1.1 Selberg zeta functions for compact Riemann surfaces

In [Reference Luo22], Luo initiated the study of the nontrivial zeros of the derivative $Z_{M}^{\prime }$ of the Selberg zeta function $Z_{M}$ associated to a compact, hyperbolic Riemann surface $M$ , proving analogues of results obtained by Spira [Reference Speiser27] and Berndt [Reference Berndt4] for the Riemann zeta function. Further refinements of results by Luo were established in [Reference Garunkŝtis11] and [Reference Garunkŝtis12]. As is standard in analytic number theory, the nontrivial zeros of $Z_{M}$ , or $Z_{M}^{\prime }$ , are its zeros which do not arise from the poles of the multiplicative factor of the functional equation. In the case of a compact Riemann surface, the nontrivial zeros of $Z_{M}$ and $Z_{M}^{\prime }$ are zeros different from negative integers. Let us summarize the three main results stemming from the aforementioned articles, which are location, vertical distribution, and an asymptotic count for the weighted vertical distribution of the nontrivial zeros of $Z_{M}^{\prime }$ .

In [Reference Luo22] it is shown that $Z_{M}^{\prime }(s)$ has at most a finite number of nontrivial zeros in the half-plane $\text{Re}(s)<1/2$ . This result was strengthened in [Reference Minamide24] and [Reference Minamide25] where it is proved that $Z_{M}^{\prime }(s)$ has no nontrivial zeros in the half-plane $\text{Re}(s)<1/2$ .

Let $\text{vol}(M)$ denote the hyperbolic volume of $M$ . Let $\ell _{M,0}$ be the length of the shortest closed geodesic on $M$ . Let $m_{M,0}$ denote the number of inconjugate geodesics whose length is $\ell _{M,0}$ . Let $N_{\text{ver}}(T;Z_{M}^{\prime })$ be the number of nontrivial zeros of $Z_{M}^{\prime }(s)$ where $s=\unicode[STIX]{x1D70E}+it$ with $\unicode[STIX]{x1D70E}\geqslant 1/2$ and $0<t<T$ , and let

$$\begin{eqnarray}N_{\text{w}}(T;Z_{M}^{\prime })=\mathop{\sum }_{{Z_{M}^{\prime }(\unicode[STIX]{x1D70E}+it)=0\atop 0<t<T,\unicode[STIX]{x1D70E}>1/2}}(\unicode[STIX]{x1D70E}-1/2)\end{eqnarray}$$

be the weighted vertical distribution with weights equal to distances of zeros to the critical line. Then, building on the results form [Reference Luo22], it is proved in [Reference Garunkŝtis11] and [Reference Garunkŝtis12] that

(1) $$\begin{eqnarray}N_{\text{ver}}(T;Z_{M}^{\prime })=\frac{\text{vol}(M)}{4\unicode[STIX]{x1D70B}}T^{2}-\frac{\ell _{M,0}}{2\unicode[STIX]{x1D70B}}T+o(T)\quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

and

(2) $$\begin{eqnarray}\displaystyle N_{\text{w}}(T;Z_{M}^{\prime }) & = & \displaystyle \frac{T}{2\unicode[STIX]{x1D70B}}\log T+\frac{T}{2\unicode[STIX]{x1D70B}}\left(\frac{1}{2}\ell _{M,0}+\log \left(\frac{\text{vol}(M)(1-e^{-\ell _{M,0}})}{m_{M,0}\ell _{M,0}}\right)-1\right)\nonumber\\ \displaystyle & & \displaystyle +\,o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

The study of the zeros of $Z_{M}^{\prime }$ is of particular interest because of the connection with spectral analysis. Recall that if $s$ is a nontrivial zero of $Z_{M}(s)$ , then $\unicode[STIX]{x1D706}=s(1-s)$ is an eigenvalue of an $L^{2}$ -eigenfunction of the hyperbolic Laplacian which acts on the space of smooth functions on $M$ . Common zeros of $Z_{M}$ and $Z_{M}^{\prime }$ are, in fact, zeros of $Z_{M}(s)$ with multiplicity greater than one. Such zeros of $Z_{M}$ correspond to multi-dimensional eigenspaces of the Laplacian. As shown in [Reference Luo22, p. 1143], all zeros of $Z_{M}^{\prime }(s)$ on the line $\text{Re}(s)=1/2$ , except possibly at $s=1/2$ , correspond to multiple zeros of $Z_{M}$ . The problem of obtaining nontrivial bounds for the dimension of eigenspaces of the Laplacian is very difficult; see [Reference Iwaniec17, p. 160]. Thus, it is possible that refined information regarding (1) possibly could shed light on this important, outstanding question.

