Subconvexity for a double Dirichlet series and non-vanishing of L -functions
aa r X i v : . [ m a t h . N T ] J un SUBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHINGOF L -FUNCTIONS ALEXANDER DAHL
Abstract.
We study a double Dirichlet series of the form P d L ( s, χ d χ ) χ ′ ( d ) d − w , where χ and χ ′ are quadratic Dirichlet characters with prime conductors N and M respectively. A functional equationgroup isomorphic to the dihedral group of order 6 continues the function meromorphically to C . Aconvexity bound at the central point is established to be ( MN ) / ε and a subconvexity bound of ( MN ( M + N )) / ε is proven. The developed theory is used to prove an upper bound for the smallestpositive integer d such that L (1 / , χ dN ) does not vanish, and further applications of subconvexitybounds to this problem are presented. Introduction
The use of multiple Dirichlet series in number theoretic problems, as well as the intrinsic structure theypossess, have evolved the subject to be a study in its own right. Indeed, Weyl group multiple Dirichletseries are considered fundamental objects whose structures are intimately linked to their analytic features[BBC + L -functionsthere is the well-known concept of convexity bound as a generic upper bound in the critical strip, andimprovements in the direction of the Lindelöf hypothesis often have deep implications. In the situationof multiple Dirichlet series, the notion of convexity is not so obvious to formalize. Nevertheless there is afairly natural candidate for a “trivial” upper bound of a multiple Dirichlet series, and it is an importantproblem to improve this estimate. This has first been carried out in [Blo11] for the Dirichlet seriesdefined by Z ( s, w ) = ζ (2) (2 s + 2 w − X d odd L (2) ( s, χ d ) d − w at s = 1 / it , w = 1 / iu simultaneously in the t, u -aspect, where the (2) superscript denotesthat we are removing the Euler factor at , and χ d is the Jacobi symbol ( d · ) . Here we study a non-archimedean analogue. We can expand the L -function in the summand as a sum indexed by n . Insteadof twisting by n it , d iu , we twist by quadratic primitive Dirichlet characters χ ( n ) , χ ′ ( d ) of conductors N, M , respectively. More precisely, we consider Z ( s, w ; χ, χ ′ ) := X d ≥ d, MN )=1 L (2 MN ) ( s, χ d χ ) χ ′ ( d ) P ( χ ) d ,d ( s ) d w for sufficiently large ℜ s, ℜ w , where we write d = d d with d squarefree. The P factors are a technicalcomplication necessary in order to construct functional equations, which we accomplish thus: if weexpand the L -function in the numerator and switch the summation, we can relate it to a similar double Mathematics Subject Classification.
Primary: 11M32, Secondary: 11F68. L -FUNCTIONS 2 Dirichlet series with the arguments and the twisting characters interchanged, with modified correctionfactors. Next, we have a functional equation obtained by application of the functional equation forDirichlet L -functions, which maps ( s, w ) to (1 − s, s + w − ) . Applying the switch of summationformula to this, we obtain a functional equation mapping ( s, w ) to ( s + w − , − w ) . These continue Z to the complex plane except for the polar lines s = 1 , w = 1 , and s + w = 3 / . They also generatea group isomorphic to D , the dihedral group of order 6, which is isomorphic to the Weyl group of theroot system of type A . For this reason Z ( s, w ; χ, χ ′ ) is considered a type A multiple Dirichlet series[BBC + GL (3) Eisenstein series of minimal parabolic type on a metaplectic cover of an algebraic group whose rootsystem is the dual root system of A [BBFH07]. In fact, it is conjectured that a similar relationshipis true between multiple Dirichlet series of type A r and GL ( r + 1) Eisenstein series. Going down indimension, type A multiple Dirichlet series can be constructed from 1/2-integral weight Eisenstein series[GH85]. As for the correction polynomials P ( χ ) d ,d ( s ) , their existence, uniqueness, and construction werestudied extensively in [BFH04] and [DGH03] in the more general setting of GL ( r ) multiple Dirichletseries. They are unique in the case of r up to 3.The notion of convexity is no longer canonical in the case of double Dirichlet series, since our initialbounds for Z ( s, w ; χ, χ ′ ) depend on what we know about bounds on their coefficients, and our knowledgehere is only partial. Nonetheless, if we use the Lindelöf hypothesis on average (cf. Theorem 3), thenthrough careful choice of initial bounds and functional equation applications (cf. §3.4), we obtain thebound Z ( , ; χ, χ ′ ) ≪ ( M N ) / ε , which we call the convexity bound. In this work, we present the following subconvexity result. Theorem 1.
For quadratic Dirichlet characters χ and χ ′ of conductors N and M which are prime orunity, and for ε > , we have the bound Z ( , + it ; χ, χ ′ ) ≪ ε (1 + | t | ) /ε ( M N ( M + N )) / ε . We first point out that the t -aspect bound can be drastically improved, but our focus here is onbounds in the moduli aspect. For purposes of comparison with the convexity bound, we point out that,via the geometric-arithmetic mean inequality, we have ( M N ) / ε ≪ ( M N ( M + N )) / ε . This resultis comparable to the subconvexity bound obtained by V. Blomer in the archimedean case, particularly | sw ( s + w ) | / ε for ℜ s = ℜ w = , an improvement upon the convexity bound with 1/4 replaced by1/6.A useful arithmetic application of the theory developed for Z ( s, w ; χ, χ ′ ) is finding a bound for theleast d such that L ( , χ dN ) does not vanish, where N is a large fixed prime. It is expected that allordinates of any zeros of L ( s, χ ) on the critical line are linearly independent over the rationals, so thatin particular it is expected that L ( , χ ) is nonzero for any χ . Random matrix theory provides furtherevidence to this conjecture: The lowest zero in families of L -functions (such as L ( s, χ d ) ) is expected tobe distributed like the “smallest” eigenvalue (i.e., closest to unity) of a certain matrix family (dependingon the family of L -function). Since the corresponding measure vanishes at zero, it suggests that thesmallest zero of a Dirichlet L -function is “repelled” from the real axis. In the case of quadratic twists of GL (2) automorphic forms, in particular twists of elliptic curves, there is a connection to Waldspurger’stheorem [Wal81] which states that L ( , f × χ ) is proportional to the squares of certain Fourier coefficientsof a half-integral weight modular form, uniformly in χ . We focus here on the simpler case of twists ofDirichlet characters. In particular, we have the following theorem which is proven using the theory ofthis double Dirichlet series. UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 3 Theorem 2.
