ON THE LOGARITHMIC DERIVATIVES OF DIRICHLET L-FUNCTIONS AT s = 1

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1 ON THE LOGARITHMIC DERIVATIVES OF DIRICHLET L-FUNCTIONS AT s = YASUTAKA IHARA, V. KUMAR MURTY AND MAHORO SHIMURA Abstract. We begin the study of how the values of L (χ, )/L(χ, ) are distributed on the comple plane C, theoretically and eperimentally, where χ runs over all non-trivial Dirichlet characters with a given large prime conductor m (L(χ, s): the associated Dirichlet L-function). In the theoretical studies, we shall first give some conditional results under the assumption of the Generalized Riemann Hypothesis (GRH), and then also give some unconditional result related to the estimation of the absolute value of the sum of L (χ, )/L(χ, ).. Introduction For each prime number m, let X m be the set of all non-principal multiplicative characters χ : (Z/m) C modulo m, and for each χ X m, let L(χ, s) denote the corresponding Dirichlet L-function. As is well-known, L(χ, ) 0. Consider the point sets DL ± (m) = {L (χ, )/L(χ, ); χ X m, χ( ) = ±}, on the comple plane C. DL(m) = DL + (m) DL (m), In this article, we begin the study of how DL ± (m) are distributed on C when m is large, theoretically and eperimentally. In the theoretical studies, we approach this mainly under the assumption of the Generalized Riemann Hypothesis (GRH), but some unconditional results will also be supplied. In, we ehibit the eplicit formula for L (χ, )/L(χ, ) that will be used later (Theorems, ). As for conditional results, we study in 3 the asymptotic behaviour of the quantities Ma{ L (χ, )/L(χ, ), χ X m }, and in 4, the moments P X m (±) χ X m (±) (L (χ, )/L(χ, )), Key words and phrases. Dirichlet L-function, gamma constant.

2 for polynomial functions P (z) = z a z b of z (Theorems 3, resp. 4). (Theorem 3 is given for any Dirichlet characters of a number field.) Some visible applications to distributions of L (χ, )/L(χ, ) will be discussed in. Finally, some unconditional results for the case P (z) = z will be given in 6. The average of values of the Dirichlet L-functions has been studied and applied to number theory by many authors. The reader may consult the monograph of K.Murty and R.Murty [9] for a discussion of results and references. As for the present subject related to that of the logarithmic derivatives of Dirichlet L-functions at s =, it arises from the work [4] of the first named author on the Euler-Kronecker invariant γ K = lim s ( ζ K (s) ζ K (s) + s associated to each global field K, where ζ K (s) is the Dedekind zeta function of K. If K denotes the Galois closure of K/Q, then γ K (which is some real number) can be epressed additively in terms of the value at s = of the logarithmic derivatives of certain Artin L-functions of Gal( K/Q) (each of which being, in general, a comple number). In particular, for the m-th cyclotomic field, γ Q(µm) = χ ) L (χ, ) L(χ, ) + γ Q, where the sum ranges over the non-trivial Dirichlet characters modulo m. Under GRH, it is shown ([4];Theorem ) that γ Q(µm) < ( + ɛ) log m, and it is conjectured ([] 3.4) that γ Q(µm) > 0 should always hold. Here, we shall give some conditional and unconditional bounds for γ Q(µm), as well as those for the Euler-Kronecker invariant of the maimal real subfield of Q(µ m ). In a more recent work of the first named author [6], the eplicit candidate (π) M σ (z)ddy for the density measure for the distribution of values of L (χ, s)/l(χ, s) for a general s (fied, with σ = Re(s) > /, m also moves) is proposed and studied. It is in fact the density measure when σ >, and (at least) under GRH, also when s =.. Eplicit Formulas.. The invariant γ K,χ Let K be a number field and χ be a primitive Dirichlet character (i.e., a primitive Hecke character with finite order) on K. Let L(χ, s) be the associated L function. When χ = χ 0,the principal character, it is the Dedekind zeta function ζ K (s) of K. As in [4], call γ K the constant term divided by the residue, of the Laurent epansion of ζ K (s) at s =. Put