1.2 Noncompact Riemann surfaces

Let $\mathbb{H}$ denote the hyperbolic upper half-plane. Let $\unicode[STIX]{x1D6E4}\subseteq \text{PSL}(2,\mathbb{R})$ be any Fuchsian group of the first kind acting by fractional linear transformations on $\mathbb{H}$ , and let $M$ be the quotient space $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ .

One realization of the spectral analysis of the Laplacian on the surface $M$ is the location of nontrivial zeros of the associated Selberg zeta function $Z_{M}$ , defined for $s\in \mathbb{C}$ with $\text{Re}(s)>1$ by the Euler product

(3) $$\begin{eqnarray}\displaystyle Z_{M}(s)=\mathop{\prod }_{n=0}^{\infty }\mathop{\prod }_{P_{0}\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})}\big(1-e^{-(s+n)\ell _{P_{0}}}\big)=\mathop{\prod }_{n=0}^{\infty }\mathop{\prod }_{P_{0}\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})}\big(1-N(P_{0})^{-(s+n)}\big). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here ${\mathcal{H}}(\unicode[STIX]{x1D6E4})$ denotes a complete set of representatives of inconjugate, primitive hyperbolic elements of $\unicode[STIX]{x1D6E4}$ , $P_{0}$ is a primitive hyperbolic element, $\ell _{P_{0}}$ is the hyperbolic length of the geodesic path in the homotopy class determined by $P_{0}$ and the norm $N(P_{0})$ is equal to $\exp (\ell _{P_{0}})$ .

The function $Z_{M}$ possesses a meromorphic continuation to the whole complex plane and satisfies the functional equation $Z_{M}(s)\unicode[STIX]{x1D719}_{M}(s)=\unicode[STIX]{x1D702}_{M}(s)Z_{M}(1-s)$ , where $\unicode[STIX]{x1D719}_{M}(s)$ denotes the determinant of the scattering matrix $\unicode[STIX]{x1D6F7}_{M}(s)$ ,

(4) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(s) & = & \displaystyle \text{vol}(M)(s-1/2)\tan (\unicode[STIX]{x1D70B}(s-1/2))-\unicode[STIX]{x1D70B}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\underset{0<\unicode[STIX]{x1D703}(R)<\unicode[STIX]{x1D70B}}{\underset{\left\{R\right\}}{\sum }}\frac{1}{M_{R}\sin \unicode[STIX]{x1D703}}\frac{\cos (2\unicode[STIX]{x1D703}-\unicode[STIX]{x1D70B})(s-1/2)}{\cos \unicode[STIX]{x1D70B}(s-1/2)}\nonumber\\ \displaystyle & & \displaystyle +\,2n_{1}\log 2+n_{1}\left(\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(1/2+s)+\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(3/2-s)\right)=\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(1-s)\end{eqnarray}$$

and where $\{R\}$ denotes a complete, finite set of inconjugate elliptic elements of $\unicode[STIX]{x1D6E4}$ so that $0<\unicode[STIX]{x1D703}(R)<\unicode[STIX]{x1D70B}$ is the uniquely determined real number such that $R$ is conjugate to the matrix

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}\cos \unicode[STIX]{x1D703}(R) & -\sin \unicode[STIX]{x1D703}(R)\\ \sin \unicode[STIX]{x1D703}(R) & \cos \unicode[STIX]{x1D703}(R)\\ \end{array}\right).\end{eqnarray}$$

