Moments of central values of quartic Dirichlet L-functions
aa r X i v : . [ m a t h . N T ] F e b MOMENTS OF CENTRAL VALUES OF QUARTIC DIRICHLET L -FUNCTIONS PENG GAO AND LIANGYI ZHAO
Abstract.
In this paper, we study moments of the central values of quartic Dirichlet L -functions and establish quanti-tative non-vanishing result for these L -values. Mathematics Subject Classification (2010) : 11A15, 11L05, 11M06, 11N37
Keywords : quartic Dirichlet character, quartic large sieve, moments of L -functions1. Introduction
Moments of L -functions at the central point over a family of characters of a fixed order have been extensively studiedin the literature. The first and second moments of quadratic Dirichlet L -functions were evaluated by M. Jutila [15].The error term for the first moment in [15] was improved in [11, 21, 22]. Asymptotic formulas for the second and thirdmoments for the same family with power savings in the error terms were obtained by K. Soundararajan [20]. The errorterm for the third moment was subsequently improved in [5] and [23]. In the number field case, the authors studied in[10] the first and second moments of quadratic Hecke L -functions in Q ( i ) and Q ( ω ), where ω = − √ i . In [17], W.Luo considered the first two moments of cubic Hecke L -functions in Q ( ω ). Moments of various families of higher orderHecke L -functions were studied in [2, 4, 6, 7], using the method of double Dirichlet series.Although many results are available on moments of quadratic Dirichlet L -functions, less is known for the momentsof higher order Dirichlet L -functions. For the family of cubic Dirichlet L -functions, this is investigated by S. Baierand M. P. Young in [1] and they also mentioned that it is plausible to extend their methods to study moments ofquartic and sextic Dirichlet L -functions. Motivated by this, it is our goal in this paper to investigate the quartic case.Writing χ for the principal character and letting w : (0 , ∞ ) → R be any smooth, compactly supported function whoseFourier transform is denoted by b w , we begin with a result that establishes an asymptotic expression for the first moment. Theorem 1.1.
With notations as above and assuming the truth of the Lindel¨of hypothesis, we have X ( q, X ∗ χ mod qχ = χ L ( , χ ) w (cid:18) qQ (cid:19) = CQ b w (0) + O ( Q / ε ) , where C is a positive explicit constant given in (3.5) and the asterisk on the sum over χ restricts the sum to primitivecharacters χ such that χ remains primitive. It is a conjecture that goes back to S. Chowla [3] that L (1 / , χ ) = 0 for all primitive Dirichlet characters χ . As aconsequence of Theorem 1.1, a partial answer can be given for this conjecture for the character under our consideration.We argue similar to the proof of [1, Corollary 1.2], using H¨older’s inequality together with the well-known bound forthe eighth moment of Dirichlet L -functions and the lower bound implied by the asymptotic formula in Theorem 1.1, toobtain following non-vanishing result on the central values of quartic Dirichlet L -functions. Corollary 1.2.
Assume that the Lindel¨of hypothesis is true. There exist infinitely many primitive Dirichlet characters χ of order such that L (1 / , χ ) = 0 . More precisely, the number of such characters with conductor ≤ Q is ≫ Q / − ε . For m ∈ Z , n ∈ Z [ i ] , ( n,
2) = 1, let ψ m (( n )) = (cid:0) mn (cid:1) where (cid:0) ·· (cid:1) is the quartic residue symbol in Q ( i ) defined inSection 2.1. It is also shown there that ψ m is a quartic Hecke character of trivial infinite type modulo 16 m . Our nextresult concerns bounds on the second moment. Theorem 1.3.
