Sharp lower bounds for moments of quadratic Dirichlet L-functions
aa r X i v : . [ m a t h . N T ] F e b SHARP LOWER BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET L -FUNCTIONS PENG GAO
Abstract.
We establish sharp lower bounds for the k -th moment in the range 0 ≤ k ≤ L -functions at the central point. Mathematics Subject Classification (2010) : 11M06
Keywords : moments, quadratic Dirichlet L -functions, lower bounds1. Introduction
As moments of families of L -functions at the central point can be applied to address important issues such as non-vanishing results concerning these central values, they have been studied intensively in the literature. For the family ofquadratic Dirichlet L -functions, asymptotic expressions are available only for the first four moments presently. Theseresults are obtained by M. Jutila [8] for the first two moments, by K. Soundararajan [16] for the third moment and Q.Shen [14] for the fourth moment. See also [4, 15, 16, 18–20] for various improvements on the error terms. Besides theabove, conjectured asymptotic formulas for various families of L -functions are made in the work of J. P. Keating and N.C. Snaith [9], building on the relation with random matrix theory. More precise formulas including lower order termsare further conjectured by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith in [2].Much progress has been made towards establishing bounds for moments of L -functions of the order of magnitude inagreement with the above conjectures. There are now several general approaches that allow one to achieve this. Forupper bounds, one can apply a method due to K. Soundararajan in [17] together with its sharpened version by A. J.Harper [5] or a principle built by M. Radziwi l l and K. Soundararajan in [11]. For lower bounds, one can make use ofa simple and powerful method developed by Z. Rudnick and K. Soundararajan in [12, 13], or a principle enunciatedby W. Heap and K. Soundararajan in [6] which can be regarded as dual to the corresponding one of Radziwi l l andSoundararajan concerning upper bounds.We now return to the case of quadratic Dirichlet L -functions. More specifically, we consider the family { L ( s, χ d ) } with χ d = (cid:0) d · (cid:1) being the Kronecker symbol such that d is odd and square-free. For this family, it is conjectured by J.C. Andrade and J. P. Keating [1] that for all positive real k , X ∗ 2. In this paper, basedon the lower bounds principle of Heap and Soundararajan in [6] together with an idea in [3], we further explore thecommensurate lower bounds for the same family. As we have indicated above, sharp bounds for the k -th moment arealready known for any real k ≥ 1. Thus, we focus on the case 0 ≤ k ≤ k ≥ Theorem 1.1. For any real number k such that ≤ k ≤ , we have X ∗ We may assume that X is a large number and we denote Φ for a smooth, non-negative function compactly supportedon [1 / , / 8] such that Φ( x ) ≤ x and Φ( x ) = 1 for x ∈ [1 / , / k by 2 k and assume that 0 < k < / < d < X into dyadic blocks, we see that in order toprove Theorem 1.1, it suffices