CENTRAL VALUES OF DERIVATIVES OF DIRICHLET L-FUNCTIONS

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1 International Journal of Number Theory Vol. 7, No. ( c World Scientific Publishing Company DOI:.4/S CENTRAL VALUES OF DERIVATIVES OF DIRICHLET L-FUNCTIONS H. M. BUI Mathematical Institute, University of Oxford Oxford, OX 3LB UK hung.bui@maths.ox.ac.uk MICAH B. MILINOVICH Department of Mathematics, University of Mississippi University, MS 38677, USA mbmilino@olemiss.edu Received November 9 Accepted 9 April Let C + be the set of even, primitive Dirichlet characters (mod. Using the mollifier method, we show that L (k (,χ for almost all the characters χ C + when k and are large. Here L(s, χ isthedirichletl-function associated to the character χ. Keywords: Dirichlet L-functions; non-vanishing of L-functions; mollifier method. Mathematics Subject Classification : M6, M6. Introduction and Statement of the Main Result An important topic in number theory is the behavior of families of L-functions and their derivatives inside the critical strip. In particular, uestions concerning the order of vanishing of L-functions at special points on the critical line have received a great deal of attention. In the case of Dirichlet L-functions, it is widely believed that L(,χ for all primitive characters χ. For uadratic characters χ, this appears to have been first conjectured by Chowla (see [3, Chap. 8]. Thougha proof of the non-vanishing of Dirichlet L-functions at the central point, s = /, has remained elusive, there has been considerable progress in showing that L(,χ is very often non-zero within various families of characters χ.in[], Iwaniec and Sarnak show that at least /3 of Dirichlet L-functions in the family of even primitive characters, to a large modulus, do not vanish at the central point. This improves upon earlier work of Balasubramanian and Murty []. Soundararajan [6] has shown that at least 7/8 of the central values in the family of uadratic Dirichlet L-functions are non-zero. More recently, Baier and Young [] consider the family 37

2 37 H. M. Bui & M. B. Milinovich of Dirichlet L-functions associated to cubic and sextic characters and show that infinitely many (though not a positive proportion of these functions are not zero at the central point. In [5], Michel and VanderKam consider the behavior of the derivatives of completed Dirichlet L-functions, Λ(s, χ, at the central point. (See Sec., below, for a definition. In particular, they show that for ε>and sufficiently large depending on ε, the ineuality (P k ε (. χ C + Λ (k (,χ χ C + holds, where the proportion P k = 3 36k c k 4 for some absolute constant c >. As k tends to infinity, the proportion P k approaches two thirds. This is analogous to a result of Conrey [4], who shows that almost all of the zeros of the kth derivative of the Riemann ξ-function are on the critical line, and to a result of Kowalski et al. [3] who show that almost half of the set { Λ (k (,f} is non-zero, where f runs over the set of primitive Hecke eigenforms of weight relative to (. This last result is best possible because half of these forms are even and half are odd. However, unlike the results in [4, 3], the ineuality in (. is not best possible since it is expected that P k =forevery positive integer k. In contrast to [5], we study the behavior of the functions L (k (s, χ, the derivatives of Dirichlet L-functions, at s =.Whenk and are sufficiently large, we show that L (k (,χ for almost all of the even, primitive characters χ. Asis the case in [4, 3], our result is asymptotically best possible as k tends to infinity. Theorem.. Let k N. Then, for ε>and sufficiently large (depending on ε, we have (Pk ε, (. χ C + L (k (,χ χ C + where the proportion Pk = 6k c k 4 (.3 for some absolute constant c>. Inparticular,P.7544,P.983, P3.964, P4.9853, P5.9935, and P Theorem. confirms a prediction of Conrey and Snaith which arises from the L-functions RatiosConjectures (see [8, 8.]. Their heuristic is based upon studying the behavior of the mollified moments of the derivatives of the Riemann zeta-