We have for any Q ≥ that (1.1) X q ≤ Q X ∗ χ mod qχ = χ (cid:12)(cid:12) L ( + it, χ ) (cid:12)(cid:12) ≪ Q / ε (1 + | t | ) / ε . Further denote for any rational integer m , (1.2) L ( s, ψ m ) = X n ∈ Z [ i ] n ≡ i ) ψ m ( n ) N ( n ) − s . Then we have (1.3) X ′ m ≤ M | L (1 / it, ψ m ) | ≪ M / ε (1 + | t | ) / ε , where P ′ means that the sum runs over square-free elements of Z . As the proof of the above result is similar to that of [1, Theorem 1.3] (using the approximate functional equationand then the appropriate version of the large sieve inequality Lemmas 2.9 and 2.10), we shall therefore omit it in thepaper.We point out here that the proof of (1.1) uses a large sieve inequality for quartic Dirichlet characters given in Lemma2.10 by using the term Q / + Q / M in the minimum there. Similarly, the proof of (1.3) follows from recalling thedefinition of C ( M, Q ) in [8, p. 907] and the bound C ( M, Q ) ≪ ( QM ) ε ( Q / + Q / M ) by Lemma 2.10. A variant ofthis for the cubic case plays a key role in the treatment of the error term in the work of Baier and Young [1] on the firstmoment of cubic Dirichlet L -functions. However, our Lemma 2.10 is just short of producing admissible error term in ourfirst moment result, even after incorporating the recursive technique of Baier and Young (see third arXiv version of thepaper [1] arXiv:0804.2233v3). This observation drives us to seek for bounds of individual quartic Dirichlet L -functionsin order to control the error term and leads us to derive our Theorem 1.1 under the Lindel¨of hypothesis since the currentbest known subconvexity bounds for the L -functions under our consideration do not provide a satisfactory error termeither.1.4. Notations.
The following notations and conventions are used throughout the paper. e ( z ) = exp(2 πiz ) = e πiz . f = O ( g ) or f ≪ g means | f | ≤ cg for some unspecified positive constant c . µ [ i ] denotes the M¨obius function on Z [ i ]. ζ Q ( i )( s ) is the Dedekind zeta function for the field Q ( i ).2. Preliminaries
In this section we provide various tools used throughout the paper.2.1.
Quartic symbol and primitive quartic Dirichlet characters.
The symbol ( (cid:0) · n (cid:1) ) is the quartic residue sym-bol in the ring Z [ i ]. For a prime ̟ ∈ Z [ i ] with N ( ̟ ) = 2, the quartic residue symbol is defined for a ∈ Z [ i ], ( a, ̟ ) = 1by (cid:0) a̟ (cid:1) ≡ a ( N ( ̟ ) − / (mod ̟ ), with (cid:0) a̟ (cid:1) ∈ {± , ± i } . When ̟ | a , we define (cid:0) a̟ (cid:1) = 0. Then the quartic residuesymbol can be extended to any composite n with ( N ( n ) ,
2) = 1 multiplicatively. We further set (cid:0) · n (cid:1) = 1 when n is aunit in Z [ i ]. We also define the quadratic residue symbol (cid:0) · n (cid:1) so that (cid:0) · n (cid:1) = (cid:0) · n (cid:1) .Note that in Z [ i ], every ideal co-prime to 2 has a unique generator congruent to 1 modulo (1 + i ) . Such a generator iscalled primary. Observe that n = a + bi, a, b ∈ Z in Z [ i ] is congruent to 1 mod (1 + i ) if and only if a ≡ , b ≡ a ≡ , b ≡ a ≡ ( − N ( n ) − (mod 4) , b ≡ − ( − N ( n ) − (mod 4) . (2.1)Recall that [16, Theorem 6.9] the quartic reciprocity law states that for two primary integers m, n ∈ Z [ i ], (cid:16) mn (cid:17) = (cid:16) nm (cid:17) ( − (( N ( n ) − / N ( m ) − / . Also, the supplement laws to the quartic reciprocity law states that for n = a + bi being primary, (cid:18) in (cid:19) = i (1 − a ) / and (cid:18) in (cid:19) = i ( a − b − − b ) / . (2.2)This implies that for any n ≡ (cid:18) in (cid:19) = (cid:18) in (cid:19) = 1 . OMENTS OF CENTRAL VALUES OF QUARTIC DIRICHLET L -FUNCTIONS 3 Hence the functions n → (cid:0) in (cid:1) and n → (cid:0) in (cid:1) can be regarded as Hecke characters (mod 16) of trivial infinite type.From this we see that ψ m is a quartic Hecke character of trivial infinite type modulo 16 m for every m ∈ Z .Using the quartic residue symbols, we have the following classification of all the primitive quartic Dirichlet charactersof conductor q co-prime to 2: Lemma 2.2.
