On interlacing of the zeros of a certain family of modular forms

Abstract

Let k = 12m(k) + s ≥ 12 for s ∈ {0, 4, 6, 8, 10, 14}, be an even integer and f be a normalised modular form of weight k with real Fourier coefficients, written asUnder suitable conditions on aj (rectifying an earlier result of Getz), we show that all the zeros of f , in the standard fundamental domain for the action of SL(2, Z) on the upper half plane, lies on the arc A := e iθ : π 2 ≤ θ ≤ 2π 3 . Further, extending a result of Nozaki, we show that for certain family {f k } k of normalised modular forms, the zeros of f k and f k+12 interlace on A • := e iθ : π 2 < θ < 2π 3 .6 1728 .

A Stieltjes Separation Property of Zeros of Eisenstein Series

“…Here, j (τ ) is the usual j-invariant and j n (τ ) is its image under the nth Hecke operator. Additionally, Saha and Saradha [10] explored the similar interlacing property of zeros of modular forms within a certain family.…”