Interlacing of Zeros of Eisenstein Series

Abstract

Let E k (z) be the normalized Eisenstein series of weight k for the full modular group S L(2, Z). Let a > 0 be an even integer. In this paper we completely determine when the zeros of E k interlace with the zeros of E k+a . This generalizes a result of Nozaki on the interlacing of zeros of E k and E k+12 .

A Stieltjes Separation Property of Zeros of Eisenstein Series

“…Expanding on Nozaki's method, Griffin et al [5] further refined Nozaki's methods and extended their results in [6] to establish necessary and sufficient conditions under which the non-trivial zeros of E k (e iθ ) and E (e iθ ) interlace.…”

“…Following [5,Section 3.1], we introduce the following notation on relevant zeros of cos(kθ /2). Definition 1.3.…”

“…Definition 1.3. [5] Let n k be the number of zeros of F k (θ ) on (π/2, 2π /3). Let β k,i be the ith zero of cos(kθ /2) on (π/2, 2π /3) counting from 2π /3, where 0 ≤ i ≤ n k − 1.…”

“…We also have an equivalent definition following [5, (2.1)], [6,Definition 2.1], and [8, (2.3)- (2.6)]. We replace k in Griffin et al's definition [5] with k. Definition 1.4. [5,6,8] Let α k,i be the ith zero of cos(kθ /2) on (π/2, 2π /3) counting from π/2, where 1 ≤ i ≤ n k .…”

“…We replace k in Griffin et al's definition [5] with k. Definition 1.4. [5,6,8] Let α k,i be the ith zero of cos(kθ /2) on (π/2, 2π /3) counting from π/2, where 1 ≤ i ≤ n k . We have that…”