3 (..) γk,χ = γ K + (χ = χ 0 ), = L (χ, )/L(χ, ) (χ χ 0 ). We shall use the following two basic formulas (Theorems,) for γ K,χ... The main formula For any >, put (..) Φ K,χ () = N(P ) k ( ) N(P ) χ(p ) k log N(P ) ( > ), k where the summation is over the pairs of a non-archimedean prime P of K and a positive integer k such that N(P ) k. (When P divides the conductor f χ of χ, we put χ(p ) = 0.) Note that Φ K,χ () is a continuous function of, and that when χ = χ 0, it is equal to the function Φ K () treated in [4]. Theorem. For any >, we have (..) γk,χ = δ χ log Φ K,χ ()+ ρ ρ( ρ) + a F ()+ a F 3()+r F (). ρ Here, δ χ = (resp. 0) for χ = χ 0 (resp. χ χ 0 ), ρ runs over all non-trivial zeros of L(χ, s) (counted with multiplicities), (..3) = lim, ρ T Im(ρ) <T a (resp. a ) is the number of real places of K where χ is unramified (resp. ramified), r = a + a (resp. r ) is the number of real (resp. comple) places of K, and F () = log log, (..4) F 3 () = log + log, + F () = (F () + F 3 ()) = log + log. Note that F i () (i=,,3) are positive real valued functions of vanishing at =. When χ = χ 0, (..) follows directly from the formulas (..),(.4.) of [4]. By letting in (..), by the same argument as in [4] (.6), we obtain the following result. Corollary... When χ χ 0, (..6) L (χ, ) L(χ, ) = lim Φ K,χ() = lim N(P ) k ( ) k χ(p ) log N(P ). N(P ) 3

4 .3. A supplementary formula Theorem. Let d K be the discriminant of K, and f χ be the conductor of χ. Put d χ = d K N(f χ ), and { α K,χ = (.3.) log d χ, β K,χ = a+r (γ Q + log 4π) a +r (γ Q + log π). Then (.3.) γ K,χ = ρ ρ α K,χ β K,χ. The equality for the real part of (.3.) can be found in some standard tets in analytic number theory, but we could not find any references for (.3.) itself. It is not a priori clear that the sum of ( ρ) over ρ in the sense of (..3) (which is the same as the sum of ρ over the non-trivial zeros ρ of L( χ, s)) converges at all. In any case, this is classical, for it can be proved by classical methods as indicated below. The readers note that (.3.) is in a sense related to (..) for, but not in an obvious sense, as ρ and lim do not commute with each other (and F i () + as (i=,,3)). Finally, if χ = χ 0, then α K,χ = α K, β K,χ = β K, and (.3.) is nothing but (.4.) of [4]..4. Indications for verifications of (..) and (.3.) Fi K and χ, and for any > and σ R, define the function Ψ (σ) () by (.4.) Ψ (σ) () = ( ) k χ(p ) log N(P ), N(P ) σ N(P ) k < when N(P ) k for any P and k, and by Ψ (σ) () = (Ψ(σ) ( + 0) + Ψ (σ) ( 0)) otherwise, so that (.4.) Φ K,χ () = ( Ψ () () Ψ (0) () ). The eplicit formulas for Ψ (σ) () (σ = 0, ) obtained from Weil s general formula [0] (specialized as indicated in [4].3) read as follows. (.4.3) Ψ (0) () = δ χ ( + log ) ρ ρ ρ + log d χ a+r (γ Q + log π + log( )) a +r 4 (γ Q + log 4π + log( + )),