Let N be an odd prime, and let D ( N ) = d denote the smallest positive integer such that L ( , χ dN ) does not vanish. Then we have D ( N ) ≪ N / ε . It is not entirely obvious what should be regarded as the trivial bound in this situation. The mostnatural approach to non-vanishing would be to prove a lower bound for the first moment P d ≍ X L ( , χ dN ) for X as small as possible in terms of N . A straightforward argument produces a main term of size X log X and an error term of size O (( N X ) / ε ) which suggests the trivial bound N ε for the firstnon-vanishing twist. This is perhaps unexpectedly weak, since the same bound holds for degree 2 L -functions L ( , f × χ d ) , where f is an automorphic form of level N . Nevertheless it is not completelyobvious how to improve this in the case of Dirichlet characters χ N . Here we follow a modified version ofa method presented in [HK10]: Let X be a large positive number, and h ( y ) be a smooth non-negativefunction with support on [1 , . By Mellin inversion, we have ˆ (2) ˜ h ( w ) Z ( , w ; χ N , ψ ) X w dw ≈ ∞ X d =1 L ( , χ dN ) h ( d/X ) , where ˜ h denotes the Mellin transform of h , and ψ denotes the trivial character. We move the contourto ℜ w = − ε , picking up a double pole at w = 1 . If we apply a symmetric functional equation (3.15) anda bound resulting from use of the Lindelöf principle on average (3.18) to the resulting integral, then wehave(1.1) ∞ X d =1 L ( , χ dN ) h ( d/X ) ≈ a N X log X + b N X + O ( N / ε ) for some coefficients a N , b N . The idea now is to bound a N from below in terms of N , and choose X sothat the main term is greater than the error term. Then it cannot be that L ( , χ dN ) vanishes for all < d < X on the left-hand side. We prove the bounds a N , b N ≫ N − ε (cf. Theorem 9), and thus we canchoose X = N / ε . The power of the asymptotic formula (1.1) is that a N , b N can be bounded belowby finding an asymptotic for P d ≍ X L ( , χ dN ) , but now we can take X to be very large.Of course, in the previous discussion there are significant details suppressed for brevity, which includethe aforementioned correction factors and the error term arising from truncation of the d -sum. Mostnontrivial, however, are the bounds for a N , b N given by Theorem 9, requiring careful treatment. Thetechniques used are similar to those in [Jut81], in which an asymptotic formula for P The variable ε will always denote a sufficiently small positive number, not necessarilythe same at each occurrence, and the variable A will denote a sufficiently large positive number, notnecessarily the same at each occurrence. The numbers M and N will always denote natural numbersthat are either odd primes or unity, possibly equal. For a real function f , we denote its Mellin transformby ˜ f . The trivial character modulo unity will be denoted by ψ , and the primitive character of conductor4 shall be denoted ψ − . As for the primitive characters modulo 8, we define ψ as the character that isunity at exactly 1 and 7, and we set ψ − = ψ ψ − . If χ is a character, we use the notation C χ to denoteits conductor. 2. Preliminaries Characters. For a positive integer d , we define a character on ( Z / d Z ) ∗ via the Jacobi symbol by χ d ( n ) = e χ n ( d ) = (cid:18) dn (cid:19) . For odd positive integers n and d , we have the following quadratic reciprocity law for the Jacobi-Kronecker symbol (cf. [Mol98], Theorem 4.2.1, page 197). χ d ( n ) = (cid:18) dn (cid:19) = (cid:16) nd (cid:17) ( − d − · n − = (e χ d ( n ) , d ≡ mod 4) ; e χ d ( − n ) = e χ d ( n ) ψ − ( n ) , d ≡ mod 4) . L -function results. Suppose that χ is a Dirichlet character. We define(2.1) L ( P ) ( s, χ ) = L ( s, χ ) Y p | P (cid:18) − χ ( p ) p s (cid:19) . We define the odd sign indicator function of a Dirichlet character χ by κ = κ ( χ ) = (1 − χ ( − . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 5 L -functional bounds and approximate functional equation. For a primitive character χ modulo q , using an absolute convergence argument for L ( s, χ ) in a right half-plane and applying thefunctional equation, we interpolate via the Phragmén-Lindelöf convexity principle to obtain(2.2) L ( s, χ ) ≪ [ q (1 + |ℑ s | )] / −ℜ s , ℜ s ≤ − ε ;[ q (1 + |ℑ s | )] (1 −ℜ s ) / ε , − ε < ℜ s < ε ;1 , ℜ s ≥ ε, away from a possible pole at s = 1 in the case where χ is trivial. This bound is known as the convexitybound for Dirichlet L -functions.We shall also need the so-called approximate functional equation for Dirichlet L -functions (cf. [IK04],Theorem 5.3). Particularly, if χ is a quadratic primitive character modulo odd q , ψ is a character withconductor dividing 8, and d is odd, squarefree and coprime to q , then we have the weighted infinitesum(2.3) L ( , χ d χψ ) = 2 ∞ X n =1 ( χ d χψ )( n ) n / G κ (cid:18) n √ c d q (cid:19) , where(2.4) κ = κ ( χ d χψ ) , c = , d ≡ mod , ψ = ψ or d ≡ mod , ψ = ψ − ;4 , d ≡ mod , ψ = ψ − or d ≡ mod , ψ = ψ ;8 , ψ = ψ or ψ − , and we have the weight function(2.5) G κ ( ξ ) = 12 πi ˆ (2) Γ( / s + κ )Γ( / κ ) ξ − s dss , satisfying the bound(2.6) G κ ( ξ ) ≪ (1 + ξ ) − A for arbitrary A ≥ (cf. [IK04], Proposition 5.4).We shall also make use of smooth weight functions. Definition 1. We say that w ( x ) is a smooth weight function if it is a smooth non-negative real functionsupported on [1 / , / and unity on [1 / , . The following bound can be shown via sufficiently many applications of integration by parts. We have(2.7) ˜ w ( z ) ≪ A, ℜ z (1 + | z | ) − A for A ≥ , where ˜ w ( z ) is the Mellin transform of w ( x ) .2.4. Short double character sums and L -function moments. We shall need the following adap-tation of Theorem 2 from [HB95] which includes a character twist. Theorem 3. Let ψ be a primitive character modulo j , and for a positive integer Q define S ( Q ) to bethe set of quadratic primitive Dirichlet characters of conductor at most Q . We have X χ ∈ S ( Q ) | L ( σ + it, χψ ) | ≪ ε { Q + ( Qj ( | t | + 1)) − σ }{ Qj ( | t | + 1) } ε for any fixed σ ∈ [1 / , and any ε > . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 6 Removing even conductors, and using the fact that − σ ≤ , we can restate this as follows. Theorem 4. If χ is a primitive character with conductor q and X is a positive real number, then X d ≤ Xd odd, squarefree | L ( s, χ d χ ) | ≪ ε ( Xq | s | ) ε , σ ≥ for all ε > . An important ingredient for the proof of the subconvexity bound Theorem 1 is a large sieve estimatefor quadratic characters due to Heath-Brown. In particular we state here Theorem 1 in [HB95]. Theorem 5 (Heath-Brown’s large sieve estimate) . Let P and Q be positive integers, and let ( a n ) be asequence of complex numbers. Then X m ≤ P ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Q ∗ a n (cid:16) nm (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ε ( P Q ) ε ( P + Q ) X n ≤ Q ∗ | a n | , for any ε > , where P ∗ denotes that the sum is over odd squarefree numbers. Due to the nature of the double Dirichlet series we shall construct, we shall need the followingnormalization of this result. Corollary 1. If ( a m ) , ( b n ) are sequences of complex numbers satisfying the bound a m , b m ≪ m − / ε for some ε > , and P and Q are positive real numbers, then X m ≤ Pm odd X n ≤ Qn odd a m b n (cid:16) n m (cid:17) ≪ ε ( P Q ) ε ( P + Q ) / ε , where we have the composition n = n n with n squarefree, uniformly in P and Q , for any ε > . Gamma identities. We shall have use for the identity(2.8) Γ( − z )Γ( z +12 ) = Γ( − z )Γ( z ) cot (cid:16) πz (cid:17) , z ∈ C . By Stirling’s formula, in particular (5.113) from [IK04], for s ∈ C with fixed real part and nonzeroimaginary part, we have(2.9) Γ( − s )Γ( s ) ≪ ℜ s (1 + | s | ) / −ℜ s , away from the poles at the odd positive