A note on the zeros of approximations of the Ramanujan $\Xi$-function

Abstract

In this paper we review the study of the distribution of the zeros of certain approximations for the Ramanujan $\Xi$-function given by Ki (Ramanujan J 17(1):123–143, 2008), and we provide new proofs of his results. Our approach is motivated by the ideas of Velásquez (J Anal Math 110:67–127, 2010) in the study of the zeros of certain sums of entire functions with some condition of stability related to the Hermite–Biehler theorem.

1 Introduction

1.1 Background

Let $\tau (n)$ be the Ramanujan’s tau-function, defined by

$$\begin{aligned} \Delta (z)=\displaystyle \sum _{n=1}^{\infty }\tau (n)q^{n}=z\prod _{n=1}^\infty (1-q^n)^{24}, \end{aligned}$$

where $q=e^{2\pi iz}$, and $\mathrm{Im}\,{z}>0$. It is well known that $\Delta (z)$ spans the space of cusp forms of dimension $-12$ associated with the unimodular group. The associated Dirichlet series and Euler product for $\Delta (z)$ are given by

$$\begin{aligned} L(s)=\displaystyle \sum _{n=1}^{\infty }\frac{\tau (n)}{n^{s}}=\prod _{p}\big (1-\tau (p)p^{-s}+p^{11-2s}\big )^{-1}, \end{aligned}$$

where the series and the product are absolutely convergent for $\mathrm{Re}\,{s}>13/2$.

Let us define the Ramanujan $\Xi$-function, denoted by $\Xi _R(s)$, as follows:

$$\begin{aligned} \Xi _R(s)=(2\pi )^{is-6}L(-is+6)\Gamma (-is+6), \end{aligned}$$

where $\Gamma (s)$ is the Gamma function. Another representation for $\Xi _R(s)$ is given by

$$\begin{aligned} \Xi _R(s)=\displaystyle \int _{-\infty }^{\infty }\phi (t)e^{ist}\,\mathrm{d}t, \end{aligned}$$

where

$$\begin{aligned} \phi (t)=e^{-2\pi \cosh (t)}\displaystyle \prod _{k=1}^{\infty }\big [\big (1-e^{-2\pi ke^{t}}\big )\big (1-e^{-2\pi ke^{-t}}\big )\big ]^{12}. \end{aligned}$$

(1.1)

In [1], Hardy highlighted the importance of the location of the zeros of $\Xi _R(s)$ in the strip $|\mathrm{Im}\,(s)|\le \tfrac{1}{2}$. The Riemann hypothesis for the Ramanujan zeta function states that all zeros of $\Xi _R(s)$ are real.

1.2 Zeros of the approximations $\Xi _F(s)$

The purpose of this paper is to study the distribution of the zeros of certain approximations for the Ramanujan $\Xi$-function. Inspired in the representation (1.1), Ki [2] defined these approximations as follows: Let F be a finite sequence of complex numbers $a_0, a_1, \ldots , a_n$ such that at least one of them is different from zero. We define the function

$$\begin{aligned} \Xi _F(s)= \displaystyle \int _{-\infty }^{\infty }\phi _{F}(t)e^{ist}\, \mathrm{d}t, \end{aligned}$$

where

$$\begin{aligned} \phi _F(t)=e^{-2\pi \cosh {t}}\Bigg (\displaystyle \sum _{m=0}^{n}a_me^{-2\pi me^{t}}\Bigg )\Bigg (\displaystyle \sum _{m=0}^{n}\overline{a_m}e^{-2\pi me^{-t}}\Bigg ). \end{aligned}$$

We recall that $\overline{\Xi _F(\overline{s})}=\Xi _F(s)$, and one can see that for some sequences $F_k$, the function $\Xi _{F_k}(s)$ converges uniformly to $\Xi _R(s)$ on all compact subsets of $\mathbb {C}$.