5 (.4.4) Ψ () () = δ χ (log + ) ρ ρ + log d ρ χ a+r (γ Q + log 4π log( + )) a +r (γ Q + log π + log( )). From (.4.), (.4.3) and (.4.4), it follows directly that ( ) Φ (.4.) K,χ () = δ χ log + ρ ρ + log d ρ( ρ) ρ χ a+r (γ Q + log 4π F ()) a +r (γ Q + log π F 3 ()). Now, since ρ ρ converges, so does ρ ρ ρ( ρ). Hence by (.4.), the sum ρ ( ρ) (in the sense of (..3)) also converges. Note that (.4.) differs from (..) just by (.3.). On the other hand, Lagarias-Odlyzko [7], (7.3) gives (.4.6) Ψ (0) () = δ χ ( + log ) γ K,χ ρ + ρ and the same method gives also (.4.7) ρ( ρ) a+r log(( )/4) a +r (log( 4( ) + ), Ψ () () = δ χ (log + ) γk,χ ρ a+r log( ) a +r log( ). + Note that the difference between (.4.3) and (.4.6), and also that of (.4.4) and (.4.7), are both eactly the same supplementary formula (.3.). This (doubly) proves (.3.) and hence also (..). The method of [], [7] is based on the formulas (.4.8) Ψ (σ) () = πi where c + σ >, and c+i c i (.4.9) Z K,χ (s) = L (χ, s) L(χ, s) = P,k ρ ρ s s Z K,χ(s + σ)ds ( χ(p ) N(P ) s ρ ρ ) k log N(P ) for Re(s) >, and on the computation of the integral Ψ (σ) () in terms of residues. The term γk,χ appears as the residue at s = σ.

6 3. A GRH-bound for L (χ, )/L(χ, ) 3. By using Theorem and Theorem, we can easily obtain an upper bound for L (χ, )/L(χ, ) under the Generalized Riemann Hypothesis (GRH). This bound makes sense when the norm N(f χ ) of the conductor of χ grows faster than the absolute value d K of the discriminant of K. Theorem 3. (Under (GRH)) Let χ be a non-principal primitive Dirichlet character of a number field K with conductor f χ. Then (3..) L (χ, )/L(χ, ) < (log log d χ + ) γ K + O( log d K + log log d χ log d χ ). Here, d χ = d K N(f χ ) and γ K = γ K +, γ K being the Euler-Kronecker invariant of K. Note that the RHS of (3..) is positive (modulo the error term), by [4], Theorem. As the proof makes evident, the implicit constant in the error term can be given eplicitly whenever needed. An eplicit bound for the case K = Q will be given later. Remark. Numerical eperiments indicate that the coefficient of (log log d χ +) on the RHS of (3..) can probably be replaced by. We assume and use (GRH) in the following proof of Theorem 3, but we are not able to make use of cancellations among the ρ-terms in the eplicit formula for Φ K,χ (), and this seems to be the cause of the difference. Figure for K = Q plots the points Q m = (m, Ma χ Xm L (χ, )/L(, χ) ) R, where X m is the set of all primitive characters mod m, and m runs over all prime numbers with 3 m 769, given together with the graph of log log m. Remark. An interesting case is when K = Q and χ is a quadratic character such that L(χ, ) > 0 and that L (χ, )/L(χ, ) is as large as one can find, in particular, L (χ, )/L(χ, ) >. Consider, in this case, the graph of the real valued function t = L(χ, s) on the real s-line. Then the tangent line at s = given by the equation t = L(χ, ) + L (χ, )(s ) meets the real ais at L(χ, )/L (χ, ) which lies between / and. So, when we look at the graph at s = towards left, the initial direction of the graph is going to violate the GRH. The larger the value of L (χ, )/L(χ, ) is, the stronger this tendency. But to our great ease, as the following graphs of t = L(χ, s) show, such 6

7 Figure. Q m and (m, log log m) violations do not easily happen... (i) m = 977, χ( ) =, L (χ, )/L(χ, ) =.6... (ii) m = , χ( ) =, L (χ, )/L(χ, ) = This point of view for (GRH) is, of course, sort of old-fashioned. But in old days, Figure. m = 977 Figure 3. m = techniques for etended computations were even more restricted. worthwhile pursuing this direction further now. So, it maybe 3.. Proof of Theorem 3 By Theorem, (3..) L (χ, )/L(χ, ) = Φ K,χ () + ρ ρ( ρ) + a + r ρ 7 F () + a + r F 3 ()