The scattering determinant $\unicode[STIX]{x1D719}_{M}(s)$ has a decomposition into a product of a general Dirichlet series and Gamma functions. Specifically, we can write

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{M}(s)=\unicode[STIX]{x1D70B}^{n_{1}/2}\left(\frac{\unicode[STIX]{x1D6E4}\left(s-\frac{1}{2}\right)}{\unicode[STIX]{x1D6E4}(s)}\right)^{n_{1}}\overset{\infty }{\underset{n=1}{\sum }}\frac{d(n)}{\mathfrak{g}_{n}^{2s}}\end{eqnarray}$$

where $n_{1}$ is the number of cusps of $M$ , and $\{d(n)\}$ and $\{\mathfrak{g}_{n}\}$ are sequences of real numbers with

$$\begin{eqnarray}0<\mathfrak{g}_{1}<\cdots <\mathfrak{g}_{n}<\mathfrak{g}_{n+1}<\cdots \,;\end{eqnarray}$$

given in terms of Kloosterman sums (see [Reference Iwaniec17, p. 160]). Let us write $\unicode[STIX]{x1D719}_{M}(s)=K_{M}(s)\cdot H_{M}(s)$ where

(5) $$\begin{eqnarray}\displaystyle K_{M}(s) & = & \displaystyle \unicode[STIX]{x1D70B}^{n_{1}/2}\left(\frac{\unicode[STIX]{x1D6E4}\left(s-\frac{1}{2}\right)}{\unicode[STIX]{x1D6E4}(s)}\right)^{n_{1}}e^{c_{1}s+c_{2}},\quad \text{with }\nonumber\\ \displaystyle & & \displaystyle c_{1}=-2\log \mathfrak{g}_{1}\qquad \text{and}\qquad c_{2}=\log d(1),\end{eqnarray}$$

and

(6) $$\begin{eqnarray}\displaystyle H_{M}(s)=1+\overset{\infty }{\underset{n=2}{\sum }}\frac{a(n)}{r_{n}^{2s}}\quad \text{with }r_{n}=\mathfrak{g}_{n}/\mathfrak{g}_{1}>1\qquad \text{and}\qquad a(n)=d(n)/d(1). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The Dirichlet series expansion for $H_{M}(s)$ converges for all $\text{Re}(s)>1$ . We call the function $H_{M}$ the Dirichlet series portion of the scattering determinant $\unicode[STIX]{x1D719}_{M}$ . In general, the function $H_{M}$ can be expressed as the determinant of a matrix whose entries are general Kloosterman sums; see [Reference Iwaniec17, Theorem 3.4]. The constants $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are explained in terms of the left lower entries of the matrices appearing in the double coset decomposition of $\unicode[STIX]{x1D6E4}$ . Therefore, the constants $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are precisely connected to the Fuchsian group $\unicode[STIX]{x1D6E4}$ and $H_{M}(s)$ is a Dirichlet series carrying the information related to parabolic subgroups of $\unicode[STIX]{x1D6E4}$ .

By nontrivial zeros of $Z_{M}(s)$ we mean all nonreal zeros and real zeros at points $s\in [0,1]$ such that $s(1-s)$ is equal to an eigenvalue of the Laplacian that is less than or equal to $1/4$ . The nontrivial zeros are related to the spectrum of the Laplacian in the sense that, according to [Reference Hejhal14, Theorem 5.3], the nontrivial zeros of $Z_{M}$ are located at points of the form $1/2\pm ir_{n}$ where $1/4+r_{n}^{2}$ is a discrete eigenvalue of the Laplacian and at points $1-\unicode[STIX]{x1D70C}$ in the half-plane $\text{Re}(s)<1/2$ which are poles of the determinant $\unicode[STIX]{x1D719}_{M}$ of the scattering matrix.