The primitive quartic Dirichlet characters of conductor q coprime to such that their squares remainprimitive are of the form χ n : m (cid:0) mn (cid:1) for some n ∈ Z [ i ] , n ≡ i ) ) , n square-free and not divisible byany rational primes, with norm N ( n ) = q . The above result is given in [8, Section 2.2], but the correspondence given there is not exact. We take this opportunityto thank Francesca Balestrieri and Nick Rome for some helpful discussions on this topic.2.3.
The Gauss sums.
For any n ∈ Z [ i ], ( n,
2) = 1, the quartic Gauss sum g ( n ) is defined by g ( n ) = X x mod n (cid:16) xn (cid:17) e e (cid:16) xn (cid:17) , where e e ( z ) = exp (cid:0) πi ( z i − z i ) (cid:1) .The following well-known formula (see [19, p. 195]) holds for all n : | g ( n ) | = (p N ( n ) if n is square-free , . (2.3)More generally, we define for n, k ∈ Z [ i ] , ( n,
2) = 1, g ( k, n ) = X x mod n (cid:16) xn (cid:17) e e (cid:18) kxn (cid:19) . We note two properties of g ( r, n ) that can be found in [4]: g ( rs, n ) = (cid:16) sn (cid:17) g ( r, n ) , ( s, n ) = 1 , (2.4) g ( r, n n ) = (cid:18) n n (cid:19) (cid:18) n n (cid:19) g ( r, n ) g ( r, n ) , ( n , n ) = 1 . (2.5)We further define the Gauss sum τ ( χ ) associated to a primitive Dirichlet character χ (mod q ) by τ ( χ ) = X ≤ x ≤ q χ ( x ) e (cid:18) xq (cid:19) . When χ is a primitive quartic Dirichlet character identified with χ n by Lemma 2.2, we have ([8, p. 894])(2.6) τ ( χ n ) = (cid:18) nn (cid:19) g ( n ) . Here we have n ≡ i ) ) and n free of rational prime divisors. If we write n = a + bi with a, b ∈ Z , thenwe deduce from this that ( a, b ) = 1 so that (cid:18) nn (cid:19) = (cid:18) a − bia + bi (cid:19) = (cid:18) aa + bi (cid:19) = − N ( n ) − a + bi ! ( − N ( n ) − aa + bi ! . (2.7)As ( − N ( n ) − a is primary according to (2.1), we have by the quartic reciprocity law, ( − N ( n ) − aa + bi ! = ( − (( N ( a ) − / N ( n ) − / (cid:18) a + bia (cid:19) = (cid:18) a + bia (cid:19) = (cid:18) bia (cid:19) = (cid:18) ba (cid:19) (cid:18) ia (cid:19) = (cid:18) ia (cid:19) , (2.8)where the last equality follows from [12, Proposition 9.8.5], which states that for a, b ∈ Z , ( a, b ) = 1 , (cid:18) ba (cid:19) = 1 . It follows from (2.2) that (cid:18) ia (cid:19) = i (1 − ( − N ( n ) − a ) / = ( − a − . (2.9) PENG GAO AND LIANGYI ZHAO
On the other hand, note that by the definition that ( − N ( n ) − a + bi ! = ( − N ( n ) − · N ( n ) − = ( − N ( n ) − = (cid:18) − n (cid:19) . (2.10)We then deduce that, putting together (2.7), (2.8), (2.9) and (2.10), (cid:18) nn (cid:19) = (cid:18) − n (cid:19) ( − a − = (cid:0) − in (cid:1) = (cid:16) ( − i ) n (cid:17) if (cid:0) − n (cid:1) = 1 ,i − (cid:0) − in (cid:1) = i − (cid:16) ( − i ) n (cid:17) if (cid:0) − n (cid:1) = − . Thus from this and (2.6), τ ( χ n ) = (cid:16) ( − i ) n (cid:17) g ( n ) if (cid:0) − n (cid:1) = 1 ,i − (cid:16) ( − i ) n (cid:17) g ( n ) if (cid:0) − n (cid:1) = − . (2.11)2.4. The approximate functional equation.