to show that X ∗ N, M be two large natural numbers (depending on k only) and denote { ℓ j } ≤ j ≤ R for a sequence of evennatural numbers such that ℓ = 2 ⌈ N log log X ⌉ and ℓ j +1 = 2 ⌈ N log ℓ j ⌉ for j ≥ 1, where R is defined to the largestnatural number satisfying ℓ R > M . We may assume that M is so chosen so that we have ℓ j > ℓ j +1 for all 1 ≤ j ≤ R − R ≪ log log ℓ , R X j =1 ℓ j ≤ ℓ R . (2.3)We denote P for the set of odd primes not exceeding X /ℓ and P j for the set of primes lying in the interval( X /ℓ j − , X /ℓ j ] for 2 ≤ j ≤ R .We set P j ( d ) = X p ∈ P j √ p χ d ( p ) , and for any real number α , N j ( d, α ) = E ℓ j ( α P j ( d )) , N ( d, α ) = R Y j =1 N j ( d, α ) . (2.4)Here we notice that it follows from [11, Lemma 1] that the quantities defined in (2.4) are all positive. Instead ofexamining (2.2), we may now look at X ∗ 1) and | L ( , χ d ) N ( d, k − | , which we find a little inconvenient.Thus, we incorporate an idea given in [3] by observing that it follows from [3, Lemma 4.1] that we have N ( d, α ) N ( d, − α ) ≥ . We deduce from this that for any real c such that 0 < c < X ∗ ( d, L ( , χ d ) N ( d, k − dX ) ≤ X ∗ ( d, | L ( , χ d ) |N ( d, k − dX ) ≤ X ∗ ( d, | L ( , χ d ) | c · | L ( , χ d ) | − c N (2 k − , d ) (1 − c ) / · N ( d, k − N ( d, − k ) (1 − c ) / Φ( dX ) . Applying H¨older’s inequality with exponents being 2 k/c, / (1 − c ) , ((1 + c ) / − c/ (2 k )) − to the last sum above, wededuce that X ∗ ( d, L ( , χ d ) N ( d, k − dX ) ≤ (cid:16) X ∗ ( d, | L ( , χ d ) | k Φ( dX ) (cid:17) c/ (2 k ) (cid:16) X ∗ ( d, | L ( , χ d ) | N ( d, k − dX ) (cid:17) (1 − c ) / × (cid:16) X ∗ ( d, N ( d, k − ((1+ c ) / − c/ (2 k )) − N ( d, − k ) (1 − c ) / · ((1+ c ) / − c/ (2 k )) − Φ( dX ) (cid:17) (1+ c ) / − c/ (2 k ) . (2.6)We now set (1 − c ) / · ((1 + c ) / − c/ (2 k )) − = 2which implies that c = ( k − − and that((1 + c ) / − c/ (2 k )) − = 2(2 − k )1 − k . One checks that the above value of c does satisfy that 0 < c < < k < / 2. We then deduce from (2.6)that in order to establish a lower bound estimation for the 2 k -th moment, it suffices to establish the following threepropositions. Proposition 2.1. With notations as above, we have X ∗ ( d, L ( 12 , χ d ) N ( d, k − (cid:16) dX (cid:17) ≫ X (log X ) (2 k )2+12 . Proposition 2.2. With notations as above, we have X ∗ ( d, N ( d, k − − k )1 − k N ( d, − k ) Φ (cid:16) dX (cid:17) ≪ X (log X ) (2 k )22 . Proposition 2.3. With notations as above, we have X ∗ ( d, L ( 12 , χ d ) N ( d, k − (cid:16) dX (cid:17) ≪ X (log X ) (2 k )2+22 . In fact, combining (2.6) with the above three propositions, we see that X (log X ) (2 k )2+12 ≪ (cid:16) X ∗ ( d, | L ( , χ d ) | k Φ( dX ) (cid:17) c/ (2 k ) (cid:16) X (log X ) (2 k )2+22 (cid:17) (1 − c ) / (cid:16) X (log X ) (2 k )22 (cid:17) (1+ c ) / − c/ (2 k ) . The desired lower bound given in (2.1) follows immediately from this. PENG GAO Notice that Proposition 2.3 can be established similar to [3, Proposition 3.1] so that it remains to prove Propositions2.1 and 2.2 in the rest of the paper. 