We may conclude that the function $Z_{M}H_{M}$ is “spectrally equivalent” to $Z_{M}$ in the following sense: The function $Z_{M}H_{M}$ can be represented as a general Dirichlet series converging in the half-plane $\text{Re}(s)>1$ and carrying information about the underlying group $\unicode[STIX]{x1D6E4}$ ; it possesses a meromorphic continuation to the entire complex plane satisfying the functional equation

(7) $$\begin{eqnarray}(Z_{M}H_{M})(s)=\unicode[STIX]{x1D702}_{M}(s)K_{M}^{-1}(s)Z_{M}(1-s);\end{eqnarray}$$

and its nontrivial zeros are at points $s=1/2\pm ir_{n}$ , where $1/4+r_{n}^{2}$ is a discrete eigenvalue of the Laplacian and at points $s=\unicode[STIX]{x1D70C}$ in the half-plane $\text{Re}(s)>1/2$ which are zeros of the determinant $\unicode[STIX]{x1D719}_{M}$ of the scattering matrix. Based on this argument, we may, loosely speaking, say that $Z_{M}H_{M}$ carries the same amount of spectral information as $Z_{M}$ does. Besides that, the function $Z_{M}H_{M}$ has no nontrivial zeros in the half-plane $\text{Re}(s)<1/2$ .

The question of studying the zeros of $Z_{M}^{\prime }$ when $M$ is not compact by applying methods presented in this paper begins with one possible technical difficulty stemming from the fact that the function $Z_{M}$ has an infinite number of zeros in the half-plane $\text{Re}(s)<1/2$ , each one of which would produce a negative weight in the weighted counting function $N_{\text{w}}$ . On the other hand, the function $Z_{M}H_{M}$ has no nontrivial zeros in the half-plane $\text{ Re}(s)<1/2$ and carries the same spectral information as $Z_{M}$ does. Therefore, as a result we shall study the zeros of $(Z_{M}H_{M})^{\prime }$ .

As we shall see below, the choice of $Z_{M}H_{M}$ instead of $Z_{M}$ is further justified by the fact that, according to the statement (a) of the Main Theorem, the only zeros of $(Z_{M}H_{M})^{\prime }$ on the critical line (with imaginary part greater than some constant depending upon the group) are the multiple zeros of $Z_{M}$ ; therefore, the study of zeros of $(Z_{M}H_{M})^{\prime }$ is related to the problem of obtaining bounds for the dimension of eigenspaces of discrete eigenvalues of the Laplacian on $M$ .

In conclusion, we followed the guide provided by the technical issues we faced and chose to study the zeros of $(Z_{M}H_{M})^{\prime }$ . To be specific, we viewed the positivity issues described above as important, thus we focused our attention on the zeros of $(Z_{M}H_{M})^{\prime }$ . Nonetheless, the problem of studying the zeros of $Z_{M}^{\prime }$ is both well-posed and remains open. It is quite possible that a successful study of the zeros of $Z_{M}^{\prime }$ when combined with the results of the present paper would yield interesting results. We leave such a study to a motivated reader.

1.3 The main result

The function $H_{M}^{\prime }/H_{M}$ admits the general Dirichlet series expansion

(8) $$\begin{eqnarray}\frac{H_{M}^{\prime }}{H_{M}}(s)=\mathop{\sum }_{i=1}^{\infty }\frac{b(q_{i})}{q_{i}^{s}},\end{eqnarray}$$

where the series on the right converges absolutely and uniformly for $\text{Re}(s)\geqslant \unicode[STIX]{x1D70E}_{0}+\unicode[STIX]{x1D716}>\unicode[STIX]{x1D70E}_{0}\gg 0$ , and where $\{q_{i}\}$ is a nondecreasing sequence of positive real numbers consisting of all finite products of numbers $r_{n}^{2}>1$ . Obviously, $q_{2}>q_{1}=\inf q_{i}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . Furthermore,

$$\begin{eqnarray}b(q_{1})=-a(2)\log q_{1}=-2(d(2)/d(1))\log (\mathfrak{g}_{2}/\mathfrak{g}_{1}).\end{eqnarray}$$

Let $\ell _{M,0}$ be the length of a shortest closed geodesic, or systole, on $M$ . With our notation from above, let

(9) $$\begin{eqnarray}A_{M}=\min \big\{e^{\ell _{M,0}},\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}\big\}.\end{eqnarray}$$