Throughout this paper, we will study the distribution of the zeros of the function $C_F(s):=\Xi _F(-is)$. Note that the zeros of $C_F(s)$ are symmetric respect to the line $\mathrm{Re}\,{s}=0$. Using the argument principle, Ki [2, Theorem 1] established for $T\ge 2$ that¹

$$\begin{aligned} N(T,C_F)=\frac{T}{\pi }\log \frac{T}{e\pi }+O(\log {T}), \end{aligned}$$

where $N(T,C_F)$ stands for the number of zeros of $C_F(s)$ such that $1\le \mathrm{Im}\,{s} < T$, counting multiplicity. In the lower half-plane a similar result holds. Moreover, using the method developed by Levinson [3], he stated that

$$\begin{aligned} \overline{N}(T, C_F)-\overline{N_1}(T,C_F)=O(T), \end{aligned}$$

(1.2)

where $\overline{N}(T, C_F)$ stands for the number of zeros of $C_F(s)$ such that $|\mathrm{Im}\,{s}| < T$, counting multiplicity and $\overline{N_1}(T,C_F)$ denotes the number of simple zeros such that $|\mathrm{Im}\,{s}|<T$ and $\mathrm{Re}\,{s}=0$. In a sense, it means that almost all zeros of $C_F(s)$ lie on the line $\mathrm{Re}\,{s}=0$ and are simple. Our first goal is to establish a refinement of (1.2).

Theorem 1

For $T \ge 2$ we have

$$\begin{aligned} 0\le \overline{N}(T, C_F)-\overline{N_1}(T,C_F)\le \bigg (32n+\dfrac{32\ln (2n+1)}{\pi }\bigg )T +O(1). \end{aligned}$$

On the other hand, Ki [2, Theorem 2] found a result about the vertical distribution of the zeros of $C_F(s)$, based on the zeros of the function $\psi _F(s)$, defined by

$$\begin{aligned} \psi _F(s)=\pi ^{-s}\displaystyle \sum _{m=0}^{n}a_m(2m+1)^{-s}. \end{aligned}$$

(1.3)

Let $k\ge 0$ be an integer such that $P(1)=P'(1)=\cdot \cdot \cdot =P^{(k-1)}(1)=0$ and $P^{(k)}(1)\ne 0$, where $P(y)=\sum _{m=0}^n{a_m}y^{m}$.

Theorem 2

Let $\Delta _{*}<\Delta _{**}$ be positive real numbers. Suppose that ${\psi _F}(s-k)$ has finitely many zeros in $-\Delta _{**}<\mathrm{Re}\,{s}<\Delta _{*}$. Let $\delta$ be such that $0<\delta <\Delta _{*}$. Then all but finitely many zeros of ${C_F}(s)$ which lie in $|\mathrm{Re}\,{s}|\le \delta$ are on the line $\mathrm{Re}\,{s}=0$. In particular, all but finitely many zeros of ${C_F}(s)$ are on the line $\mathrm{Re}\,{s}=0$, if ${\psi _F}(s-k)$ has finitely many zeros in $\mathrm{Re}\,{s}>-\Delta _{**}$.

Ki included a second proof for the second part of Theorem 2. In particular, this second proof gave information about the simplicity of the zeros of $C_F(s)$. Anyway, Ki conjectured that second case for $\psi _F(s-k)$ is not possible. On the other hand, using (2.7) it is clear that ${\psi _F}(s-k)$ has the same set of zeros of a Dirichlet polynomial in the framework of [4, Sect. 12.5]. The set of zeros of a Dirichlet polynomial is quasi-periodic (see [5, Appendix 6, p. 449]). Then, if $s_0=\sigma _0+i\tau _0$ is a zero of the Dirichlet polynomial, for any $\varepsilon >0$ we can construct a sequence $\{s_n=\sigma _n+i\tau _n\}_{n\in \mathbb {N}}$ of zeros, such that $\sigma _n\in ]\sigma _0-\varepsilon , \sigma _0+\varepsilon [$ for all $n\in \mathbb {N}$ and $\tau _n\rightarrow \pm \infty$. This implies that each open vertical strip has no zeros or has infinite zeros. Therefore, the hypothesis in Theorem 2 is reduced to ${\psi _F}(s-k)$ having no zeros in $-\Delta _{**}<\mathrm{Re}\,{s}<\Delta _{*}$. Our second goal in this paper is to give a new proof of this result.