3 Central Values of Derivatives of Dirichlet L-Functions 373 function in t-aspect which they conjecture should behave similarly to the mollified moments of the derivatives of Dirichlet L-functions at the central point in -aspect. This is in agreement with the conjectures of Keating and Snaith [, ] that suggest that both of these families of L-functions, the Riemann zeta-function in t-aspect and Dirichlet L-functions in -aspect, should have the same underlying unitary symmetry and so their (mollified moments should behave similarly. See [5] fora detailed discussion of these ideas. In particular, our Proposition. is a -analogue of a result of Conrey and Ghosh a who computed the mollified moments of the derivatives of the Riemann zeta-function on the critical line. We remark that Theorem. does not improve upon the main result of [5]. In fact, for k N, the zeros of the functions L (k (s, χ andλ (k (s, χ are expected to behave uite differently. To illustrate this point, let χ be a primitive character and assume that the Riemann Hypothesis (RH χ holds for the function L(s, χ. Then all the non-trivial zeros of L(s, χ and all the zeros of Λ(s, χ lie on the critical line Re s =. In addition, L(s, χ has an infinite number of trivial zeros on the negative real axis. Under the RH χ, one can prove that all the zeros of Λ (k (s, χ lie on the line Re s =. In contrast, it can be shown that all but possibly a finite number of the non-real zeros of L (k (s, χ are forced to lie in the half-plane Re s and it is very likely the case that none of these zeros lie on the critical line. b In particular, it is reasonable to conjecture that L (k (,χ for all primitive characters χ and all k N. However, if χ is an even, real-valued, primitive (i.e. uadratic character, then the functional euation for L(s, χ states that Λ(s, χ =Λ( s, χ. It follows from this that Λ (k (,χ = whenever k is odd. Thus, the analogous conjecture for Λ (k (,χ fails for infinitely many values of k and infinitely many characters χ... Notation and conventions We say a Dirichlet character χ (mod isevenifχ( =. We let C denote the set of primitive characters (mod andletc + denote the subset of characters in C which are even. We put ϕ + ( = ϕ ( where ϕ ( = ( ϕ(kµ = C ; k k the proof of this appears in Lemma 4., below. It is not difficult to show that C + = ϕ + ( +O(. In addition, we write + χ (mod to indicate that the summation is restricted to χ C + and we write a (mod and n to indicate that the summation is restricted to the residues a (mod which are coprime to and to n which are relatively prime to, respectively. a See [6, E. (7]. b We can show that if is sufficiently large, then the only zeros of L (s, χ on the critical line are the multiple zeros of L(s, χ. However, it is believed that the zeros of L(s, χ are simple.

4 374 H. M. Bui & M. B. Milinovich. The Mollified Moments of L (k(,χ As may be expected, we prove Theorem. by computing certain mollified first and second moments of L (k (,χ over the characters χ C + and then we use Cauchy s ineuality. For χ C +,thedirichletl-function L(s, χ satisfies the functional euation ( s/ ( s Λ(s, χ := L(s, χ =ε χ Λ( s, χ, (. where χ is conjugate character of χ, ε χ = τ(χ /,andτ(χ is the Gauss sum τ(χ = ( a χ(ae ; e(x =e ix. a(mod Note that ε χ = and, since χ is even, τ(χ =τ( χ. For each χ C +,welet ( /n, (. M(χ =M(χ, P, y := µ(nχ(n P n n y where P is an arbitrary polynomial satisfying the conditions P ( = and P ( =. The purpose of the function M(χ is to smooth out or mollify the large values of L (k (,χ as we average over χ C +.Since ε χ L (k (, χ = L(k (,χ, if we let S (k, = ( + εχ L (k, χ M(χ (.3 and S (k, = χ (mod ( + L (k,χ M(χ, (.4 χ (mod then Cauchy s ineuality implies that + S (k, χ (mod L (k (,χ S (k,. (.5 Thus, we reuire a lower bound for S (k, and an upper bound for S (k,. The following propositions provide such estimates. Proposition.. Let k N. Then, for y = ϑ and <ϑ<, we have S (k, =( k ϕ + (log k ( + O((log, where the implied constant depends on ϑ and k. Proposition.. Let k be a positive integer and ε> be arbitrary. Then, for y = ϑ and <ϑ<, we have S (k, =C k (ϑ ϕ + (log k ( + O((log +ε,

