All but finitely many non-trivial zeros of the approximations of the Epstein zeta function are simple and on the critical line

Abstract

Some errors, which include Proposition 4.1(2), and typographical mistakes are corrected. On p. 321, replace Q(m, n)s in the first equation by Q(m, n)−s. On p. 322, in (1.2), replace bk by 1 2 b k bk. On p. 322, after (1.3), replace 1 2 ( Δ / 2 π ) s by 1 2 ( Δ / 2 ) s . On p. 326, before (2.5), replace λ 1 / 2 + i z by 1 2 λ 1 / 2 + i z . On p. 326, in (2.9), replace τ 0 log 3 / 4 α k by τ 0 2 log 3 / 4 α k . On p. 327, in the equation before Preliminaries, replace Γ ( 2 s − 1 ) Γ ( 2 s ) by Γ ( s − 1 / 2 ) Γ ( s ) . On p. 327, in the definition of 2Kz(2A), replace 2Acosh t by −2Acosh t. On p. 332, in (4.2), replace ( λ π ) 1 / 2 − σ by ( λ π ) − σ . On p. 335, in the second sentence of Section 5, replace F(s) by G(s). On p. 336, in (5.6), replace λ π by π λ . On p. 337, in the second equation for W(z), replace λ 1 / 2 + i z by 1 2 λ 1 / 2 + i z . Proposition.4.1(2) is not correct. Namely, ‘1 + o(1)’ in the equation before Proposition 4.2 is not true. We modify the statement as follows. Proposition.4.1(2) There exist constants σ ˜ < 0 , a > 0 and b ≠ 0 ( b ∈ ℝ ) such that on Re ( s ) = σ ˜ , Proof.For Re ( s ) = σ < 0 , we have On p. 334, in the first equation, we missed the factor s−1/2 in the sum; nevertheless, the proof of Proposition 4.2(3) essentially follows using the same method. Set σ * = − σ ˜ + 1 2 . Then, replace 3 in Lemma 5.1(5) and the last three equations of p. 340 by σ*. In Lemma 5.1(3), replace y ⩽ slant3 by y ⩽ σ * . Also, replace β k ⩽ 3 in the last equation of p. 338 and (5.13) of p. 339 by β k ⩽ σ * . In the last equation of p. 338, replace βk > 3 by β k > σ * .