Acta Mathematica Sinica, English Series Oct., 2007, Vol. 23, No. 10, pp. 1903–1910 Published online: Feb. 15, 2007 DOI: 10.1007/s10114-005-0926-5 Http://www.ActaMath.com
Selberg’s Normal Density Theorem for Automorphic L-Functions for GLm Yan QU Department of Mathematics, Shandong University, Ji’nan 250100, P. R. China E-mail: karen
[email protected] Abstract Let π be an irreducible unitary cuspidal representation of GLm (AQ ) with m ≥ 2, and L(s, π) the L-function attached to π. Under the Generalized Riemann Hypothesis for L(s, π), we estimate the normal density of primes in short intervals for the automorphic L-function L(s, π). Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function. Keywords
automorphic L-functions, generalized Riemann Hypothesis, normal density
MR(2000) Subject Classification 11F70, 11M36, 11M41
1
Introduction
To each irreducible unitary cuspidal representation π of GLm (AQ ), one can attach a global Lfunction L(s, π), as in Godement and Jacquet [1], and Jacquet and Shalika [2]. For σ = s > 1, L(s, π) is defined by products of local factors L(s, π) = Lp (s, πp ), (1.1) p 1, ∞ L (s, π) Λ(n)aπ (n) =− , (1.2) L(s, π) ns n=1 Received October 11, 2005, Accepted January 1, 2006 Supported by NSFC Grant #10531060, and by a Ministry of Education Major Grant Program in Sciences and Technology
Qu Y.
1904
where Λ(n) is the von Mangoldt function, and απ (p, j)k . aπ (pk ) =
(1.3)
1≤j≤m
The Prime Number Theorem for L(s, π) concerns the asymptotic behavior of the counting function ψ(x, π) = n≤x Λ(n)aπ (n), and a special case of the main theorem in Liu and Ye [3] asserts that ψ(x, π) x exp(−c log x), (1.4) for some positive constant c, where π is any irreducible unitary cuspidal representation of GLm (AQ ) with m ≥ 2. In Iwaniec and Kowalski [4], Theorem 5.13, a prime number theorem is proved for general L-functions satisfying necessary axioms, from which (1.4) follows as a consequence. Under GRH, (1.4) can be improved to ψ(x, π) x1/2 log2 x,
(1.5)
a proof of which can be found in, for example, [5]. It follows from (1.5) that, under GRH, ψ(x + h(x), π) − ψ(x, π) = o(h(x)),
(1.6)
2
for increasing functions h(x) ≤ x satisfying h(x)/(x1/2 log x) → ∞. In view of Λ(n)aπ (n), ψ(x + h(x), π) − ψ(x, π) = x 1 (Jacquet and Shalika [2]). A2 The complete L-function Φ(s, π) has an analytic continuation to the whole complex plane and satisfies the functional equation Φ(s, π) = ε(s, π)Φ(1 − s, π ˜) 1/2−s επ Nπ ,
with ε(s, π) = where Nπ ≥ 1 is an integer called the conductor of π, επ is the root number satisfying |επ | = 1, and π ˜ is the representation contragradient to π (Shahidi [12, 13, 14, and 15]). A3
Φ(s, π) is entire, and bounded in vertical strips with finite width (Godement and Jacquet
[1], and Jacquet and Shalika [2]). A4
Φ(s, π) is of order one (Gelbart and Shahidi [16]).
A5 Φ(s, π) and L(s, π) are non-zero in the half-plane σ ≥ 1 (Jacquet and Shalika [2], and Shahidi [12]). A6 Let N (T, π) be the number of nontrivial zeros within the rectangular 0 ≤ σ ≤ 1, |t| ≤ T . Then N (T, π) T log T , and N (T + 1, π) − N (T, π) log T (Liu and Ye [11, Lemma 4.3], and Iwaniec and Kowalski [4, Theorem 5.8]). The following Lemma 2.1 is Theorem 3.1 of [5]: Lemma 2.1 Let π be an irreducible unitary cuspidal representation of GLm (AQ ) with m ≥ 2, and let θ be as in (1.7). Then, for x ≥ 2 and T ≥ 2, xρ x x1+θ ψ(x, π) = − + O min , (2.1) log x + O(xθ log x). ρ T 1/4 T 1/2 |γ|≤T
Explicit formulas of different forms were established by Moreno [17], [18]; under GRC, explicit formulas for general L-functions were proved in (5.53) of Iwaniec and Kowalski [4].
Qu Y.
1906
Note that our Theorem 2.1 does not require GRC. 3
Lemmas
To prove Theorem 1.1, we need the following lemmas. Lemma 3.1 is due to Gallagher [10], and is named after him in the literature. Lemma 3.1 Let S(u) = ν c(ν)e2πiνu be absolutely convergent, where the coefficients c(ν) ∈ C, and the frequencies of ν run over an arbitrary sequence of real numbers. Then
2
∞
U
1
dx. |S(u)|2 du 2 c(ν)
U −U
Lemma 3.2
−∞
x
