms in a basis for the space of weakly holomorphic modular forms of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$ interlace on most of the lower boundary of the fundamental domain.
1. Introduction and main results
A natural question in studying functions of a complex variable is to determine the location of the zeros of a function; an especially interesting case occurs when the locations of the zeros follow a strong pattern. Many modular forms have zeros satisfying such properties. The most well-known such result comes from F. Rankin and Swinnerton-Dyer Reference 8, who proved that all zeros of the classical Eisenstein series in the standard fundamental domain $\mathcal{F}$ lie on the circular arc $\mathcal{A}= \left\lbrace e^{i \theta } : \frac{\pi }{2} \leq \theta \leq \frac{2\pi }{3} \right\rbrace$ on the lower boundary of $\mathcal{F}$.
When two functions have zeros that lie on the same arc, we say that the zeros of two functions interlace if every zero of one function is contained in an open interval whose endpoints are zeros of the other function, and each such interval contains exactly one zero. Gekeler conjectured that the Eisenstein series $E_k(z)$ satisfies such interlacing properties in Reference 3, and Nozaki Reference 7 proved that the zeros of $E_k(z)$ interlace with the zeros of $E_{k+12}(z)$ by improving the bounds used by Rankin and Swinnerton-Dyer. For the modular function $j_n(z)$ given by the action of the $n$th Hecke operator on $j(z) - 744$, Jermann Reference 6 extended work of Asai, Kaneko, and Ninomiya Reference 1 to prove that the zeros of $j_n(z)$ interlace with the zeros of $j_{n+1}(z)$. In this paper, we prove interlacing for a family of holomorphic modular forms for $\mathrm{SL}_2(\mathbb{Z})$, all of whose zeros in $\mathcal{F}$ lie on the arc $\mathcal{A}$.
Denote by $M_k$ the space of holomorphic modular forms of weight $k$, and write $M_k^{!}$ for the larger space of weakly holomorphic modular forms (i.e. poles are allowed at the cusps) of weight $k$. For such weights $k$, we write $k = 12\ell + k'$, with $k' \in \{0,4,6,8,10,14\}$ and $\ell \in \mathbb{Z}$. Duke and the first author Reference 2 introduced a canonical basis $\{f_{k, m}(z)\}_{m=-\ell }^\infty$ for $M_k^{!}$ whose elements are defined by
where, as usual, $q = e^{2\pi i z}$. They approximated $f_{k,m}(z)$ on the boundary arc $\mathcal{A}$ by a trigonometric function to prove the following theorem locating the zeros of these basis elements.
The condition in Theorem 1.1 that $m \geq |\ell | - \ell$ is not sharp, but for all large enough weights $k$, the zeros of at least one of the $f_{k,m}(z)$ do not all lie on $\mathcal{A}$. The theorem is generally false when $f_{k, m}$ is a cusp form; one example is the form $f_{132,-9}(z)$.
Getz studied a subset of these basis elements in Reference 4. We call this family of modular forms “gap functions”, as they are the holomorphic modular forms with the maximum possible gap in their $q$-expansions. For any even weight $k \geq 4$, these functions are defined as
These gap functions have application to the theory of extremal lattices and questions in coding theory; see for example Reference 5.
Our first result proves that the zeros of these gap functions, all of which lie on $\mathcal{A}$, interlace.
This theorem settles a question of Nozaki, who suggested that the zeros of these functions “interlace in some range” (see Reference 7, Example 1.1).
Additionally, we are able to partially extend our results to the larger class of functions $f_{k,m}(z)$. The result is the following theorem.
As the methods used in the proof of Theorem 1.2 do not entirely apply to the case in which $m$ is nonzero, they do not give interlacing on all of $\mathcal{A}$, although preliminary computations suggest that such interlacing generally holds. We leave this as an open problem.
