A Note on the Riemann Hypothesis for the Constant Term of the Eisenstein Series

In this paper, we give an another proof of Theorem 3 in [4]. Theorem 3 in [4] is the following theorem. Theorem 1. Let ζ(s) be the Riemann zeta function and let ζ * (s) = π −s/2 Γ(s/2)ζ(s). For each y ≥ 1 the constant term of the Eisenstein series a 0 (y, s) := ζ * (2s)y s + ζ * (2 − 2s)y 1−s (1) is a meromorphic function that satisfies the modified Riemann hypothesis. More precisely , there is a critical value y * := 4πe −γ = 7.055507+, (2) such that the following hold: (1) All zeros of a 0 (y, s) lie on the critical line for 1 ≤ y ≤ y *. (2) For y > y * there are exactly two zeros off the critical line. These are real simple zeros ρ y , 1 − ρ y with 1 2 < ρ y < 1. The zero ρ y is a nondecreasing function of y, and ρ y → 1 as y → ∞. The natural entire function associated to a 0 (y, s) is G(y, s) := (2s)(2s − 2)a 0 (y, s), (3) which behaves similarly to the Riemann ξ-function, satisfying the functional equation G(y, s) = G(y, 1 − s), being real on the real axis and on ℜ(s) = 1 2. It also has G(y, 1 2) = (log 4π − γ − log y) √ y, where γ is Euler's constant. To establish Theorem 1 it proves useful to study the entire function H(y, s) := 1 2 (s − 1 2)G(y, s) = (s − 1)ξ(2s)y s + sξ(2s − 1)y 1−s. (4) The function H(y, s) has a zero at s = 1 2 and satisfies the functional equation H(y, s) = −H(y, 1 − s), but has the advantage that both terms on the right side of (4) are entire functions. We begin with the analytic part of the proof. Theorem 2. For fixed y ≥ 1, all the zeros of the entire function H(y, s) lie on the critical line ℜ(s) = 1 2 , except for possible zeros off the line lying the rectangular box B := {s = σ + it| − 19 ≤ σ ≤ 20, |t| ≤ 20}.