Theorem 3

Let $\Delta _{*}<\Delta _{**}$ be positive real numbers. Suppose that ${\psi _F}(s-k)$ has no zeros in $-\Delta _{**}<\mathrm{Re}\,{s}<\Delta _{*}$. Let $\delta$ be such that $0<\delta <\Delta _{*}$. Then all but finitely many zeros of ${C_F}(s)$ which lie in $|\mathrm{Re}\,{s}|\le \delta$ are on the line $\mathrm{Re}\,{s}=0$ and are simple.

We highlight that our proof includes information about the simplicity of the zeros for the first case. The key relation between the functions $C_F(s)$ and $\psi _F(s-k)$ is due to de Bruijn [6, p. 225], who showed that

$$\begin{aligned} C_F(s)= \displaystyle \sum _{m=k}^{\infty }b_m \psi _F(s-m)\Gamma (s-m) + \displaystyle \sum _{m=k}^{\infty }\overline{b_m\psi _F(-s-m)}\Gamma (-s-m), \end{aligned}$$

where $b_m$ are complex numbers and $b_k\ne 0$.

1.3 Strategy outline

Our approach is motivated by a result of Velásquez [7, Theorem 36], about the distribution of the zeros of a function of the form $f(s)= h(s)+h^{*}(2a-s)$, where h(s) is a meromorphic function², and $a\in \mathbb {R}$. This result can be regarded as a generalization of the necessary condition of stability for the function h(s), in the Hermite–Biehler theorem [5, 21, Part III, Lecture 27]. In our case, using an auxiliary function $W_F(s)$, we have the representation $C_F(s)=h(s)+h^*(-s)$, where $h(s)=W_F(-is-i/2)$. Some estimates of h(s) due to Ki [2, Theorem 2.1] play an important role to establish the necessary growth conditions in [7, Theorem 36]. On the other hand, the strong relation between the zeros of h(s) and $\psi _F$(s) (see (2.5)) implies that one study the distribution of zeros of $\psi _F$(s), as a set of zeros of a Dirichlet polynomial.

Throughout the paper, we fix a sequence F. For a function f(s) and the parameters $\sigma _1<\sigma _2$, and $T_1<T_2$, we denote the counting function

$$\begin{aligned}&N(\sigma _0,\sigma _1,T_1,T_2,f)=\#\{s\in \mathbb {C}: f(s)=0,\, \sigma _0<\sigma<\sigma _1, T_1<\tau<T_2\},\\&\widehat{N}(\sigma _0,\sigma _1,T_1,T_2,f)=\#\{s\in \mathbb {C}: f(s)=0,\, \sigma _0\le \sigma \le \sigma _1, T_1<\tau <T_2\}, \end{aligned}$$

where, in both cases, the counts are with multiplicity, and

$$\begin{aligned} N_{0}^{'}(T,g)=\#\{s\in \mathbb {C}: g(s)=0, \mathrm{Re}\,{s}=0, |\mathrm{Im}\,{s}|<T\}, \end{aligned}$$

where the count is without multiplicity.

2 Preliminary results

In this section we collect several results for our proof. We highlight that in [2, Proposition 2.3], Ki showed that there is a constant $\beta _0>0$ such that $C_F(s)\ne 0$, for $|\mathrm{Re}\,{s}|\ge \beta _0$. This implies that for $\beta \ge \beta _0$,

$$\begin{aligned} \overline{N}(T,C_F)=N(-\beta , \beta , -T ,T , C_F). \end{aligned}$$

(2.1)

Therefore, we can restrict our analysis of the zeros in vertical strips. Now, let us start to find a new representation for $C_F(s)$. We define the entire function

$$\begin{aligned} {W_F}(s)= \displaystyle \int _{-\infty }^{\infty }{{\tilde{\phi }}} _{F}(t)e^{ist}\, \mathrm{d}t, \end{aligned}$$

where

$$\begin{aligned} {\tilde{\phi }}_{F}(t)=\frac{e^{-2\pi \cosh {t}}}{\displaystyle {e^{t/2}}+\displaystyle {e^{-t/2}}}\Bigg (\displaystyle \sum _{m=0}^{n}a_me^{-2\pi me^{t}}\Bigg )\Bigg (\displaystyle \sum _{m=0}^{n}\overline{a_m}e^{-2\pi me^{-t}}\Bigg ). \end{aligned}$$