3. Preliminaries We include in this section a few lemmas that are needed in our proofs. In what follows, we denote the letter p for aprime number and the symbol (cid:3) for a perfect square. We define δ n = (cid:3) to be 1 when n = (cid:3) and 0 otherwise. For thefunction Φ described in Section 2, we associate its Mellin transform b Φ( s ) for any complex number s by b Φ( s ) = Z ∞ Φ( x ) x s dxx . We first recall the following two lemmas that are given in [3] as Lemma 2.2 and 2.3 there, respectively. Lemma 3.1. Let x ≥ . We have, for some constant b , X p ≤ x p = log log x + b + O (cid:16) x (cid:17) . Also, for any integer j ≥ , we have X p ≤ x (log p ) j p = (log x ) j j + O ((log x ) j − ) . Lemma 3.2. For large X and any odd positive integer n , we have X ∗ ( d, χ d ( n )Φ (cid:16) dX (cid:17) = δ n = (cid:3) b Φ(1) 2 X ζ (2) Y p | n (cid:16) pp + 1 (cid:17) + O ( X + ǫ √ n ) . (3.1)Next, similar to [3, Lemma 2.4], by setting Y = X / , M = 1 in [16, Proposition 1.1-1.2], we have the followingasymptotic formula concerning the twisted first moment of quadratic Dirichlet L -functions. Lemma 3.3. With notations as above and writing any odd l as l = l l with l square-free, we have for any ε > , X ∗ ( d, L ( , χ d ) χ d ( l )Φ( dX ) = C b Φ(1) ζ (2) 1 √ l g ( l ) X (cid:16) log √ Xl + C + X p | l C ( p ) p log p (cid:17) + O (cid:18) X + ε l 12 + ε (cid:19) , where C = Y p ≥ (cid:18) − p ( p + 1) (cid:19) and g ( l ) = Y p | l (cid:16) p + 1 p (cid:17) (cid:18) − p ( p + 1) (cid:19) . Also, C is a constant depending only on Φ and C ( p ) ≪ for all p . Lastly, we present a result which is analogue to [6, Lemma 1] and is needed in the proof of Proposition 2.2. Lemma 3.4. For ≤ j ≤ R , we have N j ( d, k − − k )1 − k N j ( d, − k ) ≤ N j ( d, k ) (cid:16) O (cid:0) e − ℓ j (cid:1)(cid:17) + Q j ( d ) , where the implied constants are absolute, and Q j ( d ) = (cid:16) P j ( d ) ℓ j (cid:17) r k ℓ j , with r k = 1 + (2 − k ) / (1 − k ) .Proof. We note first that when | z | ≤ aK/ 20 for any constant 0 < a ≤ 2, we have (cid:12)(cid:12)(cid:12) K X r =0 z r r ! − e z (cid:12)(cid:12)(cid:12) ≤ | az | K K ! ≤ (cid:16) ae (cid:17) K . We deduce from this that when |P j ( d ) | ≤ ℓ j / N j ( d, k − 1) = exp((2 k − P j ( d )) (cid:16) O (cid:16) exp((2 k − |P j ( d ) | ) (cid:16) (1 − k ) e (cid:17) ℓ j (cid:17) = exp((2 k − P j ( d )) (cid:16) O (cid:16) (1 − k ) e − ℓ j (cid:17)(cid:17) . Similarly, we have N j ( d, − k ) = exp((2 − k ) P j ( d )) (cid:16) O (cid:16) e − ℓ j (cid:17)(cid:17) . HARP LOWER BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET L -FUNCTIONS 5 The above estimations then allow us to see that when |P j ( d ) | ≤ ℓ j / N j ( d, k − − k )1 − k N j ( d, − k ) = exp(2 k P j ( d )) (cid:16) O (cid:0) e − ℓ j (cid:1)(cid:17) = N j ( d, k ) (cid:16) O (cid:0) e − ℓ j (cid:1)(cid:17) . (3.2)Next, notice that when |P j ( d ) | ≥ ℓ j / 20, we have |N j ( d, − k ) | ≤ ℓ j X r =0 | P j ( d ) | r r ! ≤ |P j ( d ) | ℓ j ℓ j X r =0 (cid:16) ℓ j (cid:17) ℓ j − r r r ! ≤ (cid:16) |P j ( d ) | ℓ j (cid:17) ℓ j . (3.3)The assertion of the lemma now