integers. We also have the cotangent bound,(2.10) cot( x + iy ) = − i sign ( y ) + O ( e − | y | ) , min k ∈ Z | z − πk | ≥ / . Structure and Analytic Properties Switch of summation formula. The object we would like to study is Z ( s, w ; χ, χ ′ ) = X d ≥ L ( s, χ d χ ) χ ′ ( d ) d w , where χ and χ ′ are quadratic characters with moduli N and M respectively. However, in order to obtainfunctional equations, we will need to augment this by some correction factors. The exact form of these UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 7 correction factors (in a more general setting) was determined in [BFH04]. Applying this theory in ourcase, we have the following theorem, which we note holds for general Dirichlet characters χ and χ ′ ,though in our case we are interested in the special case of quadratic twists. Theorem 6. Let m and d be positive integers with ( md, M N ) = 1 , and write d = d d and m = m m ,with d , m squarefree and d , m positive, and let χ and χ ′ be characters modulo lcm ( M, N ) . Thenthe Dirichlet polynomials P ( χ ) d ,d ( s ) = Y p α k d " α X n =0 χ ( p n ) p n − ns − α − X n =0 ( χ d χ )( p n +1 ) p n − (2 n +1) s ,Q ( χ ′ ) m ,m ( w ) = Y p β k m " β X n =0 χ ′ ( p n ) p n − nw − β X n =0 ( ˜ χ m χ ′ )( p n +1 ) p n − (2 n +1) w satisfy the functional equations (3.1) P ( χ ) d ,d ( s ) = d − s χ ( d ) P (¯ χ ) d ,d (1 − s ) and Q ( χ ′ ) m ,m ( w ) = m − w χ ′ ( m ) Q (¯ χ ′ ) m ,m (1 − w ) , and the interchange of summation formula (3.2) X ( d, MN )=1 L (2 MN ) ( s, χ d χ ) χ ′ ( d ) P ( χ ) d ,d ( s ) d w = X ( m, MN )=1 L (2 MN ) ( w, ˜ χ m χ ′ ) χ ( m ) Q ( χ ′ ) m ,m ( w ) m s , for ℜ s, ℜ w > .Proof. This is a straightforward but lengthy computation which we supress for brevity. We direct thereader to [BFH04] for details. (cid:3) In light of the above theorem, we now define our double Dirichlet series as follows: Let χ and χ ′ becharacters modulo lcm ( M, N ) . Then define(3.3) Z ( s, w ; χ, χ ′ ) = X ( d, MN )=1 L (2 MN ) ( s, χ d χ ) χ ′ ( d ) P ( χ ) d ,d ( s ) d w . We note at this point that it is easily shown that, for d = d d and m = m m with ℜ s, ℜ w ≥ , wehave the bounds | P ( χ ) d ,d ( s ) | ≪ d ε and | Q ( χ ′ ) m ,m ( w ) | ≪ m ε . Applying the functional equations (3.1), wetherefore have(3.4) | P ( χ ) d ,d ( s ) | ≪ ( d − ℜ s + ε , ℜ s < ; d ε , ℜ s ≥ , | Q ( χ ′ ) m ,m ( w ) | ≪ ( m − ℜ w + ε , ℜ w < ; m ε , ℜ w ≥ . Functional equations. We recall that M and N are odd prime numbers or unity, possibly equal.From this point on in the paper, we shall use the following notation: Let χ and χ ′ be quadratic primitivecharacters of squarefree conductors k and j respectively, where j, k | lcm ( M, N ) , and let ψ, ψ ′ beprimitive characters with conductors dividing 8. We first derive the following expansion of the regionsof absolute convergence of the key series involved. Lemma 1. We have the following two series representations of Z ( s, w ; χψ, χ ′ ψ ′ ) . We have Z ( s, w ; χψ, χ ′ ψ ′ ) = X ( d, MN )=1 L (2 MN ) ( s, χ d χψ ) χ ′ ψ ′ ( d ) P ( χψ ) d ,d ( s ) d w UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 8 which is absolutely convergent on the set R (1)1 := {ℜ s ≤ , ℜ w + ℜ s > / } ∪ { < ℜ s ≤ , ℜ s/ ℜ w > / } ∪ {ℜ s, ℜ w > } , except for a possible polar line { s = 1 } , and Z ( s, w ; χψ, χ ′ ψ ′ ) = X ( m, MN )=1 L (2 MN ) ( w, ˜ χ m χ ′ ψ ′ ) χψ ( m ) Q ( χ ′ ψ ′ ) m ,m ( w ) m s which is absolutely convergent on the set R (2)1 := {ℜ w ≤ , ℜ s + ℜ w > / } ∪ { < ℜ w ≤ , ℜ w/ ℜ s > / } ∪ {ℜ s, ℜ w > } , except for a possible polar line { w = 1 } .Proof. This follows from applying the bounds for the Dirichlet L -function (2.2) and the bounds for thecorrection polynomials (3.4) to the definition (3.3). (cid:3) We now proceed with derivation of functional equations for Z . Due to the summation switch formula(3.2), we have Z ( s, w ; χψ, χ ′ ψ ′ ) = X ( m, MN )=1 L (2 MN ) ( w, ˜ χ m χ ′ ψ ′ )( χψ )( m ) Q ( χ ′ ψ ′ ) m ,m ( w ) m s . We can further apply the functional equation for the Q factor (3.1) to obtain(3.5) Z ( s, w ; χψ, χ ′ ψ ′ ) = X ( m, MN )=1 L (2 MN ) ( w, ˜ χ m χ ′ ψ ′ )( χψ )( m ) χ ′ ( m ) Q ( χ ′ ψ ′ ) m ,m (1 − w ) m s m w − / , which holds for ( s, w ) ∈ R (2)1 . The next step is to apply the functional equation for Dirichlet L -functionsin order to change the w in the L -function to − w . This would allow us to switch summation again toobtain Z in its original form, but with a change in variables.We shall define the following Euler product function: For a character χ ⋆ and a positive integer P , wedefine(3.6) K P ( w ; χ ⋆ ) = Y p | P (cid:18) − χ ⋆ ( p ) p − w (cid:19) − (cid:18) − χ ⋆ ( p ) p w (cid:19) . Applying the Dirichlet functional equation along with (2.1) and the above, we now have(3.7) L (2 MN ) ( w, ˜ χ m χ ′ ψ ′ ) = π w − Γ (cid:16) − w +ˆ κ ′ (cid:17) Γ (cid:0) w +ˆ κ ′ (cid:1) K MN ( w ; ˜ χ m χ ′ ψ ′ )( C ψ ′ jm ) − w L (2 MN ) (1 − w, ˜ χ m χ ′ ψ ′ ) , where ˆ κ ′ = κ ( ˜ χ m χ ′ ψ ′ ) .We need to break down some of these parts for further manipulation. Recalling that ( m , M N ) = 1 ,if p is prime and does not divide m , we have(3.8) K p ( w ; ˜ χ m χ ⋆ ) = χ ⋆ ( p ) p − p + χ p ( m ) χ ⋆ ( p )( p − w − p w ) χ ⋆ ( p ) p w − p . If for P ∈ N and a Dirichlet character χ ⋆ we set UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 9 (3.9) F ( χ ⋆ ) P ( w ) = ( , P = 1; χ ⋆ ( P ) P − P χ ⋆ ( P ) P w − P , else , G ( χ ⋆ ) P ( w ) = ( , P = 1; χ ⋆ ( P )( P − w − P w ) χ ⋆ ( P ) P w − P , else , then from (3.8) and the definition (3.6) we have the identity(3.10) K p ( w ; ˜ χ m χ ⋆ ) = F ( χ ⋆ ) p ( w ) + χ p ( m ) G ( χ ⋆ ) p ( w ) , which holds for prime p not dividing m , or p = 1 . Noting that K P ( w ; χ ⋆ ) is multiplicative in P , andusing (3.10) above, we now have the useful expression(3.11) K MN ( w ; ˜ χ m χ ′ ψ ′ ) = F ( χ ′ ψ ′ ) M · F ( χ ′ ψ ′ ) N ( w ) + χ M ( m ) F ( χ ′ ψ ′ ) N · G ( χ ′ ψ ′ ) M ( w )+ χ N ( m ) F ( χ ′ ψ ′ ) M · G ( χ ′ ψ ′ ) N ( w ) + χ MN ( m ) G ( χ ′ ψ ′ ) M · G ( χ ′ ψ ′ ) N ( w ) . We note that this holds true even if M = N . Indeed, in this case, by the definition (3.6), we have K MN ( w ; ˜ χ m χ ′ ψ ′ ) = 1 . This is consistent with (3.11), since according to definition (3.9) we have G ( χ ′ ψ ′ ) N ( w ) = 0 and F ( χ ′ ψ ′ ) N ( w ) = 1 .Next, we see from (2.8) that(3.12) Γ (cid:16) − w +ˆ κ ′ (cid:17) Γ (cid:0) w +ˆ κ ′ (cid:1) = Γ (cid:0) − w (cid:1) Γ (cid:0) w (cid:1) cot (cid:16) πw (cid:17) ˆ κ ′ . We shall find it useful to remove the dependency of ˆ κ ′ on m , or rather, exploit that the dependencyis only on its residue modulo 4. Hence, define κ ′ = κ ( χ ′ ψ ′ ) . Now suppose that f is a function of ˆ κ ′ . Bysieving out by congruence classes modulo 4, we see that f (ˆ κ ′ ) = 12 (1 + ψ − ( m )) f ( κ ′ ) + 12 (1 − ψ − ( m )) f (1 − κ ′ ) . Hence we have(3.13) cot (cid:16) πw (cid:17) ˆ κ ′ = 12 (1 + ψ − ( m )) cot (cid:16) πw (cid:17) κ ′ + 12 (1 − ψ − ( m )) cot (cid:16) πw (cid:17) (1 − κ ′ ) . For brevity, for a character χ ⋆ , we define S ( s, w ; m, χ ⋆ ) := L (2 MN ) (1 − w, ˜ χ m χ ′ ψ ′ ) χ ⋆ ( m ) Q ( χ ′ ψ ′ ) m ,m (1 − w ) m s + w − / . Applying the functional equation for Dirichlet L -functions (3.7) to the identity (3.5) along with (3.13),and using Lemma 1 we now get(3.14) Z ( s, w ; χψ, χ ′ ψ ′ ) = 12 π w − / Γ (cid:0) − w (cid:1) Γ (cid:0) w (cid:1) ( jC ψ ′ ) / − w X ( m, MN )=1 K MN ( w ; ˜ χ m χ ′ ψ ′ ) × S ( s, w ; m, χψ ) (cid:20) (1 + ψ − ( m )) cot (cid:16) πw (cid:17) κ ′ + (1 − ψ − ( m )) cot (cid:16) πw (cid:17) (1 − κ ′ ) (cid:21) , for ( s, w ) ∈ R (2)1 except for possible polar lines { s = 1 } and { w = 1 } . There are two properties of thefunction S which we shall make use of, stated in the following lemma. UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 10 Lemma 2. For two characters χ ⋆ and χ ⋆⋆ and an integer m , the following two properties hold for ( s, w ) ∈ R (2)1 , except for possible polar lines { s = 1 } and { w = 1 } .