Then, we obtain the following relation:

$$\begin{aligned} C_F(s)=W_F\bigg (-is-\frac{i}{2}\bigg )+W_F\bigg (-is+\frac{i}{2}\bigg ). \end{aligned}$$

(2.2)

If we denote by

$$\begin{aligned} h(s)=W_F\displaystyle \bigg (-is-\frac{i}{2}\bigg ), \end{aligned}$$

(2.3)

we rewrite (2.2) as

$$\begin{aligned} C_F(s)=h(s)+h^*(-s). \end{aligned}$$

This representation allows us to use the following result (see [7, Theorem 36]).

Theorem 4

Let $\sigma _0>0$ be a parameter and h(s) be an entire function such that $h(s)\ne 0$ for $\mathrm{Re}\,{s}=\sigma _0$. We define the entire function

$$\begin{aligned}f(s)=h(s)+h^*(-s).\end{aligned}$$

Suppose that the function

$$\begin{aligned} F(s)=\frac{h^*(-s)}{h(s)} \end{aligned}$$

satisfies the following conditions.

(i):

$F(s)\ne \pm 1$ on the line $\mathrm{Re}\,{s}=\sigma _0$, and for some $\tau _0>0$ we have $|F(s)|<1$ for $s=\sigma _0+i\tau$ with $|\tau |\ge \tau _0$.

(ii):

There exist an increasing function $\varphi :\mathbb {R}\rightarrow \mathbb {R}$, a constant $K>0$ and sequences $\{T_m\}_{m\in \mathbb {N}}$, $\{T_m^{*}\}_{m\in \mathbb {N}}$ such that $\displaystyle \lim _{m\rightarrow \infty }T_m=\displaystyle \lim _{m\rightarrow \infty }T_m^{*}=\infty$,

$$\begin{aligned} T_m\le T_{m+1} \le \varphi (T_m), \quad T_m^{*}\le T_{m+1}^{*} \le \varphi (T_m^{*}) \quad \text {for m} \in \mathbb {N}, \end{aligned}$$

and $|F(s)|<e^{K|s|}$, for $s=\sigma +i\tau$ with $0\le \sigma \le \sigma _0$ and $\tau =T_m$, $\tau =-T_m^{*}$, for $m\in \mathbb {N}$.

Then, for $T\ge 2$, we have that

$$\begin{aligned} N(-\sigma _0, \sigma _0, -T, T, f)-N_{0}^{'}(T, f)\le 4 \widehat{N}(0, \sigma _0, -\varphi (2T), \varphi (2T), h) +O(1), \end{aligned}$$

(2.4)

To prove that the function h(s) defined in (2.3) satisfies the conditions of the previous theorem, we will use the estimates used by Ki. By [2, Eq. (2.1)], using the change of variable $s\mapsto -is-i/2$, we have that

$$\begin{aligned} h(s)=\Gamma (s-k)\Big (b_k\psi _{F,k}(s)+O\big (|s|^{-1/2}\big )\Big ) \end{aligned}$$

(2.5)

holds uniformly on the half-plane $\mathrm{Re}\,{s}\ge -1/4$ and |s| sufficiently large. On the other hand, by [2, Theorem 2.1] it follows using the change of variable $s\mapsto -is+i/2$: for $\Delta >0$ sufficiently large,

$$\begin{aligned} \dfrac{h^*(-s)}{\Gamma (s-k-1)|\tau |^{\mu (\sigma )}}=O(1), \end{aligned}$$

(2.6)

for $s=\sigma +i\tau$ with $0\le \sigma \le \Delta$ and $|\tau |\ge 1$, and the function $\mu (\sigma )$ is given by

$$\begin{aligned} \mu (\sigma ) = \left\{ \begin{array}{ll} 1-\sigma , &{} \mathrm {si } \quad 0\le \sigma \le 1, \\ 0, &{} \mathrm {si }\quad \sigma >1. \end{array} \right. \end{aligned}$$