follows from (3.2), (3.3) together with the observation that the same bound aboveholds for |N j ( d, k − | as well. (cid:3) Proof of Proposition 2.1 Denote Ω( n ) for the number of distinct prime powers dividing n and w ( n ) for the multiplicative function such that w ( p α ) = α ! for prime powers p α . Let b j ( n ) , ≤ j ≤ R be functions such that b j ( n ) = 1 when n is composed of at most ℓ j primes, all from the interval P j . Otherwise, we define b j ( n ) = 0. We use these notations to see that(4.1) N j ( d, k − 1) = X n j √ n j (2 k − Ω( n j ) w ( n j ) b j ( n j ) χ d ( n j ) , ≤ j ≤ R. Note that each N j ( d, k − 1) is a short Dirichlet polynomial since b j ( n j ) = 0 unless n j ≤ ( X /ℓ j ) ℓ j = X /ℓ j . It followsfrom this that N ( d, k − 1) is also a short Dirichlet polynomial of length at most X /ℓ + ... +1 /ℓ R < X / M by (2.3).We use (4.1) to expand the term N ( d, k − 1) and apply Lemma 3.2 to evaluate it. As N ( d, k − 1) is a short Dirichletpolynomial, we may ignore the error term in Lemma 3.2 to consider only the main term contribution. Upon writing n j = ( n j ) ( n j ) with ( n j ) being square-free, we see that X ∗ ( d, L ( 12 , χ d ) N ( d, k − (cid:16) dX (cid:17) ≫ X X n , ··· ,n R (cid:16) R Y j =1 p n j ( n j ) (2 k − Ω( n j ) w ( n j ) b j ( n j ) 1 g ( n j ) (cid:17)(cid:16) log (cid:16) √ X ( n ) · · · ( n R ) (cid:17) + C + X p | n ··· n R C ( p ) p log p (cid:17) . We may further ignore the contribution from the terms involving C + P p | n ··· n R C ( p ) p log p as one may show using ourarguments in the paper that the contribution is ≪ X (log X ) (2 k )2+12 − . Thus we deduce that X ∗ ( d, L ( 12 , χ d ) N ( d, k − (cid:16) dX (cid:17) ≫ X X n , ··· ,n R (cid:16) R Y j =1 p n j ( n j ) (2 k − Ω( n j ) w ( n j ) b j ( n j ) 1 g ( n j ) (cid:17)(cid:16) log (cid:16) √ X ( n ) · · · ( n R ) (cid:17)(cid:17) = S − S , where S = 12 X log X X n , ··· ,n R (cid:16) R Y j =1 p n j ( n j ) (2 k − Ω( n j ) w ( n j ) b j ( n j ) 1 g ( n j ) (cid:17) ,S = X X n , ··· ,n R (cid:16) R Y j =1 p n j ( n j ) (2 k − Ω( n j ) w ( n j ) b j ( n j ) 1 g ( n j ) (cid:17) log (cid:0) R Y i =1 ( n i ) (cid:1) . It remains to bound S from below and S from above. We bound S first by recasting it as S = 12 X log X R Y j =1 X n j (cid:16) p n j ( n j ) (2 k − Ω( n j ) w ( n j ) b j ( n j ) 1 g ( n j ) (cid:17) . We consider the sum above over n j for a fixed 1 ≤ j ≤ R . Note that the factor b j ( n j ) restricts n j to have all primefactors in P j such that Ω( n j ) ≤ ℓ j . If we remove the restriction on Ω( n j ), then the sum becomes Y p ∈ P j (cid:16) ∞ X i =0 p i (2 k − i (2 i )! g ( p i ) + ∞ X i =0 p i +1 (2 k − i +1 (2 i + 1)! 1 g ( p i +1 ) (cid:17) = Y p ∈ P j (cid:16) − (cid:0) (2 k − k − (cid:1) p (cid:17) − D ( p ) , (4.2) PENG GAO where D ( p ) = (cid:16) − (cid:0) (2 k − k − (cid:1) p (cid:17)(cid:16) ∞ X i =0 p i (2 k − i (2 i )! g ( p i ) + ∞ X i =0 p i +1 (2 k − i +1 (2 i + 1)! 1 g ( p i +1 ) (cid:17) . Note that for i ≥ 2, we have 1 p i (2 k − i (2 i )! g ( p i ) + 1 p i +1 (2 k − i +1 (2 i + 1)! 