Here, we have dropped the subscript $M$ on $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ in order to ease the notation; however, it is clear that $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ depends on $M$ . Let $m_{M,0}$ denote the number of inconjugate closed geodesics on $M$ with length $\ell _{M,0}$ . If $e^{\ell _{M,0}}\neq (\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , let

(10) $$\begin{eqnarray}a_{M}=\left\{\begin{array}{@{}cc@{}}\displaystyle \frac{m_{M,0}\ell _{M,0}}{1-e^{-\ell _{M,0}}}; & \text{if }e^{\ell _{M,0}}<\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}\\ b(\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}); & \text{if }e^{\ell _{M,0}}>\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}\end{array}\right\}.\end{eqnarray}$$

If $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , let

(11) $$\begin{eqnarray}a_{M}=\frac{m_{M,0}\ell _{M,0}}{1-e^{-\ell _{M,0}}}+b(\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}).\end{eqnarray}$$

Observe that $a_{M}$ is the sum of the two terms which appear in the two cases in (10), not the arithmetic average as one would expect from elementary Fourier analysis.

With all this, the main result of this article is the following.

As stated in the Main Theorem, the above asymptotic formulas specialize in the case $M$ is compact to give the main results in [Reference Garunkŝtis11, Reference Garunkŝtis12, Reference Luo22, Reference Minamide24] and [Reference Minamide25]. More precisely, in [Reference Minamide24] and [Reference Minamide25] it is proved that $Z_{M}^{\prime }$ in the compact case possesses no nonreal zeros in the half-plane $\text{Re}(s)<1/2$ , a statement which we believe to hold true for $(Z_{M}H_{M})^{\prime }$ .

Similar results for the zeros of higher derivatives of $Z_{M}H_{M}$ are presented in a later section. In addition, corollaries of the main theorem, analogous to results from [Reference Levinson and Montgomery21], are derived.

Since the proof of the Main Theorem is rather technical, let us present here a summary of the ideas involved in its proof.

Part (a) of the Main Theorem is proved in two parts. First, we employ the functional equation for the Selberg zeta function together with a bound for the growth of the logarithmic derivative $D_{M}(s)=Z_{M}^{\prime }(s)/Z_{M}(s)$ in the right half of the critical strip in order to deduce that $\text{Re}(Z_{M}H_{M})^{\prime }(s)\neq 0$ for sufficiently large $\text{Im}(s)$ in the half-plane $\text{Re}(s)<1/2$ . However, this method has a critical line as its boundary. Therefore, in order to show that all (but eventually finitely many) multiple zeros of $Z_{M}$ on the critical line are also zeros of $(Z_{M}H_{M})^{\prime }$ , we employ the Hadamard product representation of the completed Selberg zeta function, which was proved in [Reference Fischer7], and conduct careful analysis of the imaginary part of the logarithmic derivative of $(Z_{M}H_{M})^{\prime }$ .

Parts (b) and (c) of the Main Theorem are proved by an application of Littlewood’s theorem [Reference Titchmarsh29, p. 132] to the function

(12) $$\begin{eqnarray}X_{M}(s):=\frac{A_{M}^{s}}{a_{M}}(Z_{M}H_{M})^{\prime }(s)\end{eqnarray}$$

followed by a careful technical analysis of the integrals obtained. There are two main difficulties appearing in the noncompact case. The first one is to control the growth of $D_{M}(s)$ inside the critical strip, which is resolved by an application of Theorem 5 below. What remains is the second technical point, which is to study the growth of $\arg X_{M}(\unicode[STIX]{x1D70E}+iT)$ , for large $T$ and $\unicode[STIX]{x1D70E}\in (a,\unicode[STIX]{x1D70E}_{0})$ , where $a\in (0,1/2)$ is an arbitrary constant. In order to address this problem, we prove a Phragmen–Lindelöf type bound for $(Z_{M}H_{M})(s)$ inside the strip $-\unicode[STIX]{x1D70E}_{2}\leqslant -1\leqslant \text{Re}(s)\leqslant \unicode[STIX]{x1D70E}_{0}$ and the Lindelöf type bound for $(Z_{M}H_{M})(s)$ for $\text{Re}(s)$ close to $1/2$ . These bounds are necessary in order to apply the generalized Backlund equivalent for the Lindelöf hypothesis (see § 2.5) which will yield a sharp bound for $(Z_{M}H_{M})^{\prime }(s)$ near the critical line. The resulting estimate enables one to apply Jensen’s theorem and deduce that $\arg X_{M}(\unicode[STIX]{x1D70E}+iT)=o(T)$ , as $T\rightarrow \infty$ .