Finally, we will need to establish bounds for the right-hand side of (2.4) to estimate the number of zeros of h(s). The relation (2.5) tells us that we must study the behavior of the zeros of $\psi _F(s)$. We define $\psi _{F,k}(s):=\psi _F(s-k)$. Thus, using (1.3) this function can be written as

$$\begin{aligned} \psi _{F,k}(s) = \displaystyle \sum _{m=0}^{n}a_me^{-\ln {((2m+1)\pi )(s-k)}}=e^{-\ln ((2n+1)\pi )(s-k)}\Bigg [\displaystyle \sum _{m=0}^{n}p_me^{\beta _ms}\Bigg ], \end{aligned}$$

(2.7)

where $p_m=(a_{n-m})e^{-\beta _mk}$ and $\beta _m=\ln ((2n+1)/(2(n-m)+1))$, for $0\le m\le n$. The sum on the right-hand side of (2.7) is a Dirichlet polynomial in the framework [4, Sect. 12.5].

Proposition 5

Let $Z(\psi _{F,k})$ denote the set of zeros of $\psi _{F,k}(s)$.

• (1) There is a positive real number $c_0$ such that $Z(\psi _{F,k})\subset \{s\in \mathbb {C}: |\mathrm{Re}\,{s}|<c_0\}$.

• (2) For $T_1<T_2$ and $c\ge c_0$, we have that

  $$\begin{aligned} N(-c, c, T_1, T_2, \psi _{F,k})\le n+\dfrac{\ln (2n+1)}{2\pi }(T_2-T_1). \end{aligned}$$

• (3) Let $K\subset \mathbb {C}$ such that $|\mathrm{Re}\,{s}|\le M$ for $s\in K$, and some $M>0$. Suppose that K is uniformly bounded from the zeros of $\psi _{F,k}(s)$, i.e.,

  $$\begin{aligned}\inf \{|s-z|:s\in \textit{K}, z\in Z(\psi _{F,k})\}>0.\end{aligned}$$

  Then,  $\inf \{|\psi _{F,k}(s)|: s\in \textit{K}\}>0$.

Proof

See [4, Theorems 12.4, 12.5 and 12.6]. $\square$

3 Proofs of Theorems 1 and 3

3.1 Proof of Theorem 1

Let us define the function

$$\begin{aligned} F(s)= \frac{h^*(-s)}{h(s)}. \end{aligned}$$

(3.1)

Since that h(s) and $h^*(-s)$ are entire functions, we can choose $\sigma _0>0$ sufficiently large such that $F(s)\ne \pm 1$ and $h(s)\ne 0$ on the line $\mathrm{Re}\,{s}=\sigma _0$. Using (2.5) and (2.6) we get for $s=\sigma +i\tau$ with $0\le \sigma \le \sigma _0$ and $|\tau |$ sufficiently large,

$$\begin{aligned}&F(s) \nonumber \\&\quad =\frac{O(1)\Gamma (s-k-1)|\tau |^{\mu (\sigma )}}{\Gamma (s-k)\big (b_k\psi _{F,k}(s)+O\big (|s|^{-1/2}\big )\big )} \nonumber \\&\quad =\frac{O(1)|\tau |^{\mu (\sigma )}}{(s-k-1)\big (b_k\psi _{F,k}(s)+O\big (|s|^{-1/2}\big )\big )}. \end{aligned}$$

(3.2)

Now, we analyze the behavior of F(s) on the line $\mathrm{Re}\,{s}=\sigma _0$. Note that $\mu (\sigma _0)=0$. On another hand, the line $\mathrm{Re}\,{s}=\sigma _0$ is uniformly bounded from the zeros of $\psi _{F,k}(s)$. Then, recalling that $b_k\ne 0$, by Proposition 5 and the triangle inequality we get

$$\begin{aligned} \big |b_k\psi _{F,k}(s)+O\big (|s|^{-1/2}\big )\big |\gg 1, \end{aligned}$$

(3.3)

for $s=\sigma _0+i\tau$, with $|\tau |$ sufficiently large. Inserting this in (3.2), it follows

$$\begin{aligned} |F(s)|\ll \frac{1}{|s-k-1|}. \end{aligned}$$

Therefore, for $s=\sigma _0+i\tau$ with $|\tau |$ sufficiently large we conclude that $|F(s)|<1$. This implies (i) of Theorem 4. Let us prove (ii) of Theorem 4. For each $m\in \mathbb {Z}$ we consider the rectangle

$$\begin{aligned} R_{m}=\{s\in \mathbb {C}:-\sigma _0< \mathrm{Re}\,{s}< \sigma _0,~~ m< \mathrm{Im}\,{s}< m+1\}. \end{aligned}$$