2. Background
We begin by defining some notation. The standard fundamental domain for $\mathrm{SL}_2(\mathbb{Z})$ is
where $r = e^{2\pi i \tau }$ and $C$ is a counterclockwise circle in the $r$-plane centered at 0 with sufficiently small radius. We fix $z = e^{i \theta }$ for some $\theta$ in the interval $I =(\frac{\pi }{2},\frac{2\pi }{3})$ and change variables $r \mapsto \tau$. Then for some $A > 1$ we have
We move the contour downward to a height of $A'$. As we do so, each pole in the region
$$\begin{equation*} \left\lbrace -\frac{1}{2} < \operatorname {Re}(\tau ) < \frac{1}{2}, A' < \operatorname {Im}(\tau ) < A \right\rbrace \end{equation*}$$
contributes to the value of the integral; these poles occur when $\tau$ is equivalent to $z$ under the action of $\mathrm{SL}_2(\mathbb{Z})$. If a pole occurs with real part $-\frac{1}{2}$, we modify the contour to include small semicircles in the usual way.
We allow $A'$ to vary depending on $z$. This is to ensure that if $z$ is close to $\frac{\pi }{2}$, then the residue term from $\tau = \frac{z}{z-1}$ does not appear. We choose $A' = .75$ if $\frac{\pi }{2} < \theta \leq 1.9$, picking up residues from $\tau = z$ and $\tau = -\frac{1}{z}$, and we let $A' = .65$ if $\frac{7\pi }{12} \leq \theta < \frac{2\pi }{3}$, picking up an additional residue at $\tau = \frac{z}{z+1}$. The overlap of the two intervals is necessary to obtain interlacing of the zeros. Applying the Residue Theorem and taking absolute values as in the proof of Theorem 1.1 in Reference 2, we obtain
when $\frac{7\pi }{12} \leq \theta < \frac{2\pi }{3}$.
Note that for any modular form $g$ of weight $k$, the function $e^{\frac{ik\theta }{2}} g(e^{i\theta })$ is real-valued for $\theta \in I$, so the left-hand sides of (Equation 1) and (Equation 2) are absolute values of real-valued functions of $\theta$. Thus, these inequalities give approximations for the modular forms $f_{k,m}(z)$ by the trigonometric function $2\cos \left(\frac{k\theta }{2} - 2\pi m\cos {\theta }\right)$, whose zeros are proved to interlace in the next section. To prove Theorem 1.2, we show that the right-hand sides of equations (Equation 1) and (Equation 2) are exponentially decaying functions in the weight $k$, preserving the interlacing for $k$ sufficiently large. To prove interlacing in the first interval is straightforward. On the other hand, the additional residue term in the second interval shifts the zeros of $G_k(z)$ away from the zeros of the cosine function, necessitating more care. Computing the interlacing of the zeros of the $G_k(z)$ for all smaller $k$ proves interlacing for all $k \geq 4$.
The proof of Theorem 1.3 proceeds along similar lines, using the fact that the right-hand sides of (Equation 1) and (Equation 2) are also exponentially decaying in $m$.
3. Interlacing for cosine functions
In this section, we show that the cosine functions obtained in the residue calculation have zeros that interlace. We define
When it is clear from context, we write $b_*(\theta )$ to mean either $b_{k+12}(\theta )$ or $b_{m+1}(\theta )$. Note that $\cos (b_*(\theta ))$ has one more zero in $I$ than does $\cos (b(\theta ))$.
We first prove that the zeros of $\cos (b(\theta ))$ and $\cos (b_*(\theta ))$ interlace on $I$.
In showing interlacing for the $f_{k,m}(z)$, we will need bounds on the distances between the zeros of the approximating cosine functions. The following proposition gives a preliminary estimate on the distances between zeros.
We use Proposition 3.2 to prove a stronger result.
The lemma is clearly true when $\cos (b(\theta ))$ has only one zero in $I$ and is an immediate consequence of the following proposition.