5 Central Values of Derivatives of Dirichlet L-Functions 375 where C k (ϑ = ϑ P (x dx + k + + ϑk k and the implied constant depends on ϑ, ε and k. P (x dx, It is clear from (.5 and the propositions that in order to prove Theorem. we need to choose the polynomial P,foreachk, which minimizes the constant C k (ϑ. This is done in Sec. 6. It turns out that except for a term which is exponentially small (as a function of k, the optimal choice of P is independent of the choice of ϑ. This is not surprising, since similar phenomena have been observed when mollifying high derivatives of the Riemann zeta-function and the Riemann ξ-function on the critical line, and also when mollifying high derivatives of families of L-functions at the central point (see [4, 6, 3, 5]. 3. ProofofProposition. In this section we establish Proposition.. The result we reuire is implicit in [5, 3, p. 35] where it is shown that c ( + Λ (k(,χ M(χ =ϕ + ( ˆ / log k ˆ( + O((log (3. 4 χ (mod for k N and <ϑ<. Here ˆ = / and the implied constant depends on ϑ. From (., we see that ε χ L(s, χ =H (sλ( s, χ, where H (s = ˆ s ( s (3.. Using well-known estimates for the gamma function, it follows that ( H (k =( k ˆ / ( log k ˆ( + O k ((log (3.3 4 for each k N. Now, combining (3. (3.3 and using the Leibniz formula for differentiation, we find that ( + εχ L (k, χ M(χ χ(mod = + k χ(mod l= ( k H (l l ( ( k l Λ (k l (,χ M(χ c It follows from the functional euation for Λ(s, χ that the uantity L (P k in[5, 3] is eual to P + Λ(k` χ(mod,χ M(χ.

6 376 H. M. Bui & M. B. Milinovich = k ( ( k ( k l H (l + Λ (k l( l,χ M(χ l= =( k k l= χ(mod ( k ϕ + (log k ˆ( + O((log l =( k k ϕ + (log k ˆ( + O((log, where the implied constant depends on ϑ and k. Sincelogˆ =log + O(, we can conclude that ( + εχ L (k, χ M(χ =( k ϕ + (log k ( + O((log. χ(mod This establishes Proposition.. 4. Some Preliminary Results In this section, we collect some preliminary results which we will use to establish Proposition.. In what follows, is a large positive integer and α, β C are taken to be small shifts satisfying α, β (log. Our first lemma concerns the orthogonality of primitive characters. Lemma 4.. For (mn, =we have + χ(mχ(n = χ(mod =dr r m±n µ(dϕ(r, where the sums for the different signs ± aretobetakenseparately. Proof. Let f(h = χ(mχ(n χ(mod h where denotes summation over primitive characters χ. Thenfor(mn, =we have f(h = { ϕ(, if m n (mod, χ(mχ(n =, otherwise. h χ(mod Using Möbius inversion we obtain χ(mχ(n =f( = χ(mod h h m n ( ϕ(hµ. h

7 Central Values of Derivatives of Dirichlet L-Functions 377 It follows from this identity that C = ( = ϕ(kµ, k χ(mod which justifies an above remark. Our lemma now follows by noting that [ ] +χ( χ(mχ(n = χ(mχ(n. χ(mod χ ( = χ(mod Lemma 4.. Let G(s be an even, entire function with rapid decay as s in any fixed vertical strip A σ B and with G( =. Let W ± α,β (x = i +i i k G(sH(sg ± ds α,β (sx s s, (4. where ( ( /+α + s /+β + s g + α,β (s = ( (, /+α /+β ( ( / α + s / β + s g α,β (s = ( (, /+α /+β and ( α + β s H(s = ( (for α + β. α + β Then for χ,χ C + and α β we have that ( ( L + α, χ L + β,χ = ( χ (mχ (n mn m /+α n W + /+β α,β m,n ( α β ( χ (mχ (n mn + ε χ ε χ m / α n W / β α,β. m,n