We divide this rectangle into $2n+1$ subrectangles $R_{m,j}$ defined by

$$\begin{aligned} R_{m,j}=\Big \{s\in \mathbb {C}:-\sigma _0< \mathrm{Re}\,{s}< \sigma _0,~~ m+\dfrac{j-1}{2n+1}< \mathrm{Im}\,{s}< m+\dfrac{j}{2n+1}\Big \}, \end{aligned}$$

for $j\in \{1,2, \ldots ,2n+1\}$. By Proposition 5 we have that $N(-\sigma _0, \sigma _0, m, m+1, \psi _{F,k})\le 2n$. So, there exists $j_0$ such that $\psi _{F,k}(s)$ does not vanish in $R_{m,j_0}$. Let us write

$$\begin{aligned} T_m=m+\dfrac{j_0-\frac{1}{2}}{2n+1}. \end{aligned}$$

Note that $m<T_m<m+1$. Then, if we define $\varphi (x)=x+2$, we have that

$$\begin{aligned} m<T_m<m+1<T_{m+1}<m+2<T_m+2=\varphi (T_m). \end{aligned}$$

Let $\textit{K}=\{s\in \mathbb {C}: -\sigma _0<\mathrm{Re}\,{s}<\sigma _0, ~~\mathrm{Im}\,{s}=T_m, m\in \mathbb {Z}\}$. For any $s\in \textit{K}$, we have that $|s-z|\ge 1/{2(2n+1)}$, for all $z\in Z(\psi _{F,k})$. Then $\textit{K}$ is uniformly bounded from the zeros of $\psi _{F,k}(s)$. Using Proposition 5 we see that (3.3) holds for $s\in K$ with |m| sufficiently large. Therefore, in (3.2) we obtain that for $s=\sigma +i\tau$ with $0\le \sigma \le \sigma _0$ and $\tau =T_m$ (|m| sufficiently large) it follows

$$\begin{aligned} F(s) \ll \frac{|\tau |^{\mu (\sigma )}}{|s-k-1|}. \end{aligned}$$

Using the fact that $\mu (\sigma )\le 1$, we conclude that

$$\begin{aligned} |F(s)|\ll 1<e^{|s|}. \end{aligned}$$

Now, we choose $T_m^{*}=-T_{-m}$, for all $m\in \mathbb {N}$. Thus, we obtain (ii) of Theorem 4. Therefore

$$\begin{aligned} N(-\sigma _0, \sigma _0, -T, T, C_F)-N_{0}^{'}(T, C_F)\le 4 \widehat{N}(0, \sigma _0, -\varphi (2T), \varphi (2T), h) +O(1). \end{aligned}$$

(3.4)

To conclude we need to bound $\widehat{N}(0, \sigma _0, -\varphi (2T), \varphi (2T), h)$. Firstly, we choose $0<\varepsilon <1/4$ such that h(s) and $\psi _{F,k}(s)$ do not vanish on $\mathrm{Re}\,{s}=-\varepsilon _0$. The definition of $T_m$ implies that

$$\begin{aligned} \dfrac{1}{2n+1}\le T_{m+1}-T_m\le 2, \end{aligned}$$

(3.5)