This proposition says that as we examine an increasing sequence of intervals whose endpoints are zeros of $\cos (b_*(\theta ))$, the zero of $\cos (b(\theta ))$ in each interval moves farther from the left-hand side of the interval and closer to the right-hand side of the interval.
Recall that Theorem 1.2 is a statement about $G_k(z) = f_{k,0}(z)$, so we use Lemmas 3.1 and 3.3, when needed, with $m = 0$. We proceed with two cases, depending on the value of $\theta$.
Suppose that $\theta \in (\frac{\pi }{2},1.9]$. From Equation 1 it is clear that $G_k(z)$ may be approximated by $2\cos (\frac{k\theta }{2})$ if we convert the right-hand side into a suitably decreasing function of $k$. Lemma 3.1 shows that the zeros of $2\cos (\frac{k\theta }{2})$ interlace, so the zeros of $G_k(z)$ will also interlace if $G_k(z)$ is sufficiently close to $2\cos \left(\frac{k\theta }{2}\right)$.
Consider the quotient of $\Delta$ functions on the right-hand side of Equation 1. By Proposition 2.2 of Reference 4, we know that for $\theta \in I$, we have
Using the relation $k = 12\ell + k'$, it is clear that $\ell \geq \frac{k-14}{12}$, so we define $C(k)=2.97 (.66713)^{\frac{k-14}{12}}$. We see that $C(k) \geq 2.97 (.66713)^\ell$.
We now compute how far the zeros of $G_k(e^{i \theta })$ can stray from the zeros of $2\cos \left(\frac{k\theta }{2}\right)$. Suppose that
for some constant $C<2$, and let $\alpha$ satisfy $2\cos \left(\frac{k\alpha }{2}\right)=0.$ Then a zero of $e^{\frac{ik \theta }{2}}G_k(e^{i \theta })$ appears in the interval $(\alpha - \varepsilon ,\alpha + \varepsilon )$, where $\left|2\cos \left(\frac{k(\alpha \pm \varepsilon )}{2}\right)\right| = C.$ To get an upper bound on $\varepsilon$, consider the line intersecting $2\cos \left(\frac{k\theta }{2}\right)$ through the points $(\alpha -\frac{\pi }{k},\pm 2),(\alpha ,0),(\alpha +\frac{\pi }{k}, \mp 2).$ The concavity of $2\cos \left(\frac{k\theta }{2}\right)$ near $\alpha$ implies that this line lies between $2\cos \left(\frac{k\theta }{2}\right)$ and the $\theta$-axis. Therefore, if $\beta$ is a point at which the value of the line is $\pm C$, then $|\beta -\alpha | > \varepsilon$. The absolute value of the slope of the line is $\frac{2k}{\pi }$, and it follows that $\varepsilon < \dfrac{\pi C}{2k}$.
Lemma 3.3 gives us a lower bound on the distances between the zeros of $\cos (b(\theta ))$ and $\cos (b_*(\theta ))$; this lower bound is $\frac{\pi }{k} - \frac{\pi }{k+12}$ when $k' \in \{0,4,8\}$ and $2\left(\frac{\pi }{k} - \frac{\pi }{k+12}\right)$ when $k' \in \{ 6,10,14 \}$. This can be seen by considering what happens at the endpoints of $I$ for $k' \equiv 0,2$ mod 4. For instance, if $k \equiv 0 \pmod {4}$, then at $\theta = \frac{\pi }{2}$ we have that $\frac{k\theta }{2} = \frac{k\pi }{4}$ is an integer multiple of $\pi$ and needs to increase only by $\frac{\pi }{2}$ before $\cos (\frac{k\theta }{2})$ has a zero in $I$. When $k \equiv 2 \pmod {4}$, we see that $\frac{k\theta }{2}$ must increase by $\pi$ before a zero occurs.