Let G ( s ) be any even function which is holomorphic and bounded inthe strip − < ℜ ( s ) < G (0) = 1. It follows from [13, Theorem 5.3] that we have the following approximatefunctional equation for Dirichlet L -functions. Proposition 2.5.
Let χ be a primitive Dirichlet character of conductor q . For any α ∈ C , j ∈ {± } , let (2.12) a j = 1 − j , ǫ ( χ ) = i − a χ ( − q − / τ ( χ ) , X α,j = (cid:16) qπ (cid:17) − α Γ (cid:16) / a j − α (cid:17) Γ (cid:16) / a j + α (cid:17) . We define V α,j ( x ) = 12 πi Z (2) G ( s ) s γ α,j ( s ) x − s d s, where γ α,j ( s ) = π − s/ Γ (cid:16) / a j + α + s (cid:17) Γ (cid:16) / a j + α (cid:17) . Furthermore, let A and B be positive real numbers such that AB = q . Then for any |ℜ ( α ) | < / we have L (1 / α, χ ) = ∞ X m =1 χ ( m ) m / α V α,χ ( − (cid:16) mA (cid:17) + ǫ ( χ ) X α,χ ( − ∞ X m =1 χ ( m ) m / − α V − α,χ ( − (cid:16) mB (cid:17) . For α = 0 we set γ = γ , , γ − = γ , − , V = V , , V − = V , − .With a suitable G ( s ) (for example G ( s ) = e s ), we have for any c > V α,j ( ξ ) ≪ (cid:18) ξ | α | (cid:19) − c . On the other hand, when G ( s ) = 1, we have (see [20, Lemma 2.1]) that V ± ( ξ ) is real-valued and smooth on [0 , ∞ )and for the j -th derivative of V ± ( ξ ),(2.13) V ± ( ξ ) = 1 + O ( ξ / − ǫ ) for 0 < ξ < V ( j ) ± ( ξ ) = O ( e − ξ ) for ξ > , j ≥ . Analytic behavior of Dirichlet series associated with Gauss sums.
For any Hecke character χ (mod 16)of trivial infinite type, we let h ( r, s ; χ ) = X ( n,r )=1 n ≡ i ) χ ( n ) g ( r, n ) N ( n ) s . The following lemma gives the analytic behavior of h ( r, s ; χ ) on ℜ ( s ) > Lemma 2.7. [9, Lemma 2.5]
The function h ( r, s ; χ ) has meromorphic continuation to the entire complex plane. It isholomorphic in the region σ = ℜ ( s ) > except possibly for a pole at s = 5 / . For any ε > , letting σ = 3 / ε , thenfor σ ≥ σ ≥ σ − / , | s − / | > / , we have for t = ℑ ( s ) , h ( r, s ; χ ) ≪ N ( r ) ( σ − σ ) / ε (1 + t ) σ − σ ) / ε . OMENTS OF CENTRAL VALUES OF QUARTIC DIRICHLET L -FUNCTIONS 5 Moreover, the residue satisfies
Res s =5 / h ( r, s ; χ ) ≪ N ( r ) / ε . The large sieve with quartic Dirichlet characters.
The following large sieve inequality for quadratic residuesymbols is needed in the proof.