and using Proposition 5 we obtain $N(-\varepsilon ,\sigma _0,T_m,T_{m+1},\psi _{F,k})\le 2n$. Let us divide the rectangle $\{s\in \mathbb {C}: -\varepsilon< \mathrm{Re}\,{s}< 0 ~~\text{ and } ~~T_m< \mathrm{Im}\,{s} <T_{m+1}\}$ into $2n+1$ vertical subrectangles with horizontal length $\varepsilon /(2n+1)$. So, one of these rectangles, denoted by $I_m$, has no zeros of $\psi _{F,k}(s)$ and h(s). Suppose that the right vertical side of $I_m$ is contained on the line $\mathrm{Re}\,{s}=-\varepsilon _{m}$ that we can suppose without loss of generality that does not contain a zero of $\psi _{F,k}(s)$. Now, if we place a circle of radius $\delta >0$ sufficiently small (for instance $\delta <1/(2n+1)(16n)$) we can enclosed the zeros of the rectangle $J_m=\{s\in \mathbb {C}: -\varepsilon _m< \mathrm{Re}\,{s}< \sigma _0 ~~\text{ and } ~~ T_m< \mathrm{Im}\,{s}< T_{m+1}\}$ in a contour $C_m$ such that the distance between $C_m$ and $J_m$ is at least $1/(2n+1)(16n)$ and $C_m$ is distanced at least $1/(2n+1)(32n)$ from the zeros of $\psi _{F,k}(s)$. We set $K=\bigcup _{m} \partial J_m (\partial J_m=\text{ the } \text{ boundaries } \text{ of } J_m).$ Then, applying (2.5) and Proposition 5, we can find $M>0$ such that

$$\begin{aligned} \bigg |b_k\psi _{F,k}(s)-\dfrac{h(s)}{\Gamma (s-k)}\bigg |<M<|b_k\psi _{F,k}(s)| \end{aligned}$$

for $s\in K$, with |m| sufficiently large. If we denote $w(s)=h(s)/\Gamma (s-k)$, applying Rouché’s theorem we obtain that there is $m_0\in \mathbb {N}$ sufficiently large such that

$$\begin{aligned} N(-\varepsilon _m, \sigma _0,T_m,T_{m+1},w)=N(-\varepsilon _m,\sigma _0,T_m,T_{m+1},\psi _{F,k}), \end{aligned}$$

(3.6)

and

$$\begin{aligned} N(-\varepsilon _{-m-1}, \sigma _0, T_{-m-1},T_{-m},w)=N(-\varepsilon _{-m-1},\sigma _0,T_{-m-1},T_{-m},\psi _{F,k}) \end{aligned}$$

(3.7)

for $m\ge m_0$.

Finally, let T be a positive real parameter. If $T<T_{m_{0}}$ we obtain $N(0,\sigma _0,0,T,h)=O(1)$. If $T\ge T_{m_0}$, we choose $m_1\ge m_0\ge 1$ such that $m_1<T_{m_1}\le T<T_{m_1+1}<m_1+2$. Since that the zeros of $1/\Gamma (s)$ are the non-positive integers, by (3.6) , Proposition 5 and (3.5), we get

$$\begin{aligned} \widehat{N}(0,\sigma _0,0,T,h)&\le \displaystyle \sum _{j=m_0}^{m_1}N(-\varepsilon _j,\sigma _0,T_{j},T_{j+1},h) + \widehat{N}(0,\sigma _0,0,T_{m_0}+1,h) \\&= \displaystyle \sum _{j=m_0}^{m_1}N(-\varepsilon _j,\sigma _0,T_{j},T_{j+1},w) +O(1)\\&= \displaystyle \sum _{j=m_0}^{m_1}N(-\varepsilon _0,\sigma _0,T_{j},T_{j+1},\psi _{F,k})+O(1) \\&\le \displaystyle \sum _{j=m_0}^{m_1}\bigg (n+\dfrac{\ln (2n+1)}{2\pi }(T_{j+1}-T_{j})\bigg )+O(1) \\&\le \bigg (n+\dfrac{\ln (2n+1)}{\pi }\bigg )T +O(1). \end{aligned}$$

Similarly, for $T<0$ we use (3.7) to obtain a similar bound. Thus, we obtain for $T>0$ that

$$\begin{aligned} \widehat{N}(0,\sigma _0,-T,T,h) \le \bigg (2n+\dfrac{2\ln (2n+1)}{\pi }\bigg )T +O(1). \end{aligned}$$

We replace T by $\varphi (2T)$ in the above expression, and inserting in (3.4), and one can see that

$$\begin{aligned} N(-\sigma _0, \sigma _0, -T, T, C_F)-N_{0}^{'}(T, C_F)\le \bigg (16n+\dfrac{16\ln (2n+1)}{\pi }\bigg )T +O(1). \end{aligned}$$