8 378 H. M. Bui & M. B. Milinovich Remarks. ( An admissible choice of G in the above lemma is G(s =exp(s. ( The purpose of the function H(s in the above lemma is to cancel the poles of the functions ζ ( ± (α + β +s ats = (α + β/ which appear in the proof of the next lemma. This substantially simplifies our later calculations. A similar effect has been observed by Conrey et al. (see [7, 3]. Proof. Consider the integral I α,β = +i G(sH(s Λ(/+α + s, χ Λ(/+β + s, χ ( ( i i /+α /+β Shifting the line of integration to Re s = and using Cauchy s theorem, it follows that I α,β = R + i +i i ds s. G(sH(s Λ(/+α + s, χ Λ(/+β + s, χ ( /+α ( /+β where R is the residue of the integrand at s =.Evidently, ( (+α+β/ ( ( R = L + α, χ L + β,χ. By making the change of variables s to s and using (., we have that R = I α,β + +i G(sH(s Λ(/ α + s, χ Λ(/ β + s, χ ( ( i i /+α /+β The lemma now follows by using (. to express the Λ-functions in terms of Dirichlet series and then integrating term-by-term. Lemma 4.3. Let S + α,β (x = n= (n,= W + α,β (n /x n +α+β and S α,β (x = Then, for any ε> and α β, we have that and n= (n,= S + α,β (x =ζ ( + α + β+o(τ(x /+ε S α,β (x =g α,β (ζ ( α β+o(τ(x /+ε, W α,β (n /x n α β. where τ( is the number of divisors of and the function ζ (s is defined by ζ (s =ζ(s ( p s. p ds s, ds s.

9 Central Values of Derivatives of Dirichlet L-Functions 379 Proof. From (4., we observe that S + α,β (x = i +i i G(sH(sg + α,β (sxs ζ ( + α + β +s ds s. We now shift the line of integration left to Re s = /+ε, encountering only a simple pole of the integrand at s =. We note that the simple pole of ζ ( + α + β +s ats = (α + β/ is canceled by a zero H(s. The residue of the integrand at s =isζ ( + α + β. Also, the integral along the new contour is trivially τ(x /+ε. This implies the first claim of the lemma. The second claim can be proved in a similar manner. Lemma 4.4. Assume α β and let B(m,n ; α, β = ( ( + L + α, χ L + β,χ χ(m χ(n. χ (mod Then for (m,n =and (m n,=we have ( B(m,n ; α, β = ϕ+ ( ζ ( + α + β m n m β + nα where β(m,n satisfies + O(β(m,n + /+ε, m,n y ( α β g β(m,n m n y /+ε. α,β (ζ ( α β m α n β Proof. With χ = χ, χ = χ, Lemmas 4. and 4. imply that B(m,n ; α, β = + µ(dϕ(r =dr r mm ±nn ( α β µ(dϕ(r =dr ( mn W + α,β m /+α n /+β r mn ±nm ( mn W α,β, (4. m / α n/ β where denotes summation over all (mn, =. The main contribution to B(m,n ; α, β comes from the diagonal terms mm = nn and mn = nm in the first and second sums on the right-hand side of (4., respectively. For (m,n =,

10 38 H. M. Bui & M. B. Milinovich this contribution is ϕ + ( mm =nn = ϕ + ( ( mn W + α,β m /+α n + /+β S + α,β ( m n n /+α m /+β + ( α β mn =nm ( α β S α,β ( ( mn W α,β m / α n / β m n m / α n / β where S ± α,β (x are defined in Lemma 4.3. By Lemma 4.3, the above expression is eual to ϕ + ( ( ζ ( + α + β m n m β + nα ( α β g α,β (ζ ( α β m α n β, + O( /+ε. All the other terms in (4. contributeatmost β(m,n = ( (mm ± nn, mn W ± mn α,β. mm nn Using the estimate W ± α,β (x ( + x one can show that (see [, Sec. 4] β(m,n y /+ε ( 4. m n m,n y The lemma now follows from the above estimates. Lemma 4.5. Let S j (d = n y/d (n,d= µ(n n (log nj P ( /dn Then S j (d =M j (d+o(e j (d uniformly for d y, where ( d /d M (d = ϕ(dlogy P, M (d = d ( /d ϕ(d P, M j (d =(for j, and E j (d =(logy j (log ( + (d/y θ ( + p δ p d with θ / log and δ =/ log. Proof. Consider the Dirichlet polynomial G(z = ( µ(n /dn n +z P n y/d (n,d=..

