Abstract
Abstract. Let L(s, π) be the principal L-function attached to an irreducible unitary cuspidal automorphic representation π of GL m (A Q ). The aim of the paper is to give a simple method to show the lower bounds of mean value for automorphic L-functions over short intervals.
Introduction
An important problem in analytic number theory is to estimate the moments
for all kinds of L-functions L(s, π). For the Riemann zeta-function ζ(s), a conjecture states that the 2k-th moment should be asymptotic to C k T (log T ) . Actually, this latter result was proved by Ramachandra [11] for all positive integers k, by Heath-Brown [4] for all positive rational numbers k, and under the Riemann Hypothesis by Ramachandra [10] for all positive real numbers k. In [1] the authors propose conjectures for the full asymptotics of the moments of general L-functions. In particular, the paper provides conjectures for the moments of the Riemann zeta-function, the family of primitive Dirichlet L-functions, quadratic twists of L-functions, and automorphic L-functions attached to cusp forms.
Let π be an irreducible unitary cuspidal automorphic representation of GL m (A Q ) and s = σ + it ∈ C. Then the principal L-function [3] attached to π is given by Euler products of local factors for σ > 1,
with {α π (p, j) : 1 ≤ j ≤ m} being complex Satake parameters of π at the finite place π p according to the local Langlands correspondence. If p is unramified, its local L-factor is of the form L(s,
is a polynomial of degree at most m and P v (0) = 1. We can write the local factors at ramified places v in the same form with the convention that some of the α π (p, j) may be zero.
In this paper, we give a simple method to obtain lower bounds for integral moments of the principal L-function L(s, π) over short intervals. Our main theorem is as follows. 
Automorphic L-functions and some lemmas
Let π be an irreducible cuspidal automorphic representation of
where q is an integer called the arithmetic conductor of π and
, and {µ j : 1 ≤ j ≤ m} are local complex parameters of π at the infinite place ∞.
The following statements collect together analytic facts about principal L-functions which we will use for our proofs.
(A1) The Dirichlet series
converges absolutely in the half-plane s > 1, and we have (see [9] )
The complete L-function Λ(s, π) has an analytic continuation to the whole complex plane and satisfies the functional equation [3] 
where ε(π), a complex number of modulus 1, is the root number,π is the contragredient representation of π, and
is an entire function of order one, bounded in vertical strips with finite width with exponential decay as | s| → ∞; see [2] , [3] .
(A4) The zeros of Λ(s, π), that is, the nontrivial zeros of L(s, π), lie in the open critical strip 0 < s < 1; see [6] . In particular, L(s, π) is nonvanishing in the half-plane s ≥ 1.
(A5) Bounds toward the generalized Ramanujan conjecture, [8] , [12] , are
if π is unramified at p,
We also need the some lemmas. The following result of Littlewood establishes the connection between zeros of an analytic function and its mean-value estimates. 
where the sum runs over the zeros ρ of φ(s) in D, and Dist(ρ) is the distance from ρ to the left edge of the rectangle.
Proof. This is a revised version of the classical Littlewood lemma. See Titchmarsh [13] , Sections 9.9 and 9.15.
Lemma 2.2. (1) Let N (T ) be the number of nontrivial zeros
With the same notation as in (2), we have
Proof. For proofs of (1) and (2), see Iwaniec and Kowalski [5] , Proposition 5.7, or Liu and Ye [7] , Lemma 4.3. We integrate (2.4) along the straight line from s = σ+it to 2 + it. By (2.3), if t is not the ordinate of a zero, then 
where K(σ 0 ) is independent of T . 
where D is the rectangle with vertices σ, 2, σ + iT , and 2 + iT .
Proof. We define 

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