The Analytic Properties Of A Dirichlet Series

In general, we always definite Dirichlet series aswhere a_n is a complex number {the coefficient of Dirichlet series), s =σ+it, (?)(s) > 0In this thesis, we study a special Drichlet series.Hardy, Littlewood, and Hecke studied the analytic properties of the Dirichlet series of the formwhere q > 0 is a square-free integer, andW. Duke and O. Imamogln[3] considered Dirichlet seriesformed with coefficientswhere q and r are positive coprime square-free integers.We consider Dirichlet scries of the following form where q, r and t axe positive coprime square-free integers, andLet (?)₁ be the set of 2×2 matricesαof the following formwith integer numbers x₀, x₁,x₂,x₃,LetFor any v = z + rj∈H₃. there is only one matrix v_M∈SP₂, which corresponds with v. We definedIf we fix v_M = I,φ_q,r,t(s) =φ_q,r,t(s,I).For fixed s with R(s) large enough, we havewhereWe find thatSo we can define a kernel fimctionUsing the Selberg transform formula, we haveHence, andWe use the analytic properties ofΓ-function andζ⁺-function to get the analytic properties ofφ(s) in (3).Finely, we get the main result of this thesis.Theorem 1.1. The functionφ(s) defined in (3)(see chapter 1) has a meromorphic continuation to the whole s-plane with at most simple poles at s = 2, s= 4 and s=- 2m + 1±γn,withγn≠0; and at most double poles at s = -2m + 1. Here n is any positive integer, m is any nonncgative integer, andγn = (?). for n = 0,..., N;γn = iÏ„_n ,Ï„_n=(?)∈R for n≥N+1, whereN = max{n≥0 |λ_n < 1}.