(i) S ( s, w ; m, χ ⋆ ) χ ⋆⋆ ( m ) = S ( s, w ; m, χ ⋆ χ ⋆⋆ ) , (ii) X ( m, MN )=1 S ( s, w ; m, χ ⋆ ) = Z ( s + w − , − w ; χ ⋆ , χ ′ ψ ′ ) . We can apply the identity (3.11) and Lemma 2 to (3.14), and use the bounds (2.9) and (2.10) to cometo the following functional equation. Theorem 7. Let χ and χ ′ be primitive Dirichlet characters modulo squarefree k and j respectively,where j, k | M N , and if M = N , then j = k = M = N , and let ψ and ψ ′ be Dirichlet charactersmodulo 8. There exist functions a ( χ ′ ,ψ ′ ) n ( w ; ψ ⋆ ) for n | M N and ψ ⋆ a Dirichlet character modulo 8 whichare holomorphic except for possible poles at the positive integers, and countably many poles on the line ℜ w = 1 , bounded absolutely above by O ((16 π ) |ℜ w | (1 + | w | ) / −ℜ w ) uniformly in j and k away from thepoles such that for ( s, w ) ∈ R (2)1 away from possible polar lines { s = 1 } and { w = 1 } we have Z ( s, w ; χψ, χ ′ ψ ′ ) = j / − w X n | MN A ( χ ′ ) n ( w ) X ψ ⋆ ∈ \ ( Z / Z ) ∗ a ( χ ′ ,ψ ′ ) n ( w ; ψ ⋆ ) Z ( s + w − , − w ; χ ˜ χ n ψψ ⋆ , χ ′ ψ ′ ) , where R (2)1 is defined in Lemma 1, we have A ( χ ′ )1 ( w ) = F ( χ ′ ) M ( w ) F ( χ ′ ) N ( w ) , A ( χ ′ ) M ( w ) = F ( χ ′ ) N ( w ) G ( χ ′ ) M ( w ) ,A ( χ ′ ) N ( w ) = F ( χ ′ ) M ( w ) G ( χ ′ ) N ( w ) , A ( χ ′ ) MN ( w ) = G ( χ ′ ) M ( w ) G ( χ ′ ) N ( w ) , and the F and G functions are defined in (3.9). By similar methods, we can obtain a second functional equation under the transformation ( s, w ) (1 − s, s + w − ) by applying the functional equation (3.1) to the definition (3.3), followed by thefunctional equation for L -functions. The result of this similarly lengthy derivation is the followingsecond functional equation. Theorem 8. Let χ and χ ′ be Dirichlet characters modulo squarefree k and j respectively, where j, k | M N , and if M = N , then j = k = M = N , and let ψ and ψ ′ be Dirichlet characters modulo 8. Thereexist functions b ( χ,ψ ) n ( s ; ψ ⋆ ) for n | M N and ψ ⋆ a Dirichlet character modulo 8 which are holomorphicexcept for possible poles at the positive integers, and countably many poles on the line ℜ s = 1 , boundedabsolutely above by O ((16 π ) |ℜ s | (1 + | s | ) / −ℜ s ) uniformly in j and k away from the poles such that for ( s, w ) ∈ R (1)1 away from possible polar lines { s = 1 } and { w = 1 } we have Z ( s, w ; χψ, χ ′ ψ ′ ) = k / − s X n | MN A ( χ ) n ( s ) X ψ ⋆ ∈ \ ( Z / Z ) ∗ b ( χ,ψ ) n ( s ; ψ ⋆ ) Z (1 − s, s + w − ; χψ, χ ′ ˜ χ n ψ ′ ψ ⋆ ) , where R (1)1 is defined in Lemma 1, and the A functions are as in Theorem 7. We make note of an analytic subtlety: We note that although the a n , b n , and A n functions abovehave poles, they do not contribute poles to Z ( s, w ; χψ, χ ′ ψ ′ ) ; indeed, it will be proven in Proposition 1that the only possible poles of Z are the polar lines { s = 1 } , { w = 1 } , and { s + w = 3 / } . Looking at(3.14), though the gamma and cotangent factors together have poles at either the even or odd positiveintegers, and the K factor has countably many poles along the line ℜ w = 1 (cf. (3.8)), these polesnonetheless do not produce poles on the right-hand side. The equation (3.14) essentially results from UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 11 the application of the functional equation for L -functions (3.7) to the identity (3.5), and subsequentlysieving out by congruence classes of m modulo 4. The last step introduces coefficients with poles fromthe gamma and cotangent factors. This is a manifestation of a phenomenon that is observed in thefunctional equation for L -functions: In (3.7), we know that the L -function can only have a pole at w = 1 , yet on the right-hand side, the gamma function produces poles which are mitigated by the trivialzeros of the L -function. It is precisely these poles which appear in the coefficients of (3.14). Additionally,although the K function has poles, these are mitigated by corresponding zeros of the L function due toremoval of the Euler factors at primes dividing M N .It shall be useful to note the following properties of the A coefficients, which follow directly from thedefinitions. Lemma 3. Let χ be a Dirichlet character modulo q . Then the following properties hold. (1) For a positive integer P , if ( q, P ) > then A ( χ ) P ( w ) = 0 . (2) If q = M N then A ( χ )1 ( w ) = 1 . (3) If q = M then A ( χ )1 ( w ) = F ( χ ) N ( w ) and A ( χ ) N ( w ) = G ( χ ) N ( w ) . (4) If q = N then A ( χ )1 ( w ) = F ( χ ) M ( w ) and A ( χ ) M ( w ) = G ( χ ) M ( w ) . (5) Moreover, the following asymptotics hold, if P = 1 and ( P, q ) = 1 . | F ( χ ) P ( w ) | ≍ ( , ℜ w < − ε ; P − ℜ w , ℜ w > ε, | G ( χ ) P ( w ) | ≍ P −ℜ w , ℜ w < ; P ℜ w − , ≤ ℜ w < − ε ; P −ℜ w , ℜ w > ε. It shall also be useful to derive a somewhat symmetric functional equation, obtained by applicationof Theorem 7, then Theorem 8, followed again by Theorem 7. For quadratic Dirichlet characters ρ and ρ ′ of conductors N and M respectively which are either prime or unity, possibly equal, we have(3.15) Z ( s, w ; ρψ, ρ ′ ψ ′ ) = M / − w X n,m,r | MN C − s − wρ ˜ χ n C / − sρ ′ ˜ χ m A ( ρ ′ ) n ( w ) A ( ρ ˜ χ n ) m ( s + w − / A ( ρ ′ ˜ χ m ) r ( w ) × X ψ ⋆ ,ψ ⋆⋆ ,ψ ⋆⋆⋆ ∈ \ ( Z / Z ) ∗ c ( ρ,ρ ′ ,ψ,ψ ′ ) n,m,r ( s, w ; ψ ⋆ , ψ ⋆⋆ , ψ ⋆⋆⋆ ) Z (1 − w, − s ; ρ ˜ χ nr ψψ ⋆ ψ ⋆⋆⋆ , ρ ′ ˜ χ m ψ ′ ψ ⋆⋆ ) , where the A functions are as in Theorem 7, the c functions are holomorphic in C except for possiblepoles for s , w , or s + w − / equal to positive integers, and countably many poles on the lines ℜ w = 1 , ℜ s = 1 , and ℜ s + ℜ w = 3 / , bounded absolutely above by(3.16) O ((16 π ) |ℜ s | +2 |ℜ w | (1 + | s | ) / −ℜ s (1 + | w | ) / −ℜ w (1 + | s + w | ) −ℜ s −ℜ w ) . Analytic continuation. We continue Z ( s, w ; χ, χ ′ ) to all of C except for the polar lines s = 1 , w = 1 , and s + w = 3 / . We have Proposition 1. Let χ and χ ′ be characters modulo lcm ( M, N ) . The function ˜ Z ( s, w ; χ, χ ′ ) = ( s − w − s + w − / Z ( s, w ; χ, χ ′ ) is holomorphic in C and is polynomially bounded in the sense that, given C > , there exists C > such that ˜ Z ( s, w ; χ, χ ′ ) ≪ [ M N (1 + |ℑ s | )(1 + |ℑ w | )] C whenever |ℜ s | , |ℜ w | < C .Proof. We refer to the reader to [BFH04] and [Blo11]. (cid:3) UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 12 Convexity bound. The notion of convexity is not canonically defined for double Dirichlet seriesas it is in the case of (single) Dirichlet series; in the latter case, we have a single functional equationwhich reflects the region of absolute