Replacing $C$ with $C(k)$, we solve the inequality
which is true when $k \geq 118$. This means that when $k \geq 118$, the zeros of $G_k(z), G_{k+12}(z)$ differ from the zeros of $\cos (b(\theta )), \cos (b_*(\theta ))$ by an amount which is less than half the minimum distance between zeros of $\cos (b_*(\theta ))$ and $\cos (b(\theta ))$. The zeros of $G_k(z)$ and $G_{k+12}(z)$ therefore lie in disjoint, interlacing intervals, and must interlace on $(\frac{\pi }{2},1.9]$ for $k \geq 118$.
Now let $\theta \in [\frac{7\pi }{12},\frac{2\pi }{3})$. The method of the previous case must be modified, as we are dealing with a different approximating function for $G_k(z)$, given by
The term $(2\cos (\frac{\theta }{2}))^{-k}$ is monotonically increasing, is very small for smaller $\theta$, and tends rapidly to 1 for $\theta$ close to $\frac{2\pi }{3}$. This residue term shifts the zeros of $G_k(z)$ away from the zeros of $\cos (b(\theta ))$, but for large $k$ the effect is negligible unless $\theta$ is very near $\frac{2\pi }{3}$.
As before, we need a lower bound on the distance between the zeros of $H_k(\theta )$ and $H_{k+12}(\theta )$. An easily adapted lemma from Nozaki (Reference 7, Lemma 4.1) shows that for a zero $\alpha ^*$ of $2\cos (\frac{k\theta }{2})$ and the corresponding zero $\alpha$ of $H_k(\theta )$, we have
if $\alpha ^* \geq \frac{7\pi }{12}$. This fact follows from the observations that $|2\cos (\frac{k}{2}(\alpha ^* \pm \frac{\pi }{3k}))|=1$ and $0<(2\cos (\frac{\theta }{2}))^{-k}<1$ for $\theta \in I$. We will use this fact frequently to estimate quantities involving zeros of $H_k(\theta )$.
Let $\alpha$ denote a zero of $H_k(\theta )$ and $\beta$ an adjacent zero of $H_{k+12}(\theta )$. There are two cases to consider: intervals of type $(\beta , \alpha )$, and intervals of type $(\alpha , \beta )$. We will obtain lower bounds on the length of intervals of both types.
Consider first the $(\beta , \alpha )$ intervals. We may view these essentially as intervals defined by zeros of $2\cos (\frac{k\theta }{2})$ and $2\cos (\frac{(k+12)\theta }{2})$, along with some zero shifts due to the presence of the $(2\cos (\frac{\theta }{2}))^{-k}$ term. By Proposition 3.4, the shortest such interval is the first after $\frac{7\pi }{12}$.
We proceed by cases, according to the congruence class of $k \pmod {12}$. Suppose that $k' = 0$, so that $k = 12\ell$. We solve $\frac{7\pi (k+12)}{24} \leq \frac{\pi (2n+1)}{2}$ to find the smallest $n$ such that $\frac{\pi (2n+1)}{2}>\frac{7\pi }{12}$ is a zero of $2\cos (\frac{(k+12)\theta }{2})$ and similarly find the next zero of $2\cos (\frac{k\theta }{2})$. If $\ell$ is even, then the distance between these cosine zeros is
The method for handling the $(\beta , \alpha )$ intervals cannot be easily adapted for $(\alpha , \beta )$ intervals, so we use a different approach. Our general strategy of proof involves the function $H_k(\theta )+H_{k+12}(\theta )$. If $\beta - \alpha < \frac{\pi }{3k}$, we show that this function is monotonically increasing or decreasing on the interval $(\alpha , \beta )$. We then obtain a lower bound on $|H_{k+12}(\alpha )|+|H_k(\beta )|$, which gives the change in the value of this function over the interval, and use the trivial bound on the derivative of $H_k(\theta )+H_{k+12}(\theta )$ given by
to find a lower bound for $\beta - \alpha$. On the other hand, if $\beta - \alpha \geq \frac{\pi }{3k}$, we may simply use $\frac{\pi }{3k}$ as a lower bound.