1.4 Properties of the invariant $A_{M}$

Aspects of the spectral analysis of the Laplacian acting on smooth functions on a hyperbolic Riemann surface can be measured by studying the zeros of the Selberg zeta function. As discussed above, one equivalently can study the zeros of $Z_{M}H_{M}$ . Therefore, by slight extension, the zeros of $(Z_{M}H_{M})^{\prime }$ provide another measure of the spectral analysis of the Laplacian. In this regard, the quantity $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ is a new spectral invariant. In addition, our Main Theorem asserts that for any given surface, the spectral analysis depends on the comparison of $e^{\ell _{M,0}}$ and $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ .

In § 7, we consider various arithmetic groups and compare $e^{\ell _{M,0}}$ to $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . More precisely, we prove the following proposition.

With respect to statements (i) and (ii) of the above proposition we find it very interesting that, in the sense of our Main Theorem, not all arithmetic surfaces, even those with the same topological signature, have the same behavior.

Also in § 7, in order to prove statement (iii) of Proposition 1 we argue that if one considers a degenerating family of hyperbolic Riemann surfaces within the moduli space of surfaces of fixed topological type, one eventually has the inequality $e^{\ell _{M,0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ near the boundary. As a result, if one begins with congruence group and degenerates the corresponding surface, one will ultimately encounter a surface where $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . More generally, however, it seems as if moduli space can be separated into sets defined by the sign of $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ where most, but not all, arithmetic surfaces are in the component where $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}>0$ , and the Deligne–Mumford boundary lies in the component where $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}<0$ .

We could not explicitly construct a surface where $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}=0$ , even though we prove that such surfaces exist.

1.5 A comparison of counting functions

In [Reference Hejhal14, Theorem 2.22], D. Hejhal establishes the asymptotic behavior of the weighted vertical distribution of zeros of $\unicode[STIX]{x1D719}_{M}$ within the critical strip. In our notation, the zeros of $\unicode[STIX]{x1D719}_{M}$ within the critical strip coincide with the zeros of the Dirichlet series $H_{M}$ , so then [Reference Hejhal14, Theorem 2.22] establishes the asymptotic behavior of the weighted vertical counting function $N_{\text{w}}(T;H_{M})$ .

Let $M$ be any finite volume hyperbolic Riemann surface. We claim there exists a co-compact hyperbolic Riemann surface $\widetilde{M}$ such that $\text{vol}(M)=\text{vol}(\widetilde{M})$ , $\ell _{M,0}=\ell _{\widetilde{M},0}$ and $m_{M,0}=m_{\widetilde{M},0}$ , which we argue as follows. In the case when the number $n_{1}$ of cusps of the surface $M$ is even, we choose the surface $\widetilde{M}_{1}$ to be any co-compact surface with genus $g_{\widetilde{M}}=g_{M}+n_{1}/2$ and the same structure of elliptic points as $M$ , hence $\text{vol}(M)=\text{vol}(\widetilde{M}_{1})$ . If the number of cusps of the surface $M$ is odd, we choose the surface $\widetilde{M}_{1}$ to be any co-compact surface with genus $g_{\widetilde{M}}=g_{M}+(n_{1}-1)/2$ such that it has the same structure of elliptic points as $M$ , plus one additional elliptic point of order 2. By the Gauss–Bonnet formula, $\text{vol}(M)=\text{vol}(\widetilde{M}_{1})$ . We then deform the surface $\widetilde{M}_{1}$ in moduli space so that its shortest geodesic has the length equal to $\ell _{M,0}$ and the number of inconjugate geodesics of length $\ell _{M,0}$ is $m_{M,0}$ .

