Nonvanishing of L - functions on < ( s ) = 1

In [Ja-Sh], Jacquet and Shalika use the spectral theory of Eisenstein series to establish a new result concerning the nonvanishing of L-functions on <(s) = 1. Specifically they show that the standard L-function L(s, π) of an automorphic cusp form π on GLm is nonzero for <(s) = 1. We analyze this method, make it effective and also compare it with the more standard methods. This note is based on the letter [Sa1]. §1. Review of de la Vallée Poussin’s method A celebrated result of Hadamard and de la Vallée Poussin is the Prime Number Theorem. Their proof involved showing that the Riemann zeta function ζ(s) is not zero for <(s) = 1. In fact these two results turn out to be equivalent. de la Vallée Poussin (1899) extended this method to give a zero free region for ζ(s) of the form; ζ(s) 6= 0 for σ ≥ 1 − c log(|t|+ 2) . (1) Here c is an absolute positive constant and s = σ + it. We will call a zero free region of the type (1), a standard zero free region. Poussin’s method is based on the construction of an auxillary L-function, D(s) with positive coefficients. D(s) should be analytic in <(s) > 1, have a pole at s = 1 of order say k and if L(σ + it0) = 0, ∗ D(s) should vanish to order at least k at s = σ. † This is enough to ensure that L(1 + it0) 6= 0 and if the order of vanishing at σ is bigger than k then one obtains an effective standard zero free region for L(s). To arrange for D(s) to have positive coefficients one often uses ∗Here L(s) stands for a generic L-function whose nonvanishing we seek to establish. †Since the coefficients of D(s) are positive, the Euler product for D(s) converges absolutely for <(s) > 1. a positive definite function on an appropriate group. For example L(s, π × π̃), where π is any unitary isobaric representation of GLm (see for example [Ho-Ra] for definitions and examples) has this property. See also [De] for such positive definite functions on other groups. In the most basic case of ζ(s) and t0 ∈ R with |t0| ≥ 2, one can take D(s) = ζ(s) ζ(s+ it0) ζ (s− it0) ζ(s+ 2it0) ζ(s− 2it0) = L(s,Π× Π̃) . (2) Here Π = 1 α−it0 α0 is an isobaric representation of GL3 and α the principal quasi-character of AQ. D(s) has a pole of order 3 at s = 1 and if ζ(σ+ it0) = 0 then D(s) will have a zero of order 4 at s = σ. Hence, by a standard function theoretic argument (see [Ho-Ra] for example) we have that ζ(σ + it0) 6= 0 for σ ≥ 1 − c log |t0| . This establishes the standard zero free region (1) for ζ(s). We note that (1) is not the best zero free region that is known for ζ(s). Vinogradov [Vi] and his school have developed sophisticated techniques which lead to zero free regions of the type; ζ(s) 6= 0 for σ ≥ 1− cα/(log(|t|+ 2)) for α > 23 and cα > 0. Another well-known example of nonvanishing is that of a Dirichlet L-function L(s, χ), with χ a quadratic character of conductor q. For D(s) we can take ζ(s)L(s, χ) (or if one prefers (ζ(s)L(s, χ)) = L(s, (1 χ) × (1̃ χ)). In this case the order of zero at s = 1 is equal to the order of pole. Hence L(1, χ) 6= 0 (see Landau’s Lemma [Da , pp34]) but this does not yield a standard zero free region near s = 1 for L(s, χ) ie in terms of the conductor.‡ In fact no such zero free region is known for L(s, χ) this being the notorious problem of the exceptional, or “Landau-Siegel”, zero. In this note we will only be concerned with zero free regions for a fixed L-function (ie what is called the t-aspect). For a recent discussion of the exceptional zero problem in general, see the paper [Ho-Ra]. The Poussin method generalizes to automorphic L-functions. Let K be a number field, m ≥ 1 and let π be an automorphic cusp form on GLm(Ak). The standard (finite part) L-function associated with π namely L(s, π), has an analytic continuation to C and a functional equation s −→ 1− s, π −→ π̃ [Go-Ja]. Also well-known by now are the analytic properties (ie continuation and functional equation) of the Rankin-Selberg L-functions L(s, π × π′), where π and π′ are cusp forms on GLm(Ak) and GLm′(Ak) respectively. This follows from [Ja -PS-Sh],[Sh1], [Mo-Wa]. ‡That is to say L(σ, χ) 6= 0 for σ ≥ 1 − c log q and some c > 0.