8 holds for any >. Under (GRH), ρ( ρ) = ρ ρ > 0 and ρ + ; hence by Theorem, (3..) ρ ρ ρ( ρ) ρ( ρ) = ρ (Re(L (χ, )/L(χ, )) + α K,χ + β K,χ ) ( L (χ, )/L(χ, ) +α K,χ ). And since (3..3) and a + r F () + a + r F 3 () (3..4) Φ K,χ () Φ K (), we obtain (3..) ( = O((r + r )(log )/) = O((α K + )(log )/), ) L (χ, ) L(χ, ) < Φ K() + α K,χ + O((α K + ) log /), where the implicit constant is independent of, K and χ. On the other hand, by using Theorems, for χ = χ 0, we obtain similarly, (3..6) Φ K () < log γ K + (γ K + α K) + O( (α K + ) log ). Hence (3..7) ( ) L (χ, ) L(χ, ) < log γ K + (γ K + α K,χ + α K ) + O( (α K + ) log ). Now recall that α K,χ = log d χ, and take = αk,χ. Then the above formula gives (3..8) ( ) αk,χ 3 α K,χ L (χ, ) L(χ, ) < log α K,χ γ K + γ K α K,χ + + O(α K + ) α K,χ < (log α K,χ + ) γ K + O( α K + α K,χ ), 8

9 by Theorem of [4]. To obtain an upper bound for L (χ, )/L(χ, ) itself, we take α K,χ > 3 and add to the RHS of (6) its multiple by (α K,χ 3). But since ([4] Prop.3), we have and hence γ K < α K + β K < α K γ K/(α K,χ 3) = O((α K + )/α K,χ ), (3..9) L (χ, ) L(χ, ) < (log α K,χ + ) γ K + O( α K + log α K,χ α K,χ ), as desired The case K = Q When K = Q, we can use the inequalities Φ Q () < log ([4], (.6.36)) and F () < (log + ), (3.3.) F 3 () < + /+ log log <, to obtain the following result. Corollary (Under (GRH)) When K = Q, we have (3.3.3) ( log d χ 3 log d χ ) L (χ, ) L(χ, ) < (log log b + χ( ) log d χ + ) log d χ log log d χ+ (if χ( ) = ), (log d χ) + log +/4 (if χ( ) = ), where b = γ Q + log π = (log d χ) 4. Mean values of F (L (χ, )/L(χ, )) (χ X m ) for polynomials F of z, z. 4. Now let K = Q, and m run over the odd prime numbers. For each such m, denote by X m the collection of all non-principal characters χ : (Z/m) C. We shall study the limit, as m, of the mean value of F (L (χ, )/L(χ, )) (for χ X m ) 9

10 for the basic polynomial functions F of z, z, i.e., F (z) = P (a,b) (z) = z a z b, where (a, b) is any pair of non-negative integers. As usual, Λ(n) = log p when n is a positive integral power of a prime number p, and Λ(n) = 0 if either n = or n has at least two prime factors. For each non-negative integer k, define the arithmetic function Λ k (n) by (4..) (4..) Λ k (n) = Λ 0 (n) = (n = ), = 0 (n > ), n=n n k Λ(n ) Λ(n k ) (k > 0). For eample, Λ (n) = Λ(n), and Λ (n) = log p log q when n has eactly two prime factors p, q, Λ (n) = (r )(log p) when n = p r (r ), and Λ (n) = 0 when n has more than two prime factors. In general, Λ k (n) = 0 unless the number of prime factors of n is at most k and the sum of eponents in the prime factorization of n is at least k. It is easy to see that ( ) r (4..3) Λ k (p r ) = (log p) k k for k r, and that (4..4) Λ k (n) (log n) k. In fact, Λ k (n) is the Dirichlet coefficient determined by ( ζ (s)/ζ(s)) k Λ k (n) =. n s So, when the prime factorization of n is of the form n = p r p r j j, Λ k(n) is given as the coefficient of r r j j in the polynomial G(,, j ) k, where G(,, j ) = j i= n= (log p i )( i + r i i ). Since all coefficients of G(,, j ) k are non-negative, Λ k (n) G(,, ) k = (log n) k. For each pair (a, b) of non-negative integers, put (4..) µ (a,b) = µ (b,a) Λ a (n)λ b (n) =. n n= 0