1 g ( p i +1 ) ≥ . Observe further the estimation that 1 − p ≤ g ( p ) ≤ . (4.3)We then deduce that D ( p ) ≥ (cid:16) − (cid:0) (2 k − k − (cid:1) p (cid:17)(cid:16) (cid:0) (2 k − k − (cid:1) pg ( p ) + (2 k − p g ( p ) (cid:17) ≥ (cid:16) − (cid:0) (2 k − k − (cid:1) p (cid:17)(cid:16) (cid:0) (2 k − k − (cid:1) p + (2 k − p (cid:17) ≥ − (cid:0) (2 k − k − (cid:1) p − (cid:16) − (cid:0) (2 k − k − (cid:1) p (cid:17) (1 − k ) p ≥ − p − p , where the last estimation above follows by noting that1 − (cid:0) (2 k − k − (cid:1) p = 1 + 1 − k p ≤ . We further note that we have (2 k − / k − < < k < / − log(1 + x ) > − x for all x > Y p ∈ P j (cid:16) − (cid:0) (2 k − k − (cid:1) p (cid:17) − ≥ exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) . We then deduce from the above estimations that the left side expression in (4.2) is ≥ exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) Y p ∈ P j (1 − p ) . (4.4)On the other hand, using Rankin’s trick by noticing that 2 Ω( n ) − ℓ ≥ n ) > ℓ , we see that the error introducedthis way does not exceed X n j p n j ( n j ) | k − | Ω( n j ) w ( n j ) 2 Ω( n j ) − ℓ j g ( n j ) ≤ − ℓ j Y p ∈ P j (cid:16) ∞ X i =1 p i (2 k − i i (2 i )! 1 g ( p i ) + ∞ X i =0 p i +1 | k − | i +1 i +1 (2 i + 1)! 1 g ( p i +1 ) (cid:17) ≤ − ℓ j Y p ∈ P j (cid:16) ∞ X i =1 p i (2 k − i i (2 i )! + ∞ X i =0 p i +1 | k − | i +1 i +1 (2 i + 1)! (cid:17) ≤ − ℓ j exp (cid:16)(cid:0) k − + 2(1 − k ) (cid:1) X p ∈ P j p + X p ∈ P j R ( p ) (cid:17) , HARP LOWER BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET L -FUNCTIONS 7 where R ( p ) = ∞ X i =2 p i (2 k − i i (2 i )! + ∞ X i =1 p i +1 | k − | i +1 i +1 (2 i + 1)! ≤ p ∞ X i =2 i (2 i )! + 2 p ∞ X i =1 i (2 i + 1)! ≤ p ∞ X i =1 i (2 i )! ≤ p ∞ X i =1 i i ! ≤ e p . It follows that the error is hence ≤ − ℓ j exp (cid:16)(cid:0) k − + 2(1 − k ) (cid:1) X p ∈ P j p + X n ≥ e n (cid:17) ≤ − ℓ j exp( ( eπ ) (cid:16)(cid:0) 32 (2 k − + 3(1 − k ) (cid:1) X p ∈ P j p (cid:17) × exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) . (4.5)We notice that the expression given in (4.4) satisfies ≥ exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) Y p (1 − p ) = 6 π exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) . (4.6)We may take N large enough so that by Lemma 3.1, we have that for all 1 ≤ j ≤ R ,12 N ℓ j ≤ X p ∈ P j p ≤ N ℓ j . (4.7)We may also take M large enough to ensure that every ℓ j , ≤ j ≤ R is large. We then deduce from (4.5) and (4.6)that the error introduced above is ≤ − ℓ j / exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) Y p ∈ P j (1 − p ) . (4.8)Combining (4.4) and (4.8), we see that the sum over n j for each j , 1 ≤ j ≤ R in the expression of S is ≥ (1 − − ℓ j / ) exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) Y p ∈ P j (1 − p ) . It follows from this that we have S ≥ X log X R Y j =1 (cid:16) (1 − − ℓ j / ) exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17) Y p ∈ P j (1 − p ) (cid:17) ≥ X log X Y p (1 − p ) R Y j =1 (cid:16) − − ℓ j / (cid:17) R Y j =1 exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17)(cid:17) ≥ ζ (2) X log X (cid:16) − R X j =1 − ℓ j / (cid:17) R Y j =1 exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P j p (cid:17)(cid:17) , (4.9)where the last estimation above follows by noting that Q Ri =1 (1 − x i ) ≥ − P Ri =1 x i for positive real numbers x i satisfying P Ri =1 x i ≤ S by writing log (cid:0) Q Ri =1 ( n i ) (cid:1) as a sum of logarithms of primes dividing Q Ri =1 ( n i ) to see that S ≤ X X q ∈ S P j X l ≥ (cid:16) log qq l +1 (1 − k ) l +1 (2 l + 1)! 1 g ( q l +1 ) (cid:17) R Y i =1 (cid:16) X ( n i ,q )=1 p n i ( n i ) (2 k − Ω( n i ) w ( n i ) e b i,l ( n i ) 1 g ( n i ) (cid:17) , (4.10)where we define e b i,l ( n i ) = b i ( n i q l ) for the unique index i (1 ≤ i ≤ R ) such that b i ( q ) = 0 and e b i,l ( n i ) = b i ( n i ) otherwise. PENG GAO As above, if we remove the restriction of e b i,l on Ω( n i ), then the sum over n i becomes Y p ∈ P i ( p,q )=1 (cid:16) ∞ X m =0 p m (2 k − m (2 m )! g ( p m ) + ∞ X m =0 p m +1 (2 k − m +1 (2 m + 1)! 1 g ( p m +1 ) (cid:17) . (4.11)Note that for m ≥ 1, we have 1 p m (2 k − m (2 m )! g ( p m ) + 1 p m (2 k − m − (2 m − g ( p m − ) ≤ . It follows from this that the expression in (4.11) is ≤ Y p ∈ P i ( p,q )=1 (cid:16) (cid:0) (2 k − k − (cid:1) pg ( p ) (cid:17) ≤ exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i ( p,q )=1 pg ( p ) (cid:17) , (4.12)where the last estimation above follows from the observation that 1 + x ≤ e x for any real x.Using further the estimation given in (4.3), we deduce from (4.12) that the expression in (4.11) is ≤ exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i ( p,q )=1 p − (cid:0) (2 k − k − (cid:1) X p ∈ P i p (cid:17) ≤ exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i ( p,q )=1 p + 12 X p ∈ P i p (cid:17) . Similar to our discussions above, we see that the error introduced in this process is ≤ − ℓ j / exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i ( p,q )=1 p (cid:17) . We deduce from this that R Y i =1 (cid:16) X ( n i ,q )=1 p n i ( n i ) (2 k − Ω( n i ) w ( n i ) e b i,l ( n i ) 1 g ( n i ) (cid:17) ≤ R Y i =1 (cid:0) − ℓ j / (cid:1) exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i ( p,q )=1 p + 12 X p ∈ P i p (cid:17) ≤ A × R Y i =1 exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i p (cid:17) , where A ≤ e is a constant.It follows from this and (4.10) that S ≤ AX R Y i =1 exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i p (cid:17) × X q ∈ S P j X l ≥ (cid:16) log qq l +1 (1 − k ) l +1 (2 l + 1)! 1 g ( q l +1 ) (cid:17) ≤ AX R Y i =1 exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i p (cid:17) × X q ∈ S P j (cid:16) log qq (cid:17) ≤ AX R Y i =1 exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i p (cid:17) × (cid:16) log X M + O (1) (cid:17) , (4.13)where the last estimation above follows from Lemma 3.1.Combining (4.9) and (4.13), we deduce that, by taking M large enough, S − S ≫ X log X R Y i =1 exp (cid:16)(cid:0) (2 k − k − (cid:1) X p ∈ P i p (cid:17) . Applying Lemma 3.1 again, we see that the proof of the proposition now follows from above. HARP