In § 9.2, we show that one can compare [Reference Hejhal14, Theorem 2.22] with generalization of the part (c) of the Main Theorem to higher derivatives in order to establish a simple asymptotic relation between $N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})$ , $N_{\text{w}}(T;Z_{\widetilde{M}}^{(k)})$ for $k\geqslant 1$ and $N_{\text{w}}(T;H_{M})$ . Namely, we prove the following theorem.

Theorem 2. Let $M$ be a finite volume hyperbolic Riemann surface such that, in the notation of (9), $A_{M}=\exp (\ell _{M,0})$ . Then, for all integers $k\geqslant 1$

(13) $$\begin{eqnarray}N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})=N_{\text{w}}(T;Z_{\widetilde{M}}^{(k)})+N_{\text{w}}(T;H_{M})+o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

We find it very interesting that, in the case when $M$ is such that $\exp (\ell _{M,0})<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , the coefficients of the first two terms, namely $T\log T$ and $T$ , in the asymptotic development of the counting function $N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})$ for all $k\geqslant 1$ coincide with known results, namely Hejhal’s theorem and (2).

1.6 Further comments

Weyl’s law in its classical form evaluates the lead asymptotic behavior of the vertical counting function $N_{\text{ver}}(T;Z_{M})$ for compact $M$ . As far as is known, the expansion in $T$ involves $\text{vol}(M)$ and no other information associated to the uniformizing group $\unicode[STIX]{x1D6E4}$ . If $M$ is noncompact, the generalization of Weyl’s law addresses the asymptotic behavior of

(14) $$\begin{eqnarray}\#\{\unicode[STIX]{x1D706}_{j,M}<1/4+T^{2}\}-\frac{1}{4\unicode[STIX]{x1D70B}}\int _{-T}^{T}\unicode[STIX]{x1D719}_{M}^{\prime }/\unicode[STIX]{x1D719}_{M}(1/2+ir)\,dr\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{j,M}$ is the eigenvalue of an $L^{2}$ eigenfunction of the Laplacian on $M$ . The asymptotic expansion of (14) is recalled below (see (68)) and, as in the compact case, all terms in the expansion involve elementary quantities associated to the uniformizing group $\unicode[STIX]{x1D6E4}$ .

In § 9.1, we express the function in (14) in terms of $N_{\text{ver}}(T;Z_{M}H_{M})$ , obtaining an expression which involves the constant $\mathfrak{g}_{1}$ . As a result, we accept the appearance of the term $\mathfrak{g}_{2}/\mathfrak{g}_{1}$ in our Main Theorem as being an appropriate generalization of a version of Weyl’s law.

In a different direction, if one considers a degenerating family of finite volume hyperbolic Riemann surfaces, then it was shown in [Reference Huntley, Jorgenson and Lundelius15] that the asymptotic behavior of the associated sequence of vertical counting functions $N_{\text{ver}}(T;Z_{M})$ has lead asymptotic behavior, for fixed $T$ , which involves the lengths of the pinching geodesics; see [Reference Huntley, Jorgenson and Lundelius15, Theorem 5.5]. As a result, we do not view the appearance of the invariant $\ell _{M,0}$ in (1) and (2) as a new feature when using Weyl’s laws to understand refined information associated to the uniformizing group $\unicode[STIX]{x1D6E4}$ .

However, we find the appearance of the constants $A_{M}$ and $a_{M}$ , as defined in (9), (10) and (11) to be intriguing, specially since values of $A_{M}$ and $a_{M}$ are different for certain arithmetic groups of the same signature. In particular, for any given surface $M$ , we do not know if there are conditions which determine the value taken by $A_{M}$ . Consequently, we conclude that the study of the vertical counting function $N_{\text{ver}}(T;(Z_{M}H_{M})^{\prime })$ contains a term which provides new information associated to $\unicode[STIX]{x1D6E4}$ which we do not see as being previously detected. In other words, if we are allowed to view the vertical counting function $N_{\text{ver}}(T;(Z_{M}H_{M})^{\prime })$ as another measure of the spectral analysis of the Laplacian on $M$ , then our Main Theorem shows the existence of refined information, namely $A_{M}$ with its conditional definition (9), about the uniformizing group $\unicode[STIX]{x1D6E4}$ .