11 Note that µ (0,0) =, µ (a,0) = 0 for any a > 0, µ (a,b) > 0 in all other cases, and (4..6) µ (a,) = p In particular, (4..7) µ (,) Λ(n) = = n n= p Put Φ χ () = Φ Q,χ (), so that (4..8) Φ χ () = (log p) a+ (p ) a (a > 0). (log p) p = ( n )Λ(n)χ(n). n Theorem 4. For each pair (a, b) of non-negative integers, and m, we have ( ) (log ) (4..9) P (a,b) (Φ χ ()) = µ (a,b) a+b+ + O. X m m χ X m This remains valid if X m is replaced by the set of all even (resp. odd) characters. The implicit constant depends on a and b. We also have the following lemma. Lemma. (Under (GRH)) For >, we have ( L (χ, ) log m (4..0) L(χ, ) = Φ χ() + O In particular, choosing = m, we get L (χ, ) L(χ, ) = Φ χ(m ) + O + log ) ( log m m ). (χ X m ) Putting these together, we obtain the following result. Corollary 4... (Under (GRH)) ( ) (log m) (4..) P (a,b) (L (χ, )/L(χ, )) = ( ) a+b µ (a,b) a+b+ + O. X m m χ X m In particular, (4..3) lim P (a,b) (L (χ, )/L(χ, )) = ( ) a+b µ (a,b). m X m χ X m These remain valid if X m is replaced by the set of all even (resp. odd) characters.

12 4. Proof of Theorem 4 First, since Φ χ0 () = O(log ), we may include the principal character in proving the theorem. Write X m = X m {χ 0 }. Put (4..) µ a,b () = X m = X m χ X m χ X m P (a,b) (Φ χ ()) Φ χ () a Φ χ () b. Then the orthogonality relations for characters give directly that (4..) µ a,b () = where (4..3) (4..4) λ (k) (c, ) = m c= λ (a) (c, )λ (b) (c, ), λ (0) (c, ) = (c = ), ( ) k Now for each positive integer N, put (4..) L (k) (N, ) = so that (4..6) λ (k) (c, ) = = 0 (c > ), n,,n k < n n k c (mod m) ( ) k n,,n k < n n k =N [ k /m] l=0 k i= k i= ( ) Λ(n i ). n i ( ) Λ(n i ), n i L (k) (c + lm, ). We shall show that the terms with l > 0 are altogether negligible, and that the term with l = 0 can be epressed as the sum of a simpler quantity and a negligible one. To see these, we first note that L (k) (N, ) 0 only when N < k and that in this case (4..7) L (k) (N, ) N n,,n k <n n k =N Λ(n ) Λ(n k ) N Λ k(n) (log N)k N

13 From this, it follows immediately that the sum of terms with l > 0 in (4..6) is O((log ) k+ /m). Therefore, (4..8) λ (k) (c, ) = L (k) (c, ) + O((log ) k+ /m). Now we shall show that (4..9) L (k) (c, ) = Λ k(c) + O((log m) k /). c This is based on a very simple inequality. For any > 0 and i, j, we have ( i)( j) ( )( ij); hence for any n,, n k and > n n k, (4..0) ( ) k ( n ) ( n k ) ( ) k ( n n k ). This gives directly that k (4..) 0 i= n i ( ) k k i= ( ) c n i c( ) for any n, n k with n n k = c and m. Therefore, k ( ) k ( ) (4..) = + O, ( ) k n i n i i= which, together with (4..4) gives (4..9). Since Λ k (c) = O((log m) k ), we obtain, from (4..),(4..8) and (4..9), m ( ) (4..3) µ a,b Λ a (c)λ b (c) (log ) a+b+ () = + O. c m But since (4..4) Λ a (n)λ b (n) n n m n m c= (log n) a+b n the first statement of the theorem follows. m i= ( ) (log t) a+b (log m) a+b dt = O, t m The statement for even (and hence also for odd) characters follows from an easy estimation (4..) m c= Λ a (c)λ b (m c) c(m c) This completes the proof of Theorem 4. = O( (log m)a+b+ ). m Proof of Lemma 3