11 Central Values of Derivatives of Dirichlet L-Functions 38 Since, for n y/d, wehave ( /dn P = l we can express G(z as G(z = l a l ( l (/dnl = l a l l! ( l i +i i a l l! ( l i +i i ( s y ds A(s + z d ζ( + z + ss l+ ( s y ds dn s l+, where A(s = p d ( p +s. We note that G(z is precisely G j ( + z in Lemma of Conrey [4] (see the first expression in the proof, with x being replaced by y/d and /F (j, s being replaced by A(s. Using this, we obtain that uniformly for < z /, where [ ( /d M (d; z =A(z zp ( /d M (d; z =A(zP G (j (z =M j (d; z+o(e j (d (4.3 M j (d; z = forj. + P, and ( /d Since G(z andm j (d; z are both holomorphic in z, (4.3 alsoholdsforz =. Observing that S j (d =( j G (j ( and A( = d/ϕ(d, the lemma follows. Lemma 4.6. Suppose that f(d = p d f(p with f(p =+O(p c for some c> and that J j (y = µ(d ( d f(d j. d d y Then we have J j (y = ( ( + f(p j + p p p p ], ( + f(p ( j+ + O(( j. p Proof. We consider only the case where j. The case j = can be handled by following [4, proof of Lemma 3.]. We first express J j (y as a complex integral, namely J j (y = j! +i i i (d,= µ(d f(d d +s y s ds s j+.

12 38 H. M. Bui & M. B. Milinovich The sum over d is p ( + f(p p +s = B(sζ( + s, where B(s = p [( ( p +s + f(p p +s ] p ( + f(p p +s. Since f(p =+O(p c forsomec>, B(s is absolutely and uniformly convergent in some half-plane containing the origin. We now shift the line of integration left to Re s = δ, crossing a pole of order j + ats =.Hereδ>issomesmall, fixed constant chosen so that the arithmetical factor B(s converges absolutely for Re s δ. Using Cauchy s theorem and the bound ζ(s ( + t /+δ on the new line of integration, we obtain the estimate J j (y = j + B((j+ + O(( j. The lemma now follows. 5. Proof of Proposition. In this section, we prove Proposition.. Throughout the proof, we let y = ϑ and assume that <ϑ<. We begin by considering the mollified shifted second moment J α,β ( = ( ( + L + α, χ L + β,χ M(χ, (5. χ(mod where α, β C are small shifts satisfying α, β (log and α β. Applying Lemma 4.4, we have that J α,β ( = ( ( µ(mµ(n /m /n P P B(m, n; α, β mn m,n y =Σ (α, β+σ (α, β+o(y /+ε, (5. where Σ (α, β =ϕ + ( ζ ( + α + β d y ( /dm P P m,n y/d (m,n= ( /dn µ(dmµ(dn dm +β n +α

13 Central Values of Derivatives of Dirichlet L-Functions 383 and Σ (α, β =ϕ + ( d y ( α β g α,β (ζ ( α β m,n y/d (m,n= µ(dmµ(dn dm α n β ( ( /dm /dn P P We can remove the restriction (m, n = by writing K α,β ( := Σ (α, β + Σ (α, β as ϕ + ( µ(cµ(cd c d where cd y m,n y/cd (mn,cd= µ(mµ(n P mn ( /cdm Z,α,β (m, n, c = ζ ( + α + β c α+β m β n α + P ( α β g ( /cdn. Z,α,β (m, n, c, (5.3 α,β ( ζ ( α β c α β m α. (5.4 n β Though the function ζ (s has a simple pole at s =,wenotethatz,α,β (m, n, c is holomorphic in both α and β in a small neighborhood of α = β =(ascanbe seen, for instance, by computing the Laurent series expansion of each of the terms on the right-hand side of (5.4 aboutα = β =. Therefore, the expressions in (5. and(5.3 provide an analytic continuation of the function J α,β ( K α,β ( to the region α, β (log ; the function K, ( must be defined in terms of the limit Z,, (m, n, c = lim α ( ζ ( + α (c mn α + ( α ζ ( α (c mn α. Moreover, by the maximum modulus principle and (5., we see that J α,β ( K α,β ( ε y /+ε uniformly for α, β (log. Hence, by Cauchy s Integral Theorem, d k dα k dβ k [J α,β( K α,β (] = (k! (i C α k,ε y /+ε, C β α=β= J wα,w β ( K wα,w β ( (w α w β k+ dw α dw β