(3.8)

To obtain our desired result we will use an argument of Ki in [2, p. 131]. Following his idea, for $T>0$ we get that

$$\begin{aligned}&N(-\sigma _0, \sigma _0, -T, T, C_F) - \overline{N_{1}}(T, C_F)\nonumber \\&\quad \le 2\Bigg (N(-\sigma _0, \sigma _0, 0, T, C_F) - \displaystyle \sum _{k=1}^\infty \overline{N_{k}}(T, C_F)\Bigg ), \end{aligned}$$

(3.9)

where $\overline{N_k}(T,C_F)$ denotes the number of zeros of $C_F$ with multiplicity k with $|\mathrm{Im}\,{s}|< T$ and $\mathrm{Re}\,{s}=0$, counting with multiplicity. Note that

$$\begin{aligned} N_{0}^{'}(T, C_F) \le \displaystyle \sum _{k=1}^\infty \overline{N_{k}}(T, C_F). \end{aligned}$$

(3.10)

We conclude combining (3.8), (3.9), (3.10), and recalling by (2.1) that $\overline{N}(T,C_F)=N(-\sigma _0, \sigma _0, -T ,T , C_F)$.

3.2 Proof of Theorem 3

The proof is similar to the previous case. Using the function defined in (3.1), without loss of generality we can choose $\delta >0$ in such a way that $F(s)\ne \pm 1$, $h(s)\ne 0$ and $C_F(s)\ne 0$ when $\sigma =\delta$. By (2.5) and (2.6) it follows for $s=\sigma +i\tau$ with $0\le \sigma \le \delta$ and $|\tau |$ sufficiently large

$$\begin{aligned} F(s) =\frac{O(1)|\tau |^{\mu (\sigma )}}{(s-k-1)\big (b_k\psi _{F,k}(s)+O\big (|s|^{-1/2}\big )\big )}. \end{aligned}$$

Using the fact that the $\psi _{F,k}(s)$ has no zeros in the strip $-\Delta _{**}<\mathrm{Re}\,{s}<\Delta _{*}$, by Proposition 5 we get

$$\begin{aligned} \Big |b_k\psi (s-k)+O\big (|s|^{-1/2}\big )\Big |\gg 1, \end{aligned}$$

(3.11)

for $s=\sigma +i\tau$, with $0\le \sigma \le \delta$ and $|\tau |$ sufficiently large. Therefore

$$\begin{aligned} |F(s)|\ll \frac{|\tau |^{\mu (\sigma )}}{|s-k-1|}. \end{aligned}$$

(3.12)

Using the fact that $\mu (\delta )<1$, then

$$\begin{aligned} |F(s)|\ll \frac{|\tau |^{\mu (\delta )}}{|s-k-1|}\ll \frac{1}{|\tau |^{1-\mu (\delta )}}<1, \end{aligned}$$

for $s=\delta +i\tau$, with $|\tau |$ sufficiently large. Further, we have that $\mu (\sigma )\le 1$, which implies in (3.12) that

$$\begin{aligned} |F(s)|\ll \frac{|\tau |^{\mu (\sigma )}}{|s-k-1|}\ll 1<e^{|s|}, \end{aligned}$$

for $s=\sigma +i\tau$ with $0\le \sigma \le \delta$ and $|\tau |$ sufficiently large. Choosing $\varphi (x)=x+2$ and $T_m=T^*_{m}=m$, for m sufficiently large, we get that the hypotheses in Theorem 4 are satisfied. Then

$$\begin{aligned} N(-\delta , \delta , -T, T, C_F)-N_{0}^{'}(T, C_F)\le 4 \widehat{N}(0, \delta , -\varphi (2T), \varphi (2T), h) +O(1). \end{aligned}$$

(3.13)

Combining (2.5) and (3.11), we get a constant $L>0$ such that $|h(s)|\ge L|\Gamma (s-k)|$ for $s=\sigma +i\tau$ with $0\le \sigma \le \delta$ and $|\tau |$ sufficiently large. Then, h(s) only has finitely many zeros on the strip $0\le \sigma \le \delta$, because all possible zeros are contained in a compact set. Therefore, the right-hand side in (3.13) is bounded and this implies our desired result.