14 The main eplicit formula (..) gives for any > L (χ, ) (4..6) L(χ, ) + Φ χ() = ρ ρ( ρ) + F χ( )(). By (3.3.), the second term on the RHS has order at most O(log /). Moreover, under the GRH, the absolute value of the first term is at most ρ ρ( ρ) by Theorems and 3. = ρ ( Re L (χ, ) L(χ, ) + log m γ Q log π χ( ) log = O( log m ) ). Applications to distributions of L (χ, )/L(χ, ). In this section, we shall give several remarks on the distribution of L (χ, )/L(χ, ) which can be deduced from Theorem 4. The case a + b =. (..) (..) Since γ Q(µm) = γ Q + χ χ 0 L (χ, )/L(χ, ), γ Q(µm) + = γ Q + χ χ 0, χ( )= L (χ, )/L(χ, ), this case of Corollary 3 gives the following estimates. Corollary..3. Assuming the GRH, we have (..4) γ Q(µm), γ Q(µm) + = O ( (log m) ). For an unconditional version, see 6.. The case a + b =. If we write z = + iy (, y R), then Theorem 4 in this case says that if F (, y) = C ( + y ) + C ( y ) + C 3 y, then the average of F (L (χ, )/L(χ, )) tends to C µ (,). (Note also that the linear span of y, y is the kernel of the Laplacian.) To rephrase this, for each α, β C, let (α, β) denote the real inner product; (..) (α, β) = Re(α β) = Re(ᾱβ) = α β cos(arg(β) arg(α)). In particular, (, β) = Re(β), (i, β) = Im(β). 4

15 Corollary..6. lim m X m χ X m Let α C with α =. Then ( ) α, L (χ, ) = L(χ, ) lim m X m L (χ, ) L(χ, ) = µ(,). χ X m This remains valid if X m is replaced by the collection of all even (resp. odd) nonprincipal characters mod m. The case a + b = 3. In this case, µ (,) = This shows that the average of (Re(L (χ, )/L(χ, ))) 3 for large m is (3/4) µ (,), which is negative! The same argument shows that the average of (Re(L (χ, )/L(χ, ))) k is negative for large m, for any odd integer k 3. What this indicates is that the distribution of L (χ, )/L(χ, ) on the left half plane should be more wide-spread than accumulated near the y-ais, in contrast to the distribution on the right half plane. We do not see this so evidently from eamples..., probably because when m is relatively small, the contribution of the principal character, i.e., (log m) k /m in the theoretical computation, is already so negative that the remaining distribution need not be so scattered on the left half plane to make the total average of k-th powers negative. The case (a, b) = (, ). µ (,) =..., which is considerably larger than the square of µ (,). The result for a + b = alone does not eclude the possibility that the points L (χ, )/L(χ, ) are distributed near the circle with center O and radius µ (,) with their arguments nearly uniformly distributed. But this result for (, ) shows that this cannot be the case. We have computed other higher powers; for eample, we can show that the average of (Re(L (χ, )/L(χ, ))) 8 for large m eceeds.09. Eamples for some m points Figure 4 (resp. ) plots on the comple plane C the (..7) L (, χ)/l(, χ) (χ X m, χ( ) = resp. ) when m = 63. Figures 6 (resp. 7) are for m = 733. It seems that these m are not big enough to support the theoretic consequence of (GRH) mentioned in the paragraph related to the case a + b = 3. (For eample, the average of the cubic power of the real part of the plotted points in Figure 4 is, as can be guessed from the picture, positive. But if one adds the contribution of the principal character, i.e., -(Φ χ0 (m)) 3 / X m + ( (log m) 3 /(m/)), it becomes negative as epected theoretically. The contribution of the principal character tends to 0 as m tends to, but it is not small enough for our m.)