To see that $H_k(\theta )+H_{k+12}(\theta )$ is monotonic on the interval $(\alpha , \beta )$ when $\beta - \alpha < \frac{\pi }{3k}$, we note that the interval $(\alpha , \beta )$ is contained in the interval $(\alpha ^*-\frac{2\pi }{3k},\alpha ^*+\frac{2\pi }{3k})$. On this larger interval, the absolute value of the derivative of $2\cos \left(\frac{k\theta }{2}\right)$ ranges from $\frac{k}{2}$ to $k$. The absolute value of the derivative of $(2\cos (\theta /2))^{-k}$, on the other hand, is bounded above by its value at an upper bound for the largest possible $\beta$ of $\theta = \frac{2\pi }{3} - \frac{\pi }{k+12} + \frac{\pi }{3(k+12)}$; this value is at most $(.142)k$. Thus, the derivative of the cosine term dominates in the interval, and $H_k(\theta )$ is monotonic; it follows that $H_k(\theta )+H_{k+12}(\theta )$ is monotonic on $(\alpha , \beta )$.
We now bound $|H_{k+12}(\alpha )|+|H_k(\beta )|$ when $\beta - \alpha < \frac{\pi }{3k}$. There are three cases to consider, based on the value of $k'$ mod 12, since the behavior of $H_k(\theta )$ depends heavily on $k'$. In each case there are two subcases, since for this type of interval the zeros of $H_k(\theta )$ and $H_{k+12}(\theta )$ either both shift to the left or both shift to the right from the zeros of $2\cos (\frac{k\theta }{2})$ and $2\cos (\frac{(k+12)\theta }{2})$.
Our proof follows the outlines of the proof above; the most significant differences involve the lower bounds on the distances between zeros. We take linear approximations to $b(\theta )$ and $b_*(\theta )$ and use those approximations to derive lower bounds on the distance between zeros. We require the hypotheses for Lemma 3.3 to hold; we then need only find such a bound near $\theta =\frac{\pi }{2}$ and $\theta = \frac{2\pi }{3}$.
We first determine the bound for zeros near $\theta = \frac{\pi }{2}$. Taking the first order Taylor series approximation for $b(\theta ) = \frac{k\theta }{2} - 2\pi m \cos \theta$ gives us
When we increase $k$ by 12, the linear approximations to $b(\theta )$ and $b_{k+12}(\theta )$ have the same error term.
Write $b(\theta )=L_{k,m}(\theta )-R_{m}(\theta )$. Note that $R_{m}(\theta )$ is increasing and positive on $I$, since $b(\theta )$ is concave down. Let $\alpha _1,\alpha _2$ be the first zeros of $\cos (L_{k+12,m}(\theta ))$ and $\cos (L_{k,m}(\theta ))$ in $I$, respectively, and let $\beta _1,\beta _2$ be the first zeros on $I$ of $\cos (b_{k+12}(\theta ))$ and $\cos (b(\theta ))$. We then have, for integers $n_1$ and $n_2$,
Now we find the slopes of the lines between $(\alpha _1,b_{k+12}(\alpha _1))$ and $(\beta _1,b_{k+12}(\beta _1))$ and between $(\alpha _2,b(\alpha _2))$ and $(\beta _2,b(\beta _2))$, and apply the Mean Value Theorem. The slope can be taken to be the value of the derivative at a point in the interval, and by the proof of Proposition 3.4 the derivative of $2\cos (b_*(\theta ))$ is greater than the derivative of $2\cos (b(\theta ))$ in the appropriate intervals for $k$ large enough. Thus, for large $k$ we have that
which implies $\beta _2-\alpha _2>\beta _1-\alpha _1$. This in turn implies that $\beta _2-\beta _1 > \alpha _2 - \alpha _1$, so the distance between the zeros of $\cos (L_{k,m}(\theta ))$ and $\cos (L_{k+12,m}(\theta ))$ is less than the distance between the zeros of $\cos (b(\theta ))$ and $\cos (b_{k+12}(\theta ))$. Computing the distance between the zeros of $\cos (L_{k,m}(\theta ))$ and $\cos (L_{k+12,m}(\theta ))$, we get a lower bound of
for the distance between zeros near $\theta = \frac{\pi }{2}$.
