On the Genus of Meromorphic Functions

We define the class of Left Located Divisor (LLD) meromorphic functions and their vertical order m0(f) and their convergence exponent d(f). When m0(f) ≤ d(f) we prove that their Weierstrass genus is minimal. This explains the phenomena that many classical functions have minimal Weierstrass genus, for example Dirichlet series, the Γ-function, and trigonometric functions. 1. LLD meromorphic functions. Meromorphic functions f on C, of the variable s ∈ C, considered in this article are assumed to be of finite order o = o(f). We recall that the order o(f) is defined as o(f) = lim sup R→+∞ log log ||f ||C0(B(0,R)) logR . We study in this article Dirichlet series, and more generally the class of meromorphic of finite order with Left Located Divisor (LLD), which we call LLD meromorphic functions: Definition 1. (LLD meromorphic functions) A LLD meromorphic function is a function f of finite order and left located divisor σ1 = sup ρ∈f−1({0,∞}) Rρ < +∞ . The properties that we establish in this article are invariant by a real translation. Thus considering g(s) = f(s+ σ1) instead of f we will assume that σ1 = 0. Examples of LLD meromorphic functions are Dirichlet series, that we normalize in this article such that f(s) → 1 when Rs → +∞. A Dirichlet series is of the form (1) f(s) = 1 + ∑ n≥1 an e −λns , with an ∈ C and 0 < λ1 < λ2 < . . . 2010 Mathematics Subject Classification. Primary: 30D30. Secondary: 30B50, 30D15.