convergence, and interpolating the bounds produces a convexitybound between the two. In the case of double Dirichlet series, things are more complicated. Firstly,our bounds in the region of absolute convergence depend on our knowledge of the bounds on L ( s, χ ) onaverage. Secondly, we have 6 functional equations to choose from to apply to this region. If we use theLindelöf hypothesis on average, namely Theorem 4, then we can carefully choose a functional equationto apply in order to minimize the resulting convexity bound from application of the Phragmén-Lindelöfconvexity principle (cf. Theorem 5.53 of [IK04]).We shall require some initial bounds. We let χ and χ ′ be quadratic Dirichlet characters with con-ductors k and j respectively, and ψ, ψ ′ are characters modulo 8. We first assume that ℜ s = 1 / and ℜ w = 1 + ε . We apply the identity (2.1), the bounds (3.4), and the Cauchy-Schwarz inequality with theaverage bound of Theorem 4 to obtain(3.17) Z ( s, w ; χψ, χ ′ ψ ′ ) ≪ ( M N ) ε k / ε (1 + | s | ) / ε , ℜ s = 1 / , ℜ w = 1 + ε, and by the switch of summation formula (3.2) we also obtain(3.18) Z ( s, w ; χψ, χ ′ ψ ′ ) ≪ ( M N ) ε j / ε (1 + | w | ) / ε , ℜ s = 1 + ε, ℜ w = 1 / . We use the functional equation Theorem 8 with ℜ s = − ε and ℜ w = 1 + ε and apply (3.18) on theright-hand side in order to obtain a bound for Z ( s, w ; ρ, ρ ′ ) . Looking at the coefficient bounds in Lemma3, we pick up a factor of N / ε . Further, we see that the resulting twisting characters on the right-handside will have conductors ( k, j ) ∈ { ( N, M ) , ( N, } , so that we have(3.19) Z ( s, w ; ρ, ρ ′ ) ≪ N / ε M / ε (1 + | s | ) / ε (1 + | s + w | ) / ε , ℜ s = − ε, ℜ w = 1 + ε. Likewise with the functional equation Theorem 7 applied to (3.17), we have the symmetric bound(3.20) Z ( s, w ; ρ, ρ ′ ) ≪ M / ε N / ε (1 + | w | ) / ε (1 + | s + w | ) / ε , ℜ s = 1 + ε, ℜ w = − ε. We wish to interpolate convexly between these bounds along the two diagonal lines in C , but we mustdeal with the potential poles at s = 1 and w = 1 . To do this we can simply multiply both sides of thebounds by ( s − w − . We obtain the following bound. Proposition 2. For quadratic characters ρ and ρ ′ of prime moduli N and M respectively, we have Z ( s, w ; ρ, ρ ′ ) ≪ [(1 + | s | )(1 + | w | )(1 + | s + w | )] / ε ( M N ) / ε , ℜ s = ℜ w = 1 / , which we call the convexity bound for the function Z ( s, w ; ρ, ρ ′ ) . Approximate Functional Equations A symmetric functional equation. We introduce a succession of applications of the functionalequations in the special case of ( s, w ) = (1 / , / − z ) for some z ∈ C with ℜ z > . We recall that ρ and ρ ′ are primitive quadratic Dirichlet characters of conductors N and M respectively. We first applyTheorem 7, which after observing the coefficient properties of Lemma 3 gives(4.1) Z ( , − z ; ρ, ρ ′ ) = X ψ ⋆ ∈ \ ( Z / Z ) ∗ h M z F ( ρ ′ ) N a ( ρ ′ ,ψ )1 ( − z ; ψ ⋆ ) Z ( − z, + z ; ρψ ⋆ , ρ ′ )+ M z G ( ρ ′ ) N ( − z ) a ( ρ ′ ,ψ ) N ( − z ; ψ ⋆ ) Z ( − z, + z ; ψ ⋆ , ρ ′ ) i . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 13 We then apply the functional equation Theorem 8 (and again use Lemma 3) which further gives(4.2) Z ( , − z ; ρ, ρ ′ ) = X ψ ⋆ ,ψ ⋆⋆ ∈ \ ( Z / Z ) ∗ h ( M N ) z F ( ρ ′ ) N ( − z ) F ( ρ ) M ( − z ) c ( ρ ′ ,ρ )1 , ( z ; ψ ⋆ , ψ ⋆⋆ ) Z ( + z, ; ρψ ⋆ , ρ ′ ψ ⋆⋆ )+ ( M N ) z F ( ρ ′ ) N ( − z ) G ( ρ ) M ( − z ) c ( ρ ′ ,ρ )1 ,M ( z ; ψ ⋆ , ψ ⋆⋆ ) Z ( + z, ; ρψ ⋆ , ψ ⋆⋆ )+ M z G ( ρ ′ ) N ( − z ) F ( ψ ) M ( − z ) F ( ψ ) N ( − z ) c ( ρ ′ ,ψ ) N, ( z ; ψ ⋆ , ψ ⋆⋆ ) Z ( + z, ; ψ ⋆ , ρ ′ ψ ⋆⋆ )+ M z G ( ρ ′ ) N ( − z ) F ( ψ ) N ( − z ) G ( ψ ) M ( − z ) c ( ρ ′ ,ψ ) N,M ( z ; ψ ⋆ , ψ ⋆⋆ ) Z ( + z, ; ψ ⋆ , ψ ⋆⋆ )+ M z G ( ρ ′ ) N ( − z ) F ( ψ ) M ( − z ) G ( ψ ) N ( − z ) c ( ρ ′ ,ψ ) N,N ( z ; ψ ⋆ , ψ ⋆⋆ ) Z ( + z, ; ψ ⋆ , ρ ′ ρψ ⋆⋆ )+ M z G ( ρ ′ ) N ( − z ) G ( ψ ) M ( − z ) G ( ψ ) N ( − z ) c ( ρ ′ ,ψ ) N,MN ( z ; ψ ⋆ , ψ ⋆⋆ ) Z ( + z, ; ψ ⋆ , ρψ ⋆⋆ ) i , where c ( χ,χ ′ ) n,m ( z ; ψ ⋆ , ψ ⋆⋆ ) = a ( χ,ψ ) n ( − z ; ψ ⋆ ) b ( χ ′ ,ψ ⋆ ) m ( − z ; ψ ⋆⋆ ) . We note that, in the case where M = N = 1 , we may not apply the functional equation Theorem 8 to Z ( − z, + z ; ψ ⋆ , ρ ′ ) in (4.1) because Theorem 8 is only valid if j = k = M = N , but here we have k = 1 = N . Nonetheless, equation (4.2) still holds: Indeed, in the case where M = N , the term with Z ( − z, + z ; ψ ⋆ , ρ ′ ) of (4.1) vanishes because G ( ρ ′ ) N ( − z ) = 0 , and so only the first term of (4.2) willremain (note also that G ( ρ ) M ( − z ) = 0 in this case).Looking at (3.9), for a quadratic character χ ⋆ whose modulus is coprime to P , we see that(4.3) F ( χ ⋆ ) P ( − z ) = P − − P − z − − and G ( χ ⋆ ) P ( − z ) = P z − / (cid:18) χ ⋆ ( P )(1 − P − z ) P − z − − (cid:19) . Therefore, setting(4.4) Φ := { ( ρ, ρ ′ ) , ( ρ, ψ ) , ( ψ , ρ ′ ) , ( ψ , ψ ) , ( ψ , ρ ′ ρ ) , ( ψ , ρ ) } we have(4.5) Z ( , − z ; ρ, ρ ′ ) = X ( χ,χ ′ ) ∈ Φ ψ,ψ ′ ∈ \ ( Z / Z ) ∗ β ( χ,χ ′ ) ψ,ψ ′ ω ( χ,χ ′ ) ψ,ψ ′ ( z )( γ ( χ,χ ′ ) ψ,ψ ′ ) z Z ( + z, ; χψ, χ ′ ψ ′ ) , where we absorb the F χ ⋆ P ( − z ) factors and parenthetical expression of (4.3) for the G χ ⋆ P ( − z ) factors,as well as the c ( χ,χ ′ ) n,m ( z ; ψ ⋆ , ψ ⋆⋆ ) factors into the ω ( χ,χ ′ ) ψ,ψ ′ ( z ) functions, and collect the remaining factorsinto the β ( χ,χ ′ ) ψ,ψ ′ and γ ( χ,χ ′ ) ψ,ψ ′ coefficients. Hence we see that for ℜ z > , the ω ( χ,χ ′ ) ψ,ψ ′ ( z ) functions areholomorphic satisfying the bound(4.6) ω ( χ,χ ′ ) ψ,ψ ′ ( z ) ≪ (1 + |ℑ z | ) ℜ z uniformly in M and N . Thus, we obtain the upper bounds in Table 1. UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 14 Table 1. Coefficient upper bounds ( χ, χ ′ ) ( ρ, ρ ′ ) ( ρ, ψ ) ( ψ , ρ ′ ) ( ψ , ψ ) ( ψ , ρ ′ ρ ) ( ψ , ρ ) M = N β ( χ,χ ′ ) ψ,ψ ′ bound 1 M − / N − / M − / N − / N − M − / N − γ ( χ,χ ′ ) ψ,ψ ′ bound M N M N M N M N M N M N M = N β ( χ,χ ′ ) ψ,ψ ′ bound 1 0 0 0 0 0 γ ( χ,χ ′ ) ψ,ψ ′ bound N Approximate functional equations. The following lemma essentially takes the preceding func-tional equation a step further by opening the first sum of Z . Lemma 4. There exist smooth, rapidly decaying functions V ( ξ ; t ) and V ( χ,χ ′ ) ψ,ψ ′ ( ξ ; t ) such that for anyconstant X > one has Z ( , + it ; ρ, ρ ′ ) = X − it X ( d, MN )=1 L (2 MN ) ( , χ d ρ ) ρ ′ ( d ) P ( ρ ) d ,d ( ) d / V (cid:18) dX ; t (cid:19) + X it X ( χ,χ ′ ) ∈ Φ ψ,ψ ′ ∈ \ ( Z / Z ) ∗ β ( χ,χ ′ ) ψ,ψ ′ X ( d, MN )=1 L (2 MN ) ( , ˜ χ d χ ′ ψ ′ ) χψ ( d ) Q ( χ ′ ψ ′ ) d ,d ( ) d / V ( χ,χ ′ ) ψ,ψ ′ dXγ ( χ,χ ′ ) ψ,ψ ′ ; t ! , where β ( χ,χ ′ ) ψ,ψ and γ ( χ,χ ′ ) ψ,ψ ′ satisfy the bounds listed in Table 1, and we have the bounds V ( χ,χ ′ ) ψ,ψ ′ ( ξ ; t ) ≪ | ξ | − B (1 + | t | ) B and V ( ξ ; t ) ≪ | ξ | − B uniformly in ξ and t for any number B > .Proof. Let B > , H be an even, holomorphic function with H (0) = 1 satisfying the growth estimate H ( z ) ≪ ℜ z,A (1 + | z | ) − A for any A > . We consider the integral(4.7) I ( c, X, t ) = 12 πi ˆ (1) X cz + it + cz − + it − ! Z ( , + it + cz ; ρ, ρ ′ ) H ( z ) dzz for a real number c , a positive real number X > , and a fixed real number t . Examining the expressionwhen c = 1 , the fraction cancels the pole of the Z factor at z = 1 / − it . We apply a shift of the contourto ℜ z = − , picking up the pole at z = 0 , whence we obtain I (1 , X, t ) = Z ( , + it ; ρ, ρ ′ ) + 12 πi ˆ ( − X z + it + z − + it − ! Z ( , + it + z ; ρ, ρ ′ ) H ( z ) dzz . We now apply a change of variables z 7→ − z , arriving at Z ( , + it ; ρ, ρ ′ ) = I (1 , X, t ) + I ( − , X, t ) . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 15 Applying the functional equation (4.5) and the switch of summation formula (3.2) and expanding the Z functions, definition (3.3) gives that I ( − , X, t ) equals(4.8) X it X ( χ,χ ′ ) ∈ Φ ψ,ψ ′ ∈ \ ( Z / Z ) ∗ β ( χ,χ ′ ) ψ,ψ ′ X ( m, MN )=1 L (2 MN ) ( , ˜ χ m χ ′ ψ ′ )( χψ )( m ) Q ( χ ′ ψ ′ ) m ,m ( ) m / V ( χ,χ ′ ) ψ,ψ ′ mXγ ( χ,χ ′ ) ψ,ψ ′ ; t ! , where V ( χ,χ ′ ) ψ,ψ ′ ( ξ ; t ) = 12 πi ˆ (1) (cid:18) / it − z − / it − (cid:19) ξ − z + it ω ( χ,χ ′ ) ψ,ψ ′ ( z − it ) H ( z ) dzz . We wish to obtain an upper bound for V ( χ,χ ′ ) ψ,ψ ′ ( ξ ; t ) . Moving the contour to B (recalling that there areno poles of ω ( χ,χ ′ ) ψ,ψ ′ in this region), and bounding by taking the absolute value of the integrand and usingthe bound (4.6), we have V ( χ,χ ′ ) ψ,ψ ′ ( ξ ; t ) ≪ | ξ | − B (1 + | t | ) B uniformly in ξ and t . Changing the summation variable from m to d in (4.8), we obtain the first termin the statement of the lemma.Looking at I (1 , X, t ) , we have I (1 , X, t ) = X − it X ( d, MN )=1 L (2 MN ) ( , χ d ρ ) ρ ′ ( d ) P ( ρ ) d ,d ( ) d / it V (cid:18) dX ; t (cid:19) , where V ( ξ ; t ) = 12 πi ˆ (1) + it + z − / it − ! ξ − z H ( z ) dzz . Also, it is immediate that we have the bound V ( ξ ; t ) ≪ | ξ | − B uniformly in ξ and t . (cid:3) We now wish to truncate the sums above, accruing an error. This is the object of the following lemma. Lemma 5. Let A be a large positive constant, t be a real number, and V ( ξ ; t ) be a rapidly decayingfunction in ξ satisfying V ( ξ ; t ) ≪ | ξ | − B (1 + | t | ) B , uniformly in ξ and t for any number B > , let χ be a character modulo k , and let a be an arithmeticfunction satisfying a ( d ) ≪ d ε uniformly in d . Then we can truncate the double sum X ( d, MN )=1 L (2 MN ) ( , χ d χ ) χ ′ ( d ) a ( d ) d / V (cid:18) dY ; t (cid:19) at d < Y ε , accruing an error that is bounded above by O ((1 + | t | ) A/ε k / ε Y − A ) . Proof. The L -function is bounded asymptotically by ( d k ) / ε due to the Phragmén-Lindelöf convexitybound (2.2). The V factor is bounded by its argument to an arbitrarily large power − B . Applying thisgives an error that is bounded above by (1 + | t | ) B X d>P ε ( d k ) / ε d / − ε (cid:18) dY (cid:19) − B , and the result follows. (cid:3) UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 16 In order to bound Z ( , + it ; ρ, ρ ′ ) , by applying a smooth partition of unity as in [Blo11], it nowsuffices to bound D W,a ( Y ; t, χ ′ ψ ′ , χψ ) := X ( d, MN )=1 L (2 MN ) ( , χ d χ ′ ψ ′ ) χψ ( d ) a ( d ) d / W (cid:18) dY ; t (cid:19) for t a real number, ψ, ψ ′ ∈ \ ( Z / Z ) ∗ , a smooth function W with support on [1 , satisfying W ( x ; t ) ≪ B x − B (1 + | t | ) B uniformly in x and t for any B > , an arithmetic function a satisfying the bound a ( d ) ≪ d ε , and thefollowing conditions, according to each of the two sums in Lemma 4: either ( χ, χ ′ ) ∈ Φ (cf. (4.4)) withconductors k and j respectively, and ≤ Y ≤ (cid:16) γ ( χ,χ ′ ) ψ,ψ ′ X − (cid:17) ε , or ( χ, χ ′ ) = ( ρ ′ , ρ ) and ≤ Y ≤ X ε . Expanding according to the Dirichlet functional equation, and further truncating that sum expresses D W,a ( Y ; t, χ ′ , χ ) as a double finite character sum, allowing us to apply Heath-Brown’s large sieve estimateCorollary 1. The result of this is the following lemma. Lemma 6. We have the bound D W,a ( Y ; t, χ ′ , χ ) ≪ (1 + | t | ) /ε ( M N ) ε (cid:16) Y ε + ( Y j ) / ε (cid:17) / ε uniformly in t , Y , j , and k .Proof. Applying Lemma 5 above, we have D W,a ( Y ; t, χ ′ , χ ) = X ( d, MN )=1 d We first apply Lemma 4 which gives Z ( , + it ; ρ, ρ ′ ) = X it X ( χ,χ ′ ) ∈ Φ ψ,ψ ′ ∈ \ ( Z / Z ) ∗ β ( χ,χ ′ ) ψ,ψ ′ D V,Q γ ( χ,χ ′ ) ψ,ψ ′ X ; t, χ ′ , χ ! + X − it D V,P ( X ; t, ρ, ρ ′ ) , where the subscripts for D in the first term are V = V ( χ,χ ′ ) ψ,ψ ′ ( ξ, t ) and Q = Q ( χ ′ ψ ′ ) d ,d ( ) , and the subscriptsfor D in the second term are V = V ( ξ, t ) and P = P ( χψ ) d ,d ( ) . Applying Lemma 6 further gives Z ( , + it, ρ, ρ ′ ) ≪ (1 + | t | ) /ε ( M N ) ε (cid:20)(cid:16) X + ( N X ) / (cid:17) / ε (cid:21) + (1 + | t | ) /ε ( M N ) ε max ( χ,χ ′ ) ∈ Φ ψ,ψ ′ ∈ \ ( Z / Z ) ∗ " β ( χ,χ ′ ) ψ,ψ ′ (cid:18) γ ( χ,χ ′ ) ψ,ψ ′ X − + (cid:16) γ ( χ,χ ′ ) ψ,ψ ′ jX − (cid:17) / (cid:19) / ε . Here, Φ is given by (4.4), and j is the conductor of χ ′ .We initially set X = M a N b and eventually choose optimized values for a and b , depending on thebounds for the β ’s and γ ’s from Table 1. We come to an optimal choice of X = M / N / and obtainthe subconvexity bound of M / ε N / ε + M / ε N / ε ≍ ( M N ( M + N )) / ε (even in the case of M = N ), compared to the convexity bound of ( M N ) / ε . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 18 Proof of Theorem 2 Here we present an application of the theory of double Dirichlet series as developed. Given a fixedpositive prime N , we seek an upper bound on d such that L (1 / , χ dN ) does not vanish. We follow amodified version of a method outlined in [HK10], and we rigorously prove an important lower bound oncoefficients which arise from a residue of Z . Let h ( y ) be a smooth weight function as in Definition 1.Expanding as per (3.3) and by Mellin inversion we have(6.1) ˆ (2) ˜ h ( w ) Z ( , w ; χ N , ψ ) X w dw = X ( d, N )=1 L (2 N ) ( , χ d N ) P ( χ N ) d ,d ( ) h ( dX ) . We move the contour of the integral on the left-hand side to ℜ w = − ε , picking up a residue at w = 1 due to the double pole of Z ( , w ; χ N , ψ ) there. If we write its Laurent expansion as Z (1 / , w ; χ N , ψ ) = µ N ( w − + ν N ( w − 1) + · · · then the left-hand side of (6.1) equals [ ν N ˜ h (1) + µ N ˜ h ′ (1)] X + µ N ˜ h (1) X log X + ˆ ( − ε ) ˜ h ( w ) Z ( , w ; χ N , ψ ) X w dw. We now apply the symmetric functional equation (3.15) with ρψ = χ N , ρ ′ = ψ ′ = ψ , and ( s, w ) = (1 / , w ) with ℜ w = − ε to Z in the resulting integral. Using Lemma 3 to bound the coefficients, we see that C ρ ˜ χ n = N when n = 1 and is unity otherwise, and C / − sρ ′ ˜ χ m = 1 in every case because s = 1 / . In everycase the A factors will always be bounded above by O ( N ε ) , since ℜ w = − ε . Also, A ( ρ ˜ χ n ) m ( s + w − / vanishes when m = N and n = 1 . Hence, we have Z (1 / , w ; χ N , ψ ) = X ψ ⋆ ,ψ ⋆⋆ ,ψ ⋆⋆⋆ ∈ \ ( Z / Z ) ∗ X m,r | N B ( w ; N, m, r, ψ ⋆ , ψ ⋆⋆ , ψ ⋆⋆⋆ ) × (cid:16) N / − w Z (1 − w, / χ N ˜ χ r ψ ⋆ ψ ⋆⋆⋆ , ψ ⋆⋆ ) + Z (1 − w, / 2; ˜ χ r ψ ⋆ ψ ⋆⋆⋆ , ˜ χ m ψ ⋆⋆ ) (cid:17) , where, due also to the bound (3.16), we have | B ( w ; N, m, r, ψ ⋆ , ψ ⋆⋆ , ψ ⋆⋆⋆ ) | ≪ N ε (1 + | w | ) ε . We now apply the bound (3.18), which finally gives | Z ( , − ε + it ; χ N , ψ ) | ≪ (1 + | t | ) ε N / ε , which we shall apply to the integral. We thus have(6.2) S ( X ; χ N ) := X ( d, N )=1 L (2 N ) ( , χ d N ) P ( χ N ) d ,d ( ) h ( dX ) = a N X log X + b N X + O ( N / ε ) for certain coefficients a N and b N . More elementary analysis of S ( X ; χ N ) via Theorem 9 (cf. §7) belowgives us the lower bounds a N , b