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

In this paper, we review the study of the distribution of the zeros of certain approximations for the Ramanujan $\Xi-$function given by Haseo Ki, and we provide a new proof of his results. Our approach is motivated by the ideas of Vel\'asquez 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 τ (n) be the Ramanujan's tau-function, defined by where q = e 2πiz , and Im z > 0. It is well known that ∆(z) spans the space of cusp forms of dimension −12 associated with the unimodular group. The associated Dirichlet series and Euler product for ∆(z) is given by where the series and the product are absolutely convergent for Re s > 13/2. Let us define the Ramanujan Ξ-function, denoted by Ξ R (s), as follows Ξ R (s) = (2π) is−6 L(−is + 6)Γ(−is + 6), where Γ(s) is the Gamma function. Another representation for Ξ R (s) is given by (1.1) that at least one of them is different from zero. We define the function where φ F (t) = e −2π cosh t n m=0 a m e −2πme t n m=0 a m e −2πme −t .
We recall that Ξ F (s) = Ξ F (s), and one can see that for some sequences F k , the function Ξ F k (s) converges uniformly to Ξ R (s) on all compact subsets of C.
Throughout this paper, we will study the distribution of the zeros of the function C F (s) := Ξ F (−is).
Note that the zeros of C F (s) are symmetric respect to the line Re s = 0. Using the argument principle, Ki where N (T, C F ) stands for the number of zeros of C F (s) such that 1 ≤ Im s < T , counting multiplicity. In the lower half-plane a similar result holds. Moreover, using the method developed by Levinson [5], he stated that where N (T, C F ) stands for the number of zeros of C F (s) such that |Im s| < T , counting multiplicity and N 1 (T, C F ) denotes the number of simple zeros such that |Im s| < T and Re s = 0. In a sense, it means that almost all zeros of C F (s) lie on the line Re s = 0 and are simple. Our first goal is to establish a refinement of (1.2).
On the other hand, Ki [3, Theorem 2] a result about the vertical distribution of the zeros of C F (s), based on the zeros of the function ψ F (s), defined by Let k ≥ 0 be an integer such that P (1) = P ′ (1) = · · · = P (k−1) (1) = 0 and P (k) (1) = 0, where P (y) = n m=0 a m y m .
Theorem 2. Let ∆ * < ∆ * * be positive real numbers. Suppose that ψ F (s − k) has finitely many zeros in −∆ * * < Re s < ∆ * . Let δ be such that 0 < δ < ∆ * . Then all but finitely many zeros of C F (s) which lie in |Re s| ≤ δ are on the line Re s = 0. In particular, all but finitely many zeros of C F (s) are on the line Re s = 0, if ψ F (s − k) has finitely many zeros in Re s > −∆ * * .
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 ψ F (s−k) is not possible. On the other hand, using (2.7) is clear that ψ F (s − k) has the same set of zeros of a Dirichlet polynomial in the framework of [1,Subsection 12.5]. The set of zeros of a Dirichlet polynomial is quasiperiodic (see [4,Appendix 6,p. 449]). Then, if s 0 = σ 0 + iτ 0 is a zero of the Dirichlet polynomial, for any ε > 0 we can construct a sequence {s n = σ n + iτ n } n∈N of zeros, such that σ n ∈]σ 0 − ε, σ 0 + ε[ for all n ∈ N and τ n → ±∞. This implies that each open vertical strip has no zeros or has infinite zeros. Therefore, the hypothesis in Theorem 2 is reduced to: ψ F (s − k) has no zeros in −∆ * * < Re s < ∆ * . Our second goal in this paper is to give a new proof of this result.
Theorem 3. Let ∆ * < ∆ * * be positive real numbers. Suppose that ψ F (s − k) has no zeros in −∆ * * < Re s < ∆ * . Let δ be such that 0 < δ < ∆ * . Then all but finitely many zeros of C F (s) which lie in |Re s| ≤ δ are on the line 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 ψ F (s − k) is due by de Bruijn [2, p. 225], who showed that where b m are complex numbers and b k = 0.  Throughout the paper, we fix a sequence F . For a function f (s) and the parameters σ 1 < σ 2 , and T 1 < T 2 , we denote the counting function where, in both cases, the counts are with multiplicity, and where the count is without multiplicity.

Preliminaries results
In this section we collect several results for our proof. We highlight that in [3, Proposition 2.3], Ki showed that there is a constant β 0 > 0 such that C F (s) = 0, for |Re s| ≥ β 0 . This implies that for β ≥ β 0 , (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 Then, we obtain the following relation If we denote by This representation allows us to use the following result (see [6,Theorem 36]).
and |F (s)| < e K|s| , for s = σ + iτ with 0 ≤ σ ≤ σ 0 and τ = T m , τ = −T * m , for m ∈ N. Then, for T ≥ 2, we have that for s = σ + iτ with 0 ≤ σ ≤ ∆ and |τ | ≥ 1, and the function µ(σ) is given by Finally, we will need to establish bounds for the right-hand side of (2.4), that implies 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 ψ F (s). We define ψ F,k (s) := ψ F (s − k). Thus, using (1.3) this function we can be written as where p m = (a n−m )e −βmk and β m = ln((2n + 1)/(2(n − m) + 1)), for 0 ≤ m ≤ n. The sum on the right-hand side of (2.7) is a Dirichlet polynomial in the framework [1, Subsection 12.5].
Proposition 5. Let Z(ψ F,k ) denote the set of zeros of ψ F,k (s).
(2) For T 1 < T 2 and c ≥ c 0 , we have that (3) Let K ⊂ C such that |Re s| ≤ M for s ∈ K, and some M > 0. Suppose that K is uniformly bounded from the zeros of ψ F,k (s), i.e.
So, one of this rectangles, denoted by I m , has no zeros of ψ F,k (s) and h(s). Suppose that the right vertical side of I m is contained on the line Re s = −ε m , that we can suppose without loss of generality that doesn't contain a zero of ψ F,k (s). Now, if we place a circle of radius δ > 0 sufficiently small (for instance δ < 1/(2n+1)(16n)) we can enclosed the zeros of the rectangle J m = {s ∈ C : −ε m < Re s < σ 0 and T m < 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 ψ F,k (s). Therefore, the union of the contour C m for all m ∈ Z is uniformly bounded from the zeros. By Proposition 5 there is a constant M > 0 such that |ψ F,k (s)| > M/|b k | for each s ∈ C m . Using (2.5) we get that for s ∈ C m , with |m| sufficiently large. If we denote w(s) = h(s)/Γ(s − k), applying Rouché's theorem we obtain that there is m 0 ∈ N sufficiently large such that (3.6) and for m ≥ m 0 . On another hand, by analyticity of h(s) we have Similarly, for T < 0 we use (3.7) to obtain a similar bound. Thus, we obtain for T > 0 that N (0, σ 0 , −T, T, h) ≤ 2n + 2 ln(2n + 1) π T + O(1).
We replace T by ϕ(2T ) in the above expression, and inserting in (3.4), and one can see that To obtain our desired result we will use an argument of Ki in [3, Pag. 131]. Following his idea, for T > 0 we get that where N k (T, C F ) denotes the number of zeros of C F with multiplicity k with |Im s| < T and Re s = 0, counting with multiplicity. Note that We conclude combining (3.9), (3.10), (3.11), and recalling by (2.1) that N (T, C F ) = N (−σ 0 , σ 0 , −T, T, C F ).