Lemma 2.9. [18, Theorem 1]
Let
M, N be positive integers, and let ( a n ) n ∈ N be an arbitrary sequence of complexnumbers, where n runs over Z [ i ] . Then we have for any ε > , X ′ m ∈ Z [ i ]( m, N ( m ) ≤ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ′ n ∈ Z [ i ]( n, N ( n ) ≤ N a n (cid:16) nm (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ε ( M N ) ε ( M + N ) X N ( n ) ≤ N | a n | , where P ′ means that the sum runs over square-free elements of Z [ i ] and ( · m ) is the quadratic residue symbol. Besides the notation C ( M, Q ) mentioned in the Introduction, we also recall the norm C ( M, Q ) defined in [8, p.907]. Notice that for square-free n ∈ Z [ i ] satisfying ( n,
2) = 1, the square of the quartic symbol (cid:0) · n (cid:1) becomes thequadratic residue symbol (cid:0) · n (cid:1) . Thus, using Lemma 2.9 in [8, (39)] and proceed to the estimate [8, (40)], we see thatthe estimation given in [8, (24)] can be replaced by(2.14) C ( M, Q ) ≪ ( QM ) ε (cid:16) M + Q / (cid:17) , As we have by [8, (23)] that C ( M, Q ) ≤ C ( M, Q ), we see that the bound given in (2.14) is also valid for C ( M, Q ),which leads to the following large sieve inequality for quartic Dirichlet characters.
Lemma 2.10.
Let ( a m ) m ∈ N be an arbitrary sequence of complex numbers. Then X Q We start by setting M := X ( q, X ∗ χ mod qχ = χ L ( , χ ) w (cid:18) qQ (cid:19) . Applying Proposition 2.5 (the approximate functional equation) with G ( s ) = 1 , A q B = q yields M = M + M ,where M = X ( q, X ∗ χ mod qχ = χ ∞ X m =1 χ ( m ) √ m V χ ( − (cid:18) mA q (cid:19) w (cid:18) qQ (cid:19) , M = X ( q, X ∗ χ mod qχ = χ ǫ ( χ ) ∞ X m =1 χ ( m ) √ m V χ ( − (cid:16) mB (cid:17) w (cid:18) qQ (cid:19) . For each primitive quartic Dirichlet character χ whose square remains primitive, we write it as χ n via the correspon-dence given in Lemma 2.2. It then follows from (2.6), (2.12) and this correspondence that ǫ ( χ n ) = ǫ ( χ ) = i − a χn ( − N ( n ) − / (cid:16) ¯ nn (cid:17) g ( n ) . PENG GAO AND LIANGYI ZHAO The above allows us to further decompose M and M as M = M +1 + M − and M = M +2 + M − , where M ± = X ′′ n ≡ i ) ± χ n ( − ∞ X m =1 χ n ( m ) √ m V χ n ( − (cid:18) mA n (cid:19) w (cid:18) N ( n ) Q (cid:19) , M ± = X ′′ n ≡ i ) ± χ n ( − ǫ ( χ n ) ∞ X m =1 χ n ( m ) √ m V χ n ( − (cid:16) mB (cid:17) w (cid:18) N ( n ) Q (cid:19) . Here P ′′ means that the sum runs over square-free elements n ∈ Z [ i ] that have no rational prime divisors. Also, theparameters A n , B sastify that A n B = N ( n ). We now introduce another parameter A defined by AB = Q and we notethat we have A n = AN ( n ) /Q ≍ A for all n under consideration because of the compact support of w .It remains to evaluate M ± and M ± . As the arguments are similar, we will only evaluate M +1 and M +2 in whatfollows. We summarize the results in the following lemma. Lemma 3.1. We have M ± = 12 CQ e w (1) + O (cid:16) Q / ε A / ε + QA − / ε (cid:17) , (3.1) M ± ≪ Q / ε B / ε , (3.2) where the constant C is given explicitly in (3.5) . Theorem 