On the Selberg class of Dirichlet series: small degrees

Twists and Resonance ofL‐Functions, II

“…Moreover, Theorem 1 contains in addition a bound on the order of growth of the function G(s; f ). Note that condition d ≥ 1 is equivalent to d > 0, since the extended Selberg class is empty for degrees between 0 and 1; see Conrey-Ghosh [1] and our paper [2]. Note also that the integer J, the constants η j and the functions W j (s) and G(s; f ) may depend on F (s), f (ξ, α) and K.…”

Converse theorems: from the Riemann zeta function to the Selberg class

“…Actually, this is a simple but important special case of a duality principle, first studied by S.Bochner in the 1950's, between Dirichlet series with a general Riemann type functional equation and their Mellin transforms; see, e.g., Kaczorowski-Perelli [24]. Another instance of such a duality is represented by Hecke's correspondence, already mentioned above, between modular forms for the triangle groups G(λ), λ ∈ R, generated by z → z + λ and z → −1/z, and Dirichlet series with a functional equation of type similar to (10); the case λ = 1 corresponds to the modular forms for Γ. The more interesting part of such a correspondence is that Hecke managed to compute exactly the dimension of the resulting spaces W , again arguing on the modular forms side.…”

“…In Section 1 we briefly recalled Hecke's theory for the modular group Γ, in the case of cusp forms; we still continue with this framework for ease of exposition, since the slightly more general case of modular forms is dealt with similarly. In particular, we recall that given f ∈ S k (Γ) with Fourier series as in (8), the (normalized) Hecke L-function (9) is entire of finite order and satisfies the functional equation (10). Hecke proved that the opposite implication holds true as well, thus showing a perfect correspondence between cusp forms for Γ and Dirichlet series F (s) with analytic continuation satisfying (10).…”

“…In particular, we recall that given f ∈ S k (Γ) with Fourier series as in (8), the (normalized) Hecke L-function (9) is entire of finite order and satisfies the functional equation (10). Hecke proved that the opposite implication holds true as well, thus showing a perfect correspondence between cusp forms for Γ and Dirichlet series F (s) with analytic continuation satisfying (10). Actually, Hecke proved a more general correspondence which led him, for example, to an alternative approach to Hamburger's theorem.…”

“…The modular group Γ has two generators, which in terms of their action on the upper half-plane H are described as the shift z → z + 1 and the inversion z → −1/z; to check the modularity of f (z) for Γ, it is clearly enough to check its modularity on the generators. Now, roughly speaking, the shift-modularity is equivalent to the Fourier expansion in (8), while the inversion-modularity is equivalent to functional equation (10). Actually, this is a simple but important special case of a duality principle, first studied by S.Bochner in the 1950's, between Dirichlet series with a general Riemann type functional equation and their Mellin transforms; see, e.g., Kaczorowski-Perelli [24].…”