1.7 Computations for the modular group

After the completion of this article, W. Luo brought to our attention the unpublished article [Reference Minamide23] from 2008 in which the author undertakes a related study in the case when $\unicode[STIX]{x1D6E4}=\text{ PSL}(2,\mathbb{Z})$ . There are a number of important differences between the results in the present paper and those in [Reference Minamide23], which we now discuss.

In [Reference Minamide23], as the title of the article states, the author studies the zeros of the derivative of the zeta function $Z_{M}(s)/\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s)$ where $M=\text{ PSL}(2,\mathbb{Z})\backslash \mathbb{H}$ . If we restrict our analysis to the case when $\unicode[STIX]{x1D6E4}=\text{ PSL}(2,\mathbb{Z})$ , then the function whose derivative we study is $Z_{M}(s)\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s-1)/\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s)$ . Since the article [Reference Minamide23] studies a different function than in the present article, one would expect that the statements of the main results are different, as, indeed, is the case. More importantly, however, the asymptotic expansions obtained in [Reference Minamide23] has an error term of $O(T)$ , whereas our error term is $o(T)$ , which is significant since the coefficient of the $T$ term contains the quantity $A_{M}$ , which we view as a new spectral invariant. Finally, we note that the article [Reference Minamide23] studies the single group $\unicode[STIX]{x1D6E4}=\text{ PSL}(2,\mathbb{Z})$ .

The approach presented in [Reference Minamide23] raise the question if a similar modification of the Selberg zeta function $Z_{M}$ can occur in the general setting of the present paper. To do so, one can write the function $H_{M}(s)$ as a ratio $P_{M}(s)/Q_{M}(s)$ of two entire functions of order at most two and then consider $Z_{M}(s)/Q_{M}(s)$ . It is true that the function $Z_{M}(s)/Q_{M}(s)$ has nontrivial zeros only at zeros of $Z_{M}$ stemming from the discrete eigenvalues of the Laplacian. However, the Phillips–Sarnak conjecture/philosophy then asserts that for generic $M$ , the quotient $Z_{M}(s)/Q_{M}(s)$ would have a finite number of nontrivial zeros, hence can be written as a polynomial times Gamma-type functions. Aside from this assertion, while focusing solely from the point of view of application of techniques developed in this paper, the function $Z_{M}(s)/Q_{M}(s)$ is suitable in the sense that it possesses no nonreal zeros in the half-plane $\text{Re}(s)<1/2$ . However, there are two reasons why the investigation of $Z_{M}(s)H_{M}(s)$ is more natural. Firstly, the spectral information carried by $P_{M}(s)$ , namely, the zeros of the scattering determinant, is lost when considering $Z_{M}(s)/Q_{M}(s)$ . More importantly, unless the explicit expression for the scattering determinant is known, it is very difficult to determine $Q_{M}(s)$ explicitly and hence express $Z_{M}(s)/Q_{M}(s)$ in terms of the information related to the underlying group $\unicode[STIX]{x1D6E4}$ , whereas $Z_{M}(s)H_{M}(s)$ has a general Dirichlet series representation in terms of the group information.

1.8 Outline of the paper

This article is organized as follows. In § 2, we establish notation and recall necessary results from the literature. The zero-free region for $(Z_{M}H_{M})^{\prime }$ , as stated in part (a) of the Main Theorem, will be proved in § 3. Various lemmas leading up to the proof of parts (b) and (c) of the main Theorem will be given in § 4, the proof of parts (b) and (c) will be completed in § 5, and in § 6 we state and prove several corollaries of the Main Theorem. The examples of congruence groups and “moonshine” groups will be given in § 7. In § 8, we prove results analogous to our Main Theorem for higher derivatives of $Z_{M}H_{M}$ . Finally, in § 9, we give various concluding remarks.