16 Figure 4. m = 63 even characters Figure. m = 63 odd characters Figure 6. m = 733 even characters Figure 7. m = 733 odd characters 6. Unconditional Results: Lemmas 6. In this section, we establish some unconditional result on the sum of L (χ, )/L(χ, ). Our aim is to prove the following result. Theorem. We have X m L (χ, ) L(χ, ) χ X m = O(m ), where the implied constant is effective. For every ɛ > 0, we have the following ineffective result. L (χ, ) = O ɛ (m +ɛ ) X m L(χ, ) χ X m Corollary 6... Unconditionally, we have (6..) γ Q(µm), γ Q(µm) + = O ɛ (m ɛ ). Remark 3. The same techniques should give the average of P (a,b) (L (χ, )/L(χ, )), an unconditional version of Cor 4... However, we have not worked out the details. 6

17 6. As before, we consider the function ( ) Φ χ () = n χ(n)λ(n). n It is well known (see, for eample, []) that there is an effective and absolute constant c > 0 so that for T >, in the region c (6..) σ <, t T log mt the product χ L(s, χ) has at most one zero. This zero, if it eists, is real and simple and belongs to the (unique) quadratic character χ. We will denote it β. Proposition. We have (6..) ρ( ρ) (log m) where the sum is over zeros ρ β of L(s, χ) with 0 < Re(ρ) <. For the eceptional zero, we have the effective estimate (6..3) β ( β ) m. Moreover, for any ɛ > 0, we have the ineffective estimate (6..4) β ( β ) ɛ m ɛ Remark 4. The estimate (6..) in the above Proposition can be improved to log m provided we can get a better bound for zeros with Im(ρ) <. Proof. For the eceptional zero, we have the effective estimate β m and the ineffective estimate β m ɛ and (6..3) and (6..4) follow from that. For the other zeros, we know that for any T > 0, we have the estimate (6..) #{ρ : γ T < } log(m(t + )). Using this and the fact that the non-eceptional zeros do not satisfy (6..), we see that ρ( ρ) (log m). γ < 7

18 Moreover, for any j, we have ρ( ρ) (log mj). j j γ <j+ Now (6..) follows by summing over j. Lemma. We have for T >, ρ ρ( ρ) γ >T log mt T. Proof. By (6..) and partial summation, this is and this is This proves the lemma. Let us set T log mt dt t log mt T S(, m, T ) = χ. γ T ρ ρ( ρ) where the sum ranges over non-eceptional zeros. Then, from (3..), (3..3), Proposition and Lemma, it follows that for > and T >, (6..6) L (χ, ) L(χ, ) = Φ χ () + γ T ρ ρ( ρ) + E + E where E is the term corresponding to the eceptional zero (if χ = χ and if it eists) and Thus (6..7) where χ (6..8) E E L (, χ) L(, χ) (log m) + = χ Φ χ () + φ(m)(log m) + log mt T φ(m) log mt T 8 + log. S(, m, T ) + E + φ(m) log + m.

19 6.3. Estimation of S(, m, T ) We shall now begin the estimate of S(, m, T ). For this purpose, we shall need zero density theorems. We set ρ = β + iγ. For / < σ <, let us write N(σ, T, χ) = #{ρ = β + iγ zero of L(χ, s), β σ, γ T }. For σ > 4, we can use the result of M.N.Huley and M. Jutila [3] that N(σ, T, χ) (mt ) (+ɛ)( σ) χ where the implied constant depends on ɛ. For σ 4, we use an estimate of the form N(σ, T, χ) (mt ) A( σ) (log mt ) B χ with some postive constants A, B. Such results are known with various values of A and B (for eample, by [8] Th., A =., B = 4). Our aim is to prove the following result. Theorem 6. For we have ma(m 3, (mt ) A (log mt ) B, m 0 (log mt ) 0B, (m (log mt ) B ) A ) log mt S(, m, T ) log + + log m. In particular, if we choose T = m and = cm A+0 (log m) (A+0)B for a sufficiently large constant c, then (6.3.) S(, m, m). Proof. For a fied χ, we have (6.3.) ρ,0 γ T ρ ρ( ρ) = σ T 0 it dn(σ, t, χ)dσ (σ + it)( σ it) where the sum on the left is over zeros ρ of L(χ, s) in the specified range. By the zero-free region (6..) for L(s, χ), the upper limit on the outer integral (6.3.) can be replaced with c λ = log mt. 9