1.1 follows from Lemma 3.1 by setting B = Q / and A = Q / in Lemma 3.1. Thus the remainder of thepaper is devoted to the proof of this lemma.3.2. Evaluating M +1 , the main term. We write µ for M¨obius function and we define µ Z ( d ) = µ ( | d | ) for d ∈ Z . Thenfor any n ≡ i ) ), we note the following relation(3.3) X d | n,d ∈ Z d ≡ µ Z ( d ) = ( , n has no rational prime divisors , , otherwise . We apply the above and change variables n → dn to the sum over n . Note that any square-free d ∈ Z , d ≡ Z [ i ] and this implies that the condition that dn is square-free thenis equivalent to n being square-free and ( d, n ) = 1. We then deduce that M +1 = M +1 , + M +1 , , with M +1 , = 12 X d ∈ Z d ≡ µ Z ( d ) ∞ X m =1 (cid:0) md (cid:1) √ m X ′ n ≡ i ) ( n,d )=1 (cid:16) mn (cid:17) V (cid:18) mA QN ( nd ) (cid:19) w (cid:18) N ( nd ) Q (cid:19) , M +1 , = 12 X d ∈ Z d ≡ µ Z ( d ) ∞ X m =1 (cid:0) − md (cid:1) √ m X ′ n ≡ i ) ( n,d )=1 (cid:18) − mn (cid:19) V (cid:18) mA QN ( nd ) (cid:19) w (cid:18) N ( nd ) Q (cid:19) , where, as before, P ′ denotes a sum over the square-free elements of Z [ i ].We first evaluate M +1 , by using M¨obius inversion to detect the square-free condition that n . So M +1 , = 12 X d ∈ Z d ≡ µ Z ( d ) X l ≡ i ) ( l,d )=1 µ [ i ] ( l ) ∞ X m =1 (cid:0) mdl (cid:1) √ m M ( d, l, m ) , where M ( d, l, m ) = X n ≡ i ) ( n,d )=1 (cid:16) mn (cid:17) V (cid:18) mA QN ( ndl ) (cid:19) w (cid:18) N ( ndl ) Q (cid:19) . Now Mellin inversion gives V (cid:18) mA QN ( ndl ) (cid:19) w (cid:18) N ( ndl ) Q (cid:19) = 12 πi Z (2) (cid:18) QN ( ndl ) (cid:19) s e f ( s )d s, OMENTS OF CENTRAL VALUES OF QUARTIC DIRICHLET L -FUNCTIONS 7 where e f ( s ) = ∞ Z V (cid:16) mA x − (cid:17) w ( x ) x s − d x. Integration by parts and using (2.13) shows that e f ( s ) is a function satisfying the bound e f ( s ) ≪ (1 + | s | ) − E (1 + m/A ) − E , for any ℜ ( s ) > E > M ( d, l, m ) = 12 πi Z (2) (cid:18) QN ( dl ) (cid:19) s L ( s, ψ md ) e f ( s )d s, with L ( s, ψ md ) given in (1.2).We estimate M +1 , by moving the contour to the line ℜ s = 1 / 2. Observe that the Hecke L -function has a pole at s = 1 when m is a fourth power. We write M for the contribution to M +1 , from these residues and M ′ for theremainder.We evaluate M by noting that M = 12 X d ∈ Z d ≡ µ Z ( d ) X l ≡ i ) ( l,d )=1 µ [ i ] ( l ) ∞ X m =1 (cid:0) mdl (cid:1) √ m QN ( dl ) e f (1) Res s =1 L ( s, ψ md ) , where using the Mellin convolution formula shows that(3.4) e f (1) = ∞ Z V (cid:16) mA x − (cid:17) w ( x )d x = 12 πi Z (2) (cid:18) Am (cid:19) s e w (1 + s ) G ( s ) s γ ( s )d s, with e w ( s ) = ∞ Z w ( x ) x s − d x. From the discussions in Section 2.1, we see that ψ md is the principal character only when m is a fourth power, in whichcase L ( s, ψ md ) = ζ Q ( i ) ( s ) Y ̟ | dm (cid:0) − N ( ̟ ) − s (cid:1) , where ̟ denotes a prime in Z [ i ] in this section.Let C = π , the residue of ζ Q ( i ) ( s ) at s = 1, so that M = 12 C Q ∞ X m =1 e f (1) m Y ̟ | m (cid:0) − N ( ̟ ) − (cid:1) X d ∈ Z , ( d,m )=1 