The argument for increasing $m$ by 1 is not exactly analogous because the error terms are no longer identical. We use the Taylor series approximation for $b(\theta )$ and use the fact that near $\theta =\frac{\pi }{2}$ we have $b(\theta )$ close to its first order approximation.
Assume $k>0,$ since the case $k<0$ is similar. Additionally, assume $k \equiv 0 \pmod {4}$. If $k \equiv 2 \pmod {4}$ we find that the lower bound on the zeros is greater than the lower bound when $k \equiv 0 \pmod {4}$. Bounding from beneath the derivative of $b(\theta )$, we see that because $k \equiv 0 \pmod {4}$, the first zero of $2\cos (b(\theta ))$ in $I$ is less than $\frac{\pi }{2}+\frac{\pi }{k+2\pi \sqrt {3} m}$. By Taylor’s Theorem,
Since the first zero of $2\cos (L_{k,m}(\theta ))$ in $I$ is at $\frac{\pi (2+k+4\pi m)}{2(k+4\pi m)}$, we see that the first zero of $2\cos (b(\theta ))$ in $I$ is less than $\frac{\pi (2+k+4\pi m)}{2(k+4\pi m)}+\frac{2}{m(k+2\pi \sqrt {3}m)}$. Thus, a lower bound on the distance between the first zeros of $2\cos (b(\theta ))$ and $2\cos (b_{m+1}(\theta ))$ is given by
which is positive for $m$ large enough with respect to $k$.
We use similar arguments for the lower bound near $\theta = \frac{2\pi }{3}$ and find that the lower bound between zeros, whether we increase $k$ or increase $m$, is given by a decreasing rational function in $k$ and $m$.
Now we pick $\epsilon > 0$, which is fixed for the remainder of the proof. From equations Equation 1 and Equation 2 and the proof of Theorem 1.1 we obtain
where $\rho = \frac{2\pi }{3} - \epsilon$. We do so by comparing the bounds for the two different intervals and choosing our bound to be larger than both of them. Note that each term of the right side is exponentially decaying in both $m$ and $k$.
Suppose more generally that $|e^{\frac{ik \theta }{2}}e^{-2\pi m\sin {\theta }}f_{k,m}(e^{i\theta }) -2\cos \left(\frac{k\theta }{2} - 2\pi m\cos {\theta }\right) |<D$. We want to derive an upper bound in terms of $D$ on the distance a zero of $f_{k,m}(e^{i\theta })$ can be from a zero of $2\cos (b(\theta ))$. We will do this by bounding from beneath the absolute value of the derivative of $2\cos (b(\theta ))$ on intervals around each of its zeros. If $D$ is small enough, the zeros of $f_{k, m}(e^{i\theta })$ must lie in these intervals, and we may argue as in the proof of Theorem 1.2 to obtain a bound involving an exponentially decaying quantity.