LOWER BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET L -FUNCTIONS 9 Proof of Proposition 2.2 It follows from Lemma 3.4 that we have X ∗ ( d, N ( d, k − − k )1 − k N ( d, − k ) Φ( dX ) ≤ X ∗ ( d, (cid:16) R Y j =1 (cid:16) N j ( d, k ) (cid:16) O (cid:0) e − ℓ j (cid:1)(cid:17) + Q j ( d ) (cid:17)(cid:17) Φ( dX ) . (5.1)We now use the notations in Section 4 to write for 1 ≤ j ≤ R , N j ( d, k ) = X n j √ n j (2 k ) Ω( n j ) w ( n j ) b j ( n j ) χ d ( n j ) , P j ( d ) r k ℓ j = X Ω( n j )=2 r k ℓ j p | n j = ⇒ p ∈ P j √ n j (2 r k ℓ j )! w ( n j ) χ d ( n j ) , where r k = 1 + (2 − k ) / (1 − k ).As ( ne ) n ≤ n ! ≤ n ( ne ) n , (5.2)we deduce that (cid:16) ℓ j (cid:17) r k ℓ j (2 r k ℓ j )! ≤ r k ℓ j (cid:16) r k e (cid:17) r k ℓ j . It follows from this that we can write N j ( d, k ) (cid:16) O (cid:0) e − ℓ j (cid:1)(cid:17) + Q j ( d ) as a Dirichlet polynomial of the form D j ( d ) = X n j ≤ X rk/ℓj a n j √ n j χ d ( n j )where for some constant B ( k ) depending on k only, | a n j | ≤ B ( k ) ℓ j . We then apply Lemma 3.2 to evaluate the right side expression in (5.1) above and deduce from estimations in (2.3)that for large M, N , the contribution arising from the error term in (3.1) is ≪ B ( k ) P Rj =1 ℓ j X / ε X r k P Rj =1 1 ℓj ≪ B ( k ) Rℓ X / ε X rkℓR ≪ X − ε . We may thus focus on the main term contributions, which implies that the right side expression of (5.1) is ≪ X × X (cid:16) square term in the expansion of R Y j =1 D j ( d ) × a corresponding factor (cid:17) = X × R Y j =1 X (cid:16) square term in D j ( d ) × a corresponding factor (cid:17) , (5.3)where by “a square term” in a Dirichlet polynomial X n a n n s we mean a term corresponding to n = (cid:3) anda corresponding factor = Y p | n (cid:16) pp + 1 (cid:17) . We first note that X square term in N j ( d, k ) × a corresponding factor = X n j = (cid:3) √ n j (2 k ) Ω( n j ) w ( n j ) b j ( n j ) Y p | n j (cid:16) pp + 1 (cid:17) ≤ Y p ∈ P j (cid:16) (cid:0) k (cid:1) p (cid:16) pp + 1 (cid:17) + X i ≥ (2 k ) i p i (2 i )! (cid:16) pp + 1 (cid:17)(cid:17) ≤ Y p ∈ P j (cid:16) (cid:0) k (cid:1) p (cid:16) pp + 1 (cid:17) + 1 p (cid:17) ≤ exp (cid:16) (cid:0) k (cid:1) X p ∈ P j p + X p ∈ P j p (cid:17) , (5.4)where we apply the relation that 1 + x ≤ e x for any real x again to obtain the last estimation above. Next, we notice that X square term in Q j ( d ) × a corresponding factor ≪ (cid:16) ℓ j (cid:17) r k ℓ j (2 r k ℓ j )! X n j = (cid:3) Ω( n j )=2 r k ℓ j p | n j = ⇒ p ∈ P j √ n j w ( n j ) Y p | n j (cid:16) pp + 1 (cid:17) ≪ (cid:16) ℓ j (cid:17) r k ℓ j (2 r k ℓ j )!( r k ℓ j )! (cid:16) X p ∈ P j p (cid:17) r k ℓ j . We apply (4.7) and (5.2) to estimate the right side expression above to see that for some constant B ( k ) depending on k only, X square term in Q j ( d ) × a corresponding factor ≪ B ( k ) ℓ j e − r k ℓ j log( r k ℓ j ) (cid:16) X p ∈ P j p (cid:17) r k ℓ j ≪ B ( k ) ℓ j e − r k ℓ j log( r k ℓ j ) e r k ℓ j log(2 ℓ j /N ) ≪ e − ℓ j exp (cid:16) (2 k ) X p ∈ P j p (cid:17) . 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