On zeros of approximate functions of the Rankin-Selberg L-functions

Notations. As usual, Z is the ring of rational integers, ℤ>o the set of positive integers, ℂ the field of complex numbers. We denote by η the upper half-plane, and by r the full modular group PSL 2(ℤ). For a complex variable s, we put e(s) = e 2πis, Γ ℝ(s) = π -s/2/Γ(s/ 2) and Γ ℂ(s) = 2(2π) -sΓ(s). We denote by Ç(s) and Ç*(*) = Γ ℝ(S) Ç(s) Riemann zeta-function and the completed Riemann zeta-function, respectively, and denote by o- v(n) = Σ d/n d/v the divisor function. Throughout the paper, z = x + iy (x ε ℝ, y > 0) is a variable on h, and s = σ + it (σ, t €ℝ) is a complex variable. A sum over the empty set is meant to be zero. © Instytut Matematyczny PAN, 2009.