N ≫ N − ε . If we now assume that L (2 N ) ( , χ dN ) = L ( , χ d N ) Y p | d N (1 − p − / ) vanishes for d ≪ X , then we get a contradiction as long as a N X log X is greater than the error term.We see we can choose X = N / ε , as required. UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 19 We essentially combine two asymptotic formulas for S ( X ; χ N ) : Looking at it elementarily via Theorem9 from §7 gives us a bad error term, but lower bounds on the coefficients. Looking at it analytically asabove allows us to take advantage of the bound (3.18) for Z in order to obtain a smaller error term.7. An Asymptotic Formula for an L -function sum Result. Of particular importance for the smallest nonvanishing quadratic central value of twisted L -functions result is an asymptotic formula for the weighted and twisted L -function sum given by S ( X ; χ ) in (6.2). Indeed, we already have an asymptotic formula, but it is important that we have a lower boundon the main term coefficients.Let N be a natural number, X a large positive real number, and let χ be a quadratic primitivecharacter modulo N . Let h be a smooth weight function as defined in Definition 1, and define S ( X ; χ ) = X ( d, N )=1 L (2 N ) ( , χ d χ ) P ( χ ) d ,d ( ) h ( d/X ) . We seek to obtain information on the main term coefficients of the asymptotic formula (6.2). We shallprove the following Theorem. Theorem 9. There exists δ > such that we have the asymptotic formula S ( X ; χ ) = a N X log X + b N X + O ( N / ε X − δ ) , where N − ε ≪ a N , b N ≪ N ε , uniformly in N . Proof of Theorem 9. According to the approximate functional equation for Dirichlet L -functions(2.3), we have L ( , χ d χ ) = 2 ∞ X n =1 ( χ d χ )( n ) n / G κ (cid:18) n √ c d N (cid:19) , where G κ is given in (2.5), and c is given in (2.4). From the definition of P ( χ ) d ,d in Theorem 6 we have P ( χ ) d ,d ( ) = X f | d µ ( f )( χ d χ )( f ) f / , where we write f = f f with f squarefree. We shall also use the expansion − ( χ d χ )(2)2 / = X g | µ ( g )( χ d χ )( g ) g / . Applying these expressions to S ( X ; χ ) , we have(7.1) S ( X ; χ ) = 2 X ( d , N )=1 X ( d , N )=1 X f | d X g | f µ ( g )( χ d χ )( g ) h / ∞ X n =1 ( χ d χ )( n ) n / G κ (cid:18) n √ c d N (cid:19) h (cid:18) d d X (cid:19) . For a subset H ⊂ N , we define S H ( X ; χ ) to be the same as the expression (7.1) with the addedcondition in the n -sum of ng ∈ H . We use Mellin inversion for G κ to separate the variables, and by UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 20 moving the d -sum to the inside, we get that S H ( X ; χ ) equals(7.2) ˆ ( + ε ) N s X ( d , N )=1 X f | d X g | f ( g,N )=1 µ ( g ) χ ( g ) g / X ng ∈ H ( n,N )=1 χ ( n ) n s + X ( d , N )=1 ˜ G κ ( s ) χ d ( ng ) c s/ d s/ h (cid:18) d d X (cid:19) ds. The variables κ and c depend on the residues of d and N modulo 4. Thus, given ℓ ∈ {± } , we define κ ( ℓ ) and c ( ℓ ) to be the corresponding values for d ≡ ℓ ( mod . For convenience, for ι ∈ {± } , wedefine S H ( X ; χ, ι, ℓ ) to be the same as (7.2) except that κ and c are replaced with κ ( ℓ ) and c ( ℓ ) , and ψ ι ( d ) is multiplied to the summand in the d -sum.Observing that (1 ± ψ − ( d )) is the characteristic function of d ≡ ± mod , the d -sum in theintegrand of (7.2) is X ( d , N )=1 X ± (1 ± ψ − ( d )) ˜ G κ ( ± ( s ) c ( ± s/ ˜ χ ng ( d ) d s/ h (cid:18) d d X (cid:19) , whence we see that(7.3) S H ( X ; χ ) = 12 X ι = ± X ℓ = ± sgn (1 + ι + ℓ ) S H ( X ; χ, ι, ℓ ) . For treatment of the d -sum, for a positive real number Y and a Dirichlet character ψ we further define T ( s ; Y, ψ ) = ∞ X d =1 ψ ( d ) d s/ h (cid:18) d Y (cid:19) , where the sum is over squarefree d . With these simplifications, we obtain that S H ( X, χ, ι, ℓ ) equals(7.4) ˆ (1 / ε ) ( N c ( ℓ )) s/ ˜ G κ ( ℓ ) ( s ) X ( d , N )=1 X f | d X g | f ( g,N )=1 µ ( g ) χ ( g ) g / X ng ∈ H ( n,N )=1 χ ( n ) n s +1 / T ( s ; X/d , χ (2 N )0 ˜ χ ng ψ ι ) ds. We shall choose H = (cid:3) which we use to denote the set of positive squares, and denoting the complementof H in N by ¯ H , it is clear that S ( X ; χ ) = S (cid:3) ( X ; χ ) + S ¯ (cid:3) ( X ; χ ) . In order to estimate the size of T ( s ; Y, ψ ) , we observe that there are order Y squarefree numbers upto Y . If ψ is a principal character and ℜ s = ε , we therefore expect this sum to be roughly of size Y ε .If ψ is non-principal, the oscillations will give us a Pólya-Vinogradov type estimate.Indeed, with this last point in mind, we step back to explain the main idea of the proof: The sum S H ( X, χ, ψ ι , ℓ ) will hence only be large when ˜ χ ng ψ ι is principal, that is, precisely when H = (cid:3) and ι = 1 , and will be small otherwise. To this end, we have the following asymptotic formula. Lemma 7. For ℜ s > , we have T ( s ; Y, χ ( m )0 ) = ˜ h (1 + s/ ζ (2) Y p | m (cid:18) p (cid:19) − Y s/ + U ( s ; Y, m ) with U ( s ; Y, m ) holomorphic and U ( s ; Y, m ) ≪ | s | / −ℜ s/ ε m ε Y / ℜ s/ ε , UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 21 uniformly in Y and ℑ s . Further, if ψ = χ ( m )0 ˜ ψ , where ˜ ψ is a nontrivial quadratic primitive characterwith conductor c , then T ( s ; Y, ψ ) ≪ | cs | / −ℜ s/ ε m ε Y / ℜ s/ ε uniformly in ℑ s , m, and Y .Proof. Via Mellin inversion, we have T ( s ; Y, χ ( m )0 ˜ ψ ) = ˆ (1+ ℜ s/ ε ) T s ( z )˜ h ( z ) Y z dz, where we have the generating function T s ( z ) = ∞ X d =1 µ ( d )( χ ( m )0 ˜ ψ )( d ) d z − s/ = L ( z − s/ , ˜ ψ ) L (2 z − s, ˜ ψ ) Y p | m (cid:18) p z − s/ (cid:19) − . If ˜ ψ is the trivial character then the L -function in the numerator is just the zeta function, and thereforehas a pole at z = 1 + s/ . Due to the /L (2 z − s, ˜ ψ ) factor, all other poles lie in ℜ z < ℜ s/ . We movethe contour to (1 / ℜ s/ ε ) , and in the case where ˜ ψ is trivial, we pick up the residue(7.5) ˜ h (1 + s/ ζ (2) Y p | m (cid:18) p (cid:19) − Y s/ . Due to (2.7) along with the convexity bound L ( z − s/ , ˜ ψ ) ≪ [ c (1 + |ℑ z − ℑ s/ | )] / −ℜ s/ ε obtained from (2.2), we bound the resulting integral by | cs | / −ℜ s/ ε m ε Y / ℜ s/ ε . (cid:3) Main term. As explained above, the main contribution will come from S (cid:3) ( X ; χ, , ℓ ) , which wewill bound from below. Looking at the expansion (7.4) with ι = ℓ = 1 and H = (cid:3) , since ng is square,and recalling that g is squarefree, we have n = gm for m ∈ N , so the inner sum is X ng ∈ (cid:3) ( n,N )=1 n s +1 / T ( s ; X/d , χ (2 ngN )0 ) = 1 g s +1 / X ( m,N )=1 m s +1 T ( s ; X/d , χ (2 gmN )0 ) . By Lemma 7, the above expression is g s +1 / X ( m,N )=1 m s +1 Y p | gmN (cid:18) p (cid:19) − ˜ h (1 + s/ ζ (2) (cid:18) Xd (cid:19) s/ + U ( s ; X/d , gmN ) where U ( s ; X/d , gmN ) is holomorphic for ℜ s > and U ( s ; X/d , gmN ) ≪ | s | / −ℜ s/ ε ( gmN ) ε (cid:0) X/d (cid:1) / ℜ s/ . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 22 Referring to (7.4), and moving the contour to ( ε ) , the error term is bounded above by(7.6) X ( d , N )=1 ˆ ( ε ) | N c ( ℓ ) | ℜ s/ | ˜ G κ ( ℓ ) ( s ) || s | / −ℜ s/ ε X f | d X g | f ( g,N )=1 g s +1 X ( m,N )=1 ( gmN ) ε m s +1 (cid:18) Xd (cid:19) / ℜ s/ ds. Bounding the sums absolutely and using the fact that ˜ G κ ( ℓ ) ( s ) decays rapidly in fixed vertical strips, wesee that this is bounded above by N ε X / ε . As for the main term, through a calculation we have thefollowing result. Lemma 8. We have the identity X ( m,N )=1 m s +1 Y p | gmN (cid:18) p (cid:19) − = ζ (2 s + 1) E ( s ) E ( s ; g ) , where E ( s ) = 49 (1 − − s − ) Y p (1 + p − )(1 + p − (1 − p − s − ) − ) − Y p | N (1 + p − ) − (1 + p − − p − s − ) ,E ( s ; g ) = Y p | gg =2 (1 + p − ) − (1 − p − s − ) − (1 + p − − p − s − ) . Proof. This is a straightforward but monotonous calculation which we omit. (cid:3) Applying this, we now have that S (cid:3) ( X, χ, , ℓ ) equals, up to an error O ( N ε X / ε ) , ζ (2) ˆ ( ε ) ˜ h (1+ s )( c ( ℓ ) N ) s/ X s/ ˜ G κ ( ℓ ) ( s ) ζ (2 s +1) E ( s ) X ( d , N )=1 d − − s X f | d X g | f ( g,N )=1 µ ( g ) g s +1 E ( s ; g ) ds. Since E ( s ; g ) is multiplicative in g , we can further collapse the g -sum above into an Euler product.Hence we have(7.7) S (cid:3) ( X, χ, , ℓ ) = 2 ζ (2) ˆ ( ε ) ˜ h (1 + s/ c ( ℓ ) N ) s/ X s/ ˜ G κ ( ℓ ) ( s ) ζ (2 s + 1) E ( s ) H ( s ) ds + O ( N ε X / ε ) , where H ( s ) = X ( d , N )=1 d − − s X f | d Y p | f (cid:0) − p − s − E ( s ; p ) (cid:1) . We have the following estimates for H . Lemma 9. There exists K > such that / ≤ H (0) ≤ K and | H ′ (0) | ≤ K .Proof. Let ℜ s ≥ , and for convenience define H ( s ; d ) = X f | d Y p | f (cid:0) − p − s − E ( s ; p ) (cid:1) . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 23 Because < E (0; p ) ≤ / , we have / ≤ − p − E (0; p ) < , and so taking d = 1 , we have H (0) ≥ / . Hence by the same reasoning, | H (0; d ) | ≤ X n | d ≪ d ε , and so we see that H (0) is absolutely bounded above. In order to show that H ′ (0) is absolutely bounded,because we have H ′ (0) = X ( d , N )=1 d − ( H ′ (0; d ) − H (0; d ) log d ) , it suffices to show that H ′ (0; d ) ≪ d ε , and for this, it suffices to show that E ′ (0; p ) is absolutelybounded, which is easily seen via taking the logarithmic derivative. (cid:3) We shall also need the following bounds for the E function. Lemma 10. We have the bounds N − ε ≪ E (0) , E ′ (0) ≪ N ε . Proof. We have N − ε ≪ d ( N ) ≪ ζ (2) Y p | N (1 + p − ) − = E (0) ≪ , and the result follows.To treat the derivative, it is easily shown that the logarithmic derivative at s = 0 is bounded belowby a constant and above by N ε , as required. (cid:3) We wish to move the contour of integration of (7.7) from ( ε ) to ( − ε ) . In doing so, we pick up a doublepole, since ˜ G κ ( ℓ ) ( s ) and ζ (2 s + 1) have simple poles at s = 0 . The residue of this pole shall be our mainterm. In order to calculate it, we shall need further analysis of the integrand. We define A ( s ) = ˜ h (1 + s/ N s/ X s/ E ( s ) H ( s ) which is the part of the integrand which is holomorphic at s = 0 . If the Laurent coefficients of ζ (2 s + 1) and ˜ G κ ( ℓ ) (centred at ) are given by e n and g n respectively, then the residue is(7.8) R := ( g − e + g e − ) A (0) + g − e − A ′ (0) , and(7.9) A (0) = ˜ h (1) E (0) H (0) X, (7.10) A ′ (0) = 12 ˜ h ′ (1) E (0) H (0) X + 12 ˜ h (1) E (0) H (0)(log N ) X + 12 ˜ h (1) E (0) H (0) X log X + ˜ h (1) E ′ (0) H (0) X + ˜ h (1) E (0) H ′ (0) X. We now prove the following expression of the residue R from the above results. Corollary 2. For sufficiently large N , we have R = a N X log X + b N X, where N − ε ≪ a N , b N ≪ N ε . UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 24 Proof. The first term comes from the X log X term in (7.10). We see that g − = 1 due to the definitionof ˜ G κ ( s ) given in (2.5), and we also easily see that e − = 1 / , so that the coefficient for A ′ (0) in (7.8)is positive. Now the bounds for a N follow from those for E (0) and H (0) from Lemmas 9 and 10, andthe fact that ˜ h (1) is simply a positive constant.Next, we prove the bounds for the X term coefficient. The upper bound follows from those for E (0) and H (0) in Lemmas 9 and 10, and again from the fact that ˜ h (1) and ˜ h ′ (1) are constants. As forthe lower bound, there are two difficulties: First, we do not know if the coefficient for A (0) in (7.8) isnegative, which would result in a negative X term contribution from (7.9). Secondly, we do not knowwhether the ˜ h ′ (1) factor in the first term of (7.10) is positive. Nonetheless, we can simply compare the(positive) X coefficient in the second term of (7.10) to that of the two terms just mentioned, and choose N large enough so that the log N factor dominates. The lower bound then follows again from Lemmas9 and 10. (cid:3) Now we just need to bound the remaining integral (7.7) with contour ( − ε ) . Using the triangleinequality and observing that ˜ G κ ( ℓ ) decays rapidly in fixed vertical strips, we therefore see that theintegral with contour ( − ε ) is bounded above by N ε X − ε/ . Error term. Using the same method for bounding the error term in the previous section, wecan bound S (cid:3) ( X ; χ, − , ℓ ) from above, since its expansion according to (7.4) will have the factor T ( s ; X/d , χ (2 ngN )0 ψ − ) . By Lemma 7 this becomes (7.6). It now remains to bound S ¯ (cid:3) ( X ; χ, ι, ℓ ) fromabove.According to (7.2) with the integral contour moved to (3 / ε ) , we see that S ¯ (cid:3) ( X ; χ, ι, ℓ ) equals ˆ (3 / ε ) ( c ( ℓ ) N ) s/ ˜ G κ ( ℓ ) ( s ) X ( d , N )=1 X f | d X g | f ( g,N )=1 µ ( g ) χ ( g ) g / X ng ∈ ¯ (cid:3) ( n,N )=1 χ ( n ) n s +1 / T ( s ; X/d , χ (2 N )0 ˜ χ ng ψ ι ) ds. Applying Lemma 7, since ng is not square, the character χ (2 N )0 ˜ χ ng ψ is never principal, so we have T ( s ; X/d , χ (2 N )0 ˜ χ ng ψ ι ) ≪ ( | s | ng ) / ε N ε ( X/d ) / ε . Now we can absolutely bound the sums and ignore the condition that ng is not square, as we did for(7.6). We note that the n -sum will absolutely converge since ℜ s = 3 / ε , as long as we select a smallenough ε in the bound for the T -function above. We then arrive at a bound of S ¯ (cid:3) ( X ; χ, ι, ℓ ) ≪ N / ε X / ε , whence by (7.3) we have sufficiently bounded S ¯ (cid:3) ( X ; χ ) .8. Appendix: Subconvexity bounds applied to non-vanishing results Though the non-vanishing result Theorem 2 proven in §6 does not require the subconvexity boundTheorem 1, one might ask whether a subconvexity bound could be applied to such a problem. Wepositively answer this here, outlining another method presented in [HK10].In §6, we start with (6.1) and move the contour of the integral from (2) to ( − ε ) , picking up a residueat w = 1 . Instead, we can move the contour to (1 / . Now it is clear that a good subconvexitybound for Z (1 / , w ; χ N , ψ ) with ℜ w = 1 / will produce a non-vanishing result. In our case, thesubconvexity bound Theorem 1 will only yield an upper bound of N / ε , which is worse than thatalready presented. However, this formulation gives motivation for developing subconvexity bounds for UBCONVEXITY FOR A DOUBLE DIRICHLET SERIES AND NON-VANISHING OF L -FUNCTIONS 25 double Dirichlet series. In particular, an improvement to Theorem 2 would result from an improvementin the N exponent lower than / ε in Theorem 1. Funding This work was supported by the Ontario Graduate Scholarship Program; and the European ResearchCouncil Starting Grant [258713]. 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Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario,Canada M3J 1P3 E-mail address ::