d ≡ µ Z ( d ) d Y ̟ | d (cid:0) − N ( ̟ ) − (cid:1) X ( l,md )=1 l ≡ i ) µ [ i ] ( l ) N ( l ) . Computing the sum over l explicitly, we obtain that M = 12 C ζ − Q ( i ) (2) Q ∞ X m =1 e f (1) m Y ̟ | m (cid:0) − N ( ̟ ) − (cid:1) X d ∈ Z , ( d,m )=1 d ≡ µ Z ( d ) d Y ̟ | d (cid:0) − N ( ̟ ) − (cid:1) Y ̟ | md (cid:0) − N ( ̟ ) − (cid:1) − = 12 C ζ − Q ( i ) (2) Q ∞ X m =1 e f (1) m Y ̟ | m (cid:0) N ( ̟ ) − (cid:1) − X d ∈ Z , ( d,m )=1 d ≡ µ Z ( d ) d Y ̟ | d (cid:0) N ( ̟ ) − (cid:1) − . We define C = X d ∈ Z d ≡ µ Z ( d ) d Y ̟ | d (cid:0) N ( ̟ ) − (cid:1) − . PENG GAO AND LIANGYI ZHAO It is clear that C is a constant. Using this, we have that M = 12 C C ζ − Q ( i ) (2) Q ∞ X m =1 e f (1) m Y ̟ | m (cid:0) N ( ̟ ) − (cid:1) − Y p | m/ ( m, − p − Y ̟ | p (1 + N ( ̟ ) − ) − − . Let Z ( u ) = ∞ X m =1 m − u Y ̟ | m (cid:0) N ( ̟ ) − (cid:1) − Y p | m/ ( m, − p − Y ̟ | p (1 + N ( ̟ ) − ) − − , which is holomorphic and bounded for ℜ ( u ) ≥ δ > 1. Then M = 12 C C ζ − Q ( i ) (2) Q πi Z (1) A s Z (2 + 4 s ) e w (1 + s ) G ( s ) s γ ( s )d s. We move the contour of integration to − / ε , crossing a pole at s = 0 only. The integral over the new contour is O ( A − / ε Q ), while residue of the pole at s = 0 gives(3.5) 12 CQ e w (1) , where C = C C ζ − Q ( i ) (2) Z (2) . Note that Z ( u ) converges absolutely at u = 2 so that one may express Z (2) explicitly as an Euler product if interested.We then conclude that M = 12 CQ e w (1) + O (cid:16) QA − / ε (cid:17) . (3.6)3.3. Evaluating M +1 , the remainder term. In this section, we estimate M ′ and M +1 , . Since the arguments aresimilar, we shall only estimate M ′ here. We bound everything with absolute values to see that for some large E ∈ N , |M ′ | ≪ X d ≤ c √ Q X N ( l ) ≤ c √ Q p N ( dl ) X m √ Q √ m (1 + m/A ) − E ∞ Z | L (1 / it, ψ m ) | (1 + | t | ) − E d t. Here c and c are constants, chosen according to the size of the support the weight function w . In view of the factor(1 + m/A ) − E , we may truncate the sum over m above to m ≤ M ≪ A ε for ε > ε > | L (1 / it, ψ m ) | ≪ ( | m | (1 + | t | )) ǫ . We apply this to bound the sum over m as X m ≤ M √ m | L (1 / it, ψ m ) | ≪ M / ε (1 + | t | ) ε . Since d ∈ Z , we have N ( d ) = d so that summing trivially over d and l , we obtain |M ′ | + |M +1 , | ≪ Q / ε A / ε . This combined with (3.6) gives (3.1).3.4. Estimating M +2 . Using (2.11), we have M +2 = M +2 , + M +2 , , where M +2 , = 12 ∞ X m =1 √ m V (cid:16) mB (cid:17) X ′ n ≡ i ) χ n (8 im ) g ( n ) p N ( n ) w (cid:18) N ( n ) Q (cid:19) , M +2 , = 12 ∞ X m =1 √ m V (cid:16) mB (cid:17) X ′ n ≡ i ) χ n ( − im ) g ( n ) p N ( n ) w (cid:18) N ( n ) Q (cid:19) . To estimate the above expressions, we need the following result. Lemma 3.5. For any l ∈ Z [ i ] , we have H ′ ( l, Q ) := X ′ n ∈ Z [ i ] n ≡ i ) χ n ( l ) g ( n ) p N ( n ) w (cid:18) N ( n ) Q (cid:19) ≪ Q / ε N ( l ) / ε . OMENTS OF CENTRAL VALUES OF QUARTIC DIRICHLET L -FUNCTIONS 9 To prove Lemma 3.5, we first use (3.3) to remove the condition that n has no rational prime