Suppose that $2\cos (b(\alpha ^*))=0$ for some $\alpha ^*$. We calculate that
Using the above bounds, we must bound $|\frac{d}{d\theta }\left(2\cos (b(\theta )) \right)|$ from below on a suitable interval. Trivially bounding the derivative of $2\cos (b(\theta ))$, we see that $\frac{\pi }{2(|k|+4\pi m)}$ is an insufficient change in the absolute value for $2\cos (b(\theta ))$ to reach an extreme. Thus we consider the interval $(\alpha ^*- \frac{\pi }{2(|k|+4\pi m)}, \alpha ^*+ \frac{\pi }{2(|k|+4\pi m)})$. Standard formulas give
$$\begin{equation*} \sin \left(b\left(\alpha ^* \pm \frac{\pi }{2(|k|+4\pi m)}\right)\right)=\sin x\cos y \pm \cos x\sin y, \end{equation*}$$
where $x = \frac{k\alpha ^*}{2} - 2\pi m\cos \alpha ^* \cos (\frac{\pi }{2(|k|+4\pi m)})$ and $y = 2\pi m\sin \alpha ^* \sin (\frac{\pi }{2(|k| + 4\pi m)}) + \frac{k\pi }{4(|k| + 4\pi m)}$. When $k$ or $m$ is large enough, $\cos (\frac{\pi }{2(|k|+4\pi m)})$ is very near 1, so $\cos (x) \approx 0$ and $|\sin (x)| \approx 1$. When $m$ is fixed and $k$ increases, we see that $y$ approaches $\frac{\pi }{4}$, so $|\sin (b(\alpha ^* \pm \frac{\pi }{2(|k|+4\pi m)}))|$ is close to $\frac{\sqrt {2}}{2}$. When $k$ is fixed and $m$ increases, $y$ approaches $\frac{\pi \sin \alpha ^*}{4}$ and $|\sin (b(\alpha ^* \pm \frac{\pi }{2(k+4\pi m)}))|$ is close to or greater than $\cos (\frac{\pi }{4}) = \frac{\sqrt {2}}{2}$, since $\alpha ^*$ may range from $\frac{\pi }{2}$ to $\frac{2\pi }{3}$. With this bound at the endpoints of our interval, we let $E$ be a positive constant smaller than $\frac{\sqrt {2}}{2}$ such that for all $k,m$ under consideration with $k$ fixed and $m$ increasing (or $m$ fixed and $k$ increasing) and $\theta \in (\alpha ^*-\frac{\pi }{2(|k|+4\pi m)},\alpha ^*+\frac{\pi }{2(|k|+4\pi m)})$, we have
Letting $M(k,m)$ be the minimum of the four lower bounds on the distance between zeros we calculated above and replacing $D$ with our exponential quantities from before, we find that the zeros interlace when
Because the left-hand side has exponential decay in $k$ and $m$ while the right-hand side is a rational function of $k$ and $m$, we see that the inequality holds for all but finitely many $k$ and $m$, so the zeros of $f_{k,m}(z)$ interlace with the zeros of $f_{k+12,m}(z)$ or $f_{k,m+1}(z)$ on $(\frac{\pi }{2},\frac{2\pi }{3} - \epsilon )$ for $k,m$ sufficiently large.■
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author={Nozaki, Hiroshi},
title={A separation property of the zeros of Eisenstein series for ${\rm SL}(2,\mathbb {Z})$},
journal={Bull. Lond. Math. Soc.},
volume={40},
date={2008},
number={1},
pages={26--36},
issn={0024-6093},
review={\MR {2409175 (2009d:11070)}},
doi={10.1112/blms/bdm117},
}
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author={Rankin, F. K. C.},
author={Swinnerton-Dyer, H. P. F.},
title={On the zeros of Eisenstein series},
journal={Bull. London Math. Soc.},
volume={2},
date={1970},
pages={169--170},
issn={0024-6093},
review={\MR {0260674 (41 \#5298)}},
}
Show rawAMSref\bib{3211795}{article}{
author={Jenkins, Paul},
author={Pratt, Kyle},
title={Interlacing of zeros of weakly holomorphic modular forms},
journal={Proc. Amer. Math. Soc. Ser. B},
volume={1},
number={7},
date={2014},
pages={63-77},
issn={2330-1511},
review={3211795},
doi={10.1090/S2330-1511-2014-00010-9},
}
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