14 384 H. M. Bui & M. B. Milinovich where C α (respectively, C β denotes the positively oriented circle in the complex plane centered at α = (respectively, β = with radius (log. Thus, we have shown that S (k, = dk dα k dβ k K α,β( + O k,ε (y /+ε. (5.5 α=β= Writing d k dα k dβ k Z,α,β(m, n, c α=β= = (a h,i,j (log c h + b h,i,j (log /c h (log m i (log n j h+i+j k+ for certain constants a h,i,j and b h,i,j,weseethat d k dα k dβ k K α,β( = ϕ + ( α=β= h+i+j k+ cd y (ah,i,j (log c h + b h,i,j (log c h µ(cµ(cd c S i (cds j (cd, d where S i and S j are defined in Lemma 4.5. It follows from Lemma 4.5 that S i (cd i cd ϕ(cd (i, (5.6 from which it can be seen that the contribution of the terms with h + i + j k to thesumontheright-handsideof(5.6 is k (log k ϕ + (/ϕ( k,ε ϕ + ((log k +ε. The last estimate holds since /ϕ( log log. It remains to consider the contribution of the terms with h + i + j =k +. In the notation of Lemma 4.5, it can be shown that cd y S i (cde j (cd c d i,j,ε ( i+j +ε and cd y E i (cde j (cd c d i,j,ε ( i+j 3+ε.

15 Central Values of Derivatives of Dirichlet L-Functions 385 Hence the contribution of the error terms E i and E j, arising from Lemma 4.5, to the terms in (5.6 withh + i + j =k +is k,ε ϕ + ((log k +ε.thus, d k dα k dβ k K α,β( = ϕ + ( α=β= h+i+j=k+ cd y (ah,i,j (log c h + b h,i,j (log c h µ(cµ(cd c M i (cdm j (cd d + O k,ε (ϕ + ((log k +ε. Since M i (cd =fori>, we need only to consider the terms with i, j. Moreover, the terms involving powers of log c can be ignored, as they contribute (due to the presence of c in the sum an amount which is k,ε (log k +ε. Therefore, the above expression simplifies to d k dα k dβ k K α,β( = T +T + T 3 + O k,ε (ϕ + ((log k +ε, (5.7 α=β= where T = ϕ + ( k+ µ(cµ(cd bk+,, (log c M (cd, d cd y T = ϕ + ( k µ(cµ(cd bk,, (log c M (cdm (cd d cd y and T 3 = ϕ + ( k µ(cµ(cd bk,, (log c M (cd. d cd y We first evaluate T. Using Lemma 4.5, we have that T = ϕ + ( b k+,, (log k+ ϕ( ( = ϕ + ( b k+,, (log k+ ϕ( ( Now Lemma 4.6 implies that µ(n ( /n ϕ(n P Hence n y T = ϕ + ( b k+,,(log k+ ϕ(logy cd y n y = ϕ( µ(cµ(cd ( d /cd ϕ(cd P µ(n ( /n ϕ(n P. ( + O( P (x dx. P (x dx + O k,ε (ϕ + ((log k +ε. (5.8

16 386 H. M. Bui & M. B. Milinovich Similarly, it can be shown that T = ϕ + ( b k,,(log k ϕ( P (xp (xdx + O k,ε (ϕ + ((log k +ε = ϕ + ( b k,,(log k + O k,ε (ϕ + ((log k +ε (5.9 ϕ( and that T 3 = ϕ + ( b k,,(log k P (x dx + O k,ε (ϕ + ((log k +ε. ϕ( (5. Thus, combining (5.5, (5.7 (5., and noting that b k+,, = ϕ( (k +, b k,, = ϕ( it follows that, for y = ϑ and <ϑ<, ( ϑ S (k, = P (x dx + k + + ϑk k + O k,ε (ϕ + ((log k +ε. This completes the proof of Proposition.. and b k,, = ϕ(k (k, P (x dx ϕ + ((log k 6. Completing the Proof of Theorem.: Optimizing the Mollifier We are now in a position to complete the proof of Theorem.. By Propositions. and., for<ϑ<,weseethat P k [ ϑ k + P (x dx + + ϑk k P (x dx]. (6. For each choice of k N, we wish to find a polynomial P satisfying P ( = and P ( = that maximizes the expression on the right-hand side of the above ineuality. Euivalently, we wish to minimize the expression F k (P := ϑ k + P (x dx + ϑk k P (x dx. (6. This optimization problem is solved explicitly in [5, Sec. 7] and, independently, in [6, p. 97]. We recall the argument given by Michel and Vanderkam in [5]. Using a standard approximation argument, the polynomial P can be replaced by any infinitely differentiable function with a rapidly convergent Taylor series on [, ]. In this case, using the calculus of variations, the optimization problem can be explicitly solved and, for k >, the optimal choice of P is k + P (t = sinh(λt sinh(λ, where Λ = ϑk k.