divisors. Note that itfollows from (2.5) and the quartic reciprocity that for d ∈ Z , n ∈ Z [ i ] , ( dn, 2) = 1 , ( d, n ) = 1 and d primary, g ( dn ) = (cid:16) nd (cid:17) (cid:18) dn (cid:19) g ( d ) g ( n ) = χ n ( d ) g ( d ) g ( n ) . We use the notation e g ( d ) = g ( d ) N ( d ) − / so that | e g ( d ) | ≤ g ( n ) = 0unless n is square-free. This gives that H ′ ( l, Q ) = X d ∈ Z d ≡ µ Z ( d )˜ g ( d ) χ d ( l ) H ( d l, Q/d ) , where H ( d l, X ) = X n ∈ Z [ i ] n ≡ i ) χ n ( d l ) g ( n ) p N ( n ) w (cid:18) N ( n ) X (cid:19) . We estimate H with the next lemma. Lemma 3.6. For any l ∈ Z [ i ] , write l = l l where l is a unit times a power of i , and l ≡ i ) ) . Thenwe have H ( l, X ) ≪ X / ε N ( l ) / + X / N ( l ) / ε . Proof. Writing l = l l as above, we set (cid:18) ln (cid:19) = (cid:18) l n (cid:19) · (cid:18) l n (cid:19) . From the discussion in Section 2.1, the function λ ( n ) = (cid:0) l n (cid:1) is a Hecke character (mod 16) of trivial infinite type.Thus H ( l, X ) = X n ∈ Z [ i ] n ≡ i ) λ ( n ) (cid:0) l n (cid:1) g ( n ) p N ( n ) w (cid:18) N ( n ) X (cid:19) . Note that the identity (2.4) implies (cid:0) l n (cid:1) g ( n ) = g ( l , n ) for ( n, l ) = 1. Upon introducing the Mellin transform of w ,we get(3.7) H ( l, X ) = 12 πi Z (2) e w ( s ) X s h ( l , / s ; λ )d s, where e w ( s ) is defined as in (3.4).We move the line of integration in (3.7) to ℜ ( s ) = 1 / ε , crossing a pole at s = 3 / 4, which contributes ≪ X / N ( l ) / ε . The contribution from the new line of integration is ≪ X / ε N ( l ) / . This completes the proof of Lemma 3.6. (cid:3) Now, to prove Lemma 3.5, we treat | d | ≤ Y and | d | > Y separately, where Y is a parameter to be chosen later. For | d | ≤ Y we use Lemma 3.6, while for | d | > Y we use the trivial bound H ( l, X ) ≪ X . Thus H ′ ( l, Q ) ≪ X | d |≤ Y (cid:18) Qd (cid:19) / ε N ( d l ) / + X | d |≤ Y (cid:18) Qd (cid:19) / N ( d l ) / ε + X | d | >Y Qd , which simplifies as H ′ ( l, Q ) ≪ Q / ε Y − ε N ( l ) / + QY − + Q / N ( l ) / ε Y ε . The optimal choice of Y is Y = Q / N ( l ) − / and gives Lemma 3.5. Applying this lemma and summing trivially over m in the expressions for M +2 , and M +2 , , one easily deduces (3.2). This completes the proof of Lemma 3.1. Acknowledgments. P. G. is supported in part by NSFC grant 11871082 and L. Z. by the FRG grant PS43707 and theFaculty Goldstar Award PS53450. Parts of this work were done when P. G. visited the University of New South Wales(UNSW). He wishes to thank UNSW for the invitation, financial support and warm hospitality during his pleasant stay. References [1] S. Baier and M. P. Young, Mean values with cubic characters , J. 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(2009), no. 1, 73–99.[23] , The third moment of quadratic Dirichlet L-functions , Selecta Math. (N.S.) (2013), no. 2, 509–543. School of Mathematical Sciences School of Mathematics and StatisticsBeihang University University of New South WalesBeijing 100191 China Sydney NSW 2052 AustraliaEmail: [email protected]