Bagchi’s Theorem for Families of Automorphic Forms

Abstract

Abstract. We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight 2 and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic L-functions.

A spectral universality theorem for Maass L-functions

“…(iii) Use suitable equidistribution theorems to show that the approximations in (i) become, as |U | → ∞, arbitrarily close to the random Dirichlet polynomials in (ii), uniformly over K and for all u in a subset of A of (small) positive measure. The above triple approximation procedure leads the required universality result, and can be traced in the probabilistic setting as well; see in particular Kowalski [12]. We have chosen to present our result in the classical language; our approach might therefore be conceptually less satisfactory, but probably is more transparent.…”

“…Besides the family of vertical shifts considered in Voronin's theorem (and extended to other Lfunctions as observed shortly in the beginning), there are genuinely GL(1) results for the family of Dirichlet characters, see Gonek [5] and Bagchi [2], or the family of quadratic characters, see Mishou-Nagoshi [15]. More recently, Kowalski [12] proved a universality theorem for modular L-functions in the level aspect, which can therefore be considered as a genuinely GL(2) statement of this type. Moreover, he uses the probabilistic setting to give a conceptual explanation of universality theorems for families of L-functions.…”

“…Lemma 2.4 ([12, Lemma 6]). For any given Z > 2 the set of all sums of type Lemma 2.4 is an immediate consequence of Lemma 6 in[12], where the numbers ω p are viewed as Tr(x p ) with x p ∈ SU 2 (C). From Lemma 2.4 we deduce the required density result in the following form.Proposition 2.2.…”

An effective universality theorem for the Riemann zeta function

“…The assumption that f (z) = 0 is necessary: if f were allowed to vanish then an application of Rouche's theorem would produce at least ≍ T zeros ρ = β + iγ of ζ(s) with β > 1 2 + ε and T ≤ γ ≤ 2T , contradicting the simplest zero-density theorems. Subsequent work of Bagchi [1] clarified Voronin's universality theorem by setting it in the context of probability theory (see [7] for a streamlined proof). Viewing ζ( with {X(p)} p a sequence of independent random variables uniformly distributed on the unit circle (and with p running over prime numbers).…”

“…The only analytic ingredients that are needed are zero density estimates, and bounds on the coefficients of these L-functions (the so-called Ramanujan conjecture). In particular, the techniques of this paper can be used to obtain an effective version of a recent result of Kowalski [7], who proved an analogue of Voronin's universality theorem for families of L-functions attached to GL 2 automorphic forms. In fact, using the zero-density estimates near 1 that are known for a very large class of L-functions (including those in the Selberg class by Kaczorowski and Perelli [6], and for families of L-functions attached to GL n automorphic forms by Kowalski and Michel [8]), one can prove an analogue of Theorem 1.1 for these L-functions, where we replace 3/4 by some σ < 1 (and r < 1 − σ).…”