17 Central Values of Derivatives of Dirichlet L-Functions 387 Table. In the table, lower bounds for the proportions P k and Pk, defined in Es. (. and (., respectively. These calculations were performed by using the expression for F k (P given in (6.3 with ϑ = 8. k Lower bound for P k Lower bound for P k With this choice of P, it follows that F k (P = ΛcothΛ ϑ(k + = k coth Λ 4k. (6.3 As k gets large, the function coth Λ and so asymptotically (as k wehave F k (P = + ( 6k + O k 4. When combined with (6. and(6., this asymptotic formula is enough to establish theestimateforpk in (.3 and, thus, completes the proof of Theorem.. Acknowledgments Much of this research was completed while the first author was visiting the University of Rochester and the University of Mississippi. He would like to thank these institutions for their hospitality. He was supported by the EPSRC Postdoctoral Fellowship grant EP/F4748/. Both authors would like to thank Professor K. Soundararajan for pointing out an error in the original version of this manuscript and Professor Steven M. Gonek for his support and encouragement. References [] R. Balasubramanian and V. Kumar Murty, Zeros of Dirichlet L-functions, Ann. Sci. École Norm. Sup. 5 ( [] S. Baier and M. P. Young, Mean values with cubic characters, J. Number Theory 3 ( [3] S.D.Chowla,The Riemann Hypothesis and Hilberts Tenth Problem (Gordon and Breach Science Publishers, 965.

18 388 H. M. Bui & M. B. Milinovich [4] J. B. Conrey, Zeros of derivatives of Riemann s ξ-function on the critical line, J. Number Theory 6 ( [5] J.B.ConreyandD.W.Farmer,MeanvaluesofL-functions and symmetry, Int. Math. Res. Not. 7 ( [6] J. B. Conrey and A. Ghosh, Zeros of derivatives of the Riemann zeta-function near the critical line, in Analytic Number Theory (Allerton Park, IL, 989, Progr. Math., Vol. 85 (Birkhäuser Boston, 99, pp. 95. [7] J. B. Conrey, H. Iwaniec and K. Soundararajan, The sixth power moment of Dirichlet L-functions, preprint; arxiv at [8] J. B. Conrey and N. C. Snaith, Applications of the L-functions ratios conjectures, Proc. London Math. Soc. 94 ( [9] H.IwaniecandE.Kowalski,Analytic Number Theory, Amer.Math.Soc.Collo. Publ., Vol. 53 (Amer. Math. Soc., 4. [] H. Iwaniec and P. Sarnak, Dirichlet L-functions at the central point, in Number Theory in Progress, Vol. (de Gruyter, 999, pp [] J. P. Keating and N. C. Snaith, Random matrix theory and ζ(/+it, Comm. Math. Phys. 4 ( [] J. P. Keating and N. C. Snaith, Random matrix theory and L-functions at s =/, Comm. Math. Phys. 4 ( 9. [3] E. Kowalski, P. Michel and J. VanderKam, Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip, J. Reine Angew. Math. 56 ( 34. [4] N. Levinson, More than one third of zeros of Riemann s zeta-function are on σ = /, Adv. Math. 3 ( [5] P. Michel and J. VanderKam, Non-vanishing of high derivatives of Dirichlet L-functions at the central point, J. Number Theory 8 ( [6] K. Soundararajan, Nonvanishing of uadratic Dirichlet L-functions at s =, Ann. Math. 5 (