20 Now, integrating by parts, we have (6.3.3) T it dn(σ, t, χ) 0 (σ + it)( σ it) [ it = N(σ, t, χ) (σ + it)( σ it) ] T 0 T 0 N(σ, t, χ) d dt ( ) it dt. (σ + it)( σ it) We consider the contribution of the first term of (6.3.3) in (6.3.). Integrating over σ, it is λ [ ] σ it T N(σ, t, χ) dσ (σ + it)( σ it) 0 (6.3.4) + 4 λ σ N(σ, T, χ)dσ + T σ N(σ, 0, χ)dσ. σ( σ) λ 4 σ N(σ, T, χ)dσ T Using the zero density theorems quoted above, we have λ σ N(σ, T, χ)dσ λ σ (mt ) (+ɛ)( σ) dσ 4 T T 4 χ (6.3.) = λ ( ) σ ( (mt )+ɛ dσ m (mt ) ɛ T (mt ) +ɛ (mt ) +ɛ Assuming that the above is 4 (mt ) +ɛ T. (Here, note that (mt ) λ = e c ). Similarly, we have (6.3.6) 4 σ T χ (mt )A (log mt ) B T N(σ, T, χ)dσ T 4 ( ) 4 (mt ) A log ) λ log σ (mt ) A( σ) (log mt ) B dσ (mt ) A. (mt ) +ɛ Assuming = (mt )A/ T 4/ (log mt )B log. (mt ) A (mt ) A (log mt ) B 0

21 this is T. Using similar arguments, we also see that the final term is λ σ σ( σ) N(σ, 0, χ)dσ χ Now, we estimate the other term. λ T (6.3.7) σ N(σ, t, χ) d ( dt 0 χ log mt log /m. A it (σ + it)( σ it) ) dtdσ First, note that ( ) d it = d ( ) it + d ( ) it dt (σ + it)( σ it) dt σ + it dt σ it (6.3.8) ( ) ( ) i it = (log ) + i it (σ + it)( σ it) ( σ it). (σ + it) The integral from 4/ to λ in being small, we have only to deal with the integral from / to 4/. We have λ T σ i it (log ) N(σ, t, χ) (6.3.9) 4 log log 0 χ λ 4 λ 4 (σ + it)( σ it) dtdσ T σ (m( t + )) (+ɛ)( σ) (σ + it)( σ it) dtdσ 0 T σ m (+ɛ)( σ) t (+ɛ)( σ) dtdσ λ ( ) σ = (log )m +ɛ m +ɛ 4 (log )m +ɛ λ 4 T ( m +ɛ ) σ dσ t ɛ (+ɛ)σ dtdσ Note that + ɛ ( + ɛ)σ < 0. The above is seen to be ( ) λ (6.3.0) m +ɛ log m +ɛ log and this is m +ɛ

22 provided Finally, we have m 3. (6.3.) 4 T σ i it log N(σ, t, χ) (σ + it)( σ it) dtdσ log log χ T σ (m( t + )) A( σ) (log m( t + )) B (σ + it)( σ it) dtdσ T σ m A( σ) t A( σ) (log mt) B dtdσ. As A( σ) < 0 for σ >, this is A m A (log )(log mt ) B A ( T A m A ) σ T A dσ + and this is majorized by the sum of two epressions and If we require m A (log )(log mt ) B log /(mt ) A m A/ 4/ (log mt ) B log log /m. A 4/ A ma(m 0 (log mt ) 0B, (m (log mt ) B ) A ), then the sum of the above quantities is Moreover, we find that for (log )/m. ( m A ) σ dσ ma(m 3, (mt ) A (log mt ) B, m 0 (log mt ) 0B, (m (log mt ) B ) A ) we have log mt S(, m, T ) log This completes the estimation of S(, m, T ). + + log m.

23 6.4. Proof of Theorem From (6..7), it follows that L (, χ) = Φ χ () + L(, χ) χ χ S(, m, m) + E(, m, m). Now using the estimate (6..8), as well as (6.3.), it follows that for a large power of m as in Theorem 6, we have L (, χ) = Φ χ () + O( m). L(, χ) χ χ Applying Theorem 4 and dividing through by X m, the result follows. References [] H.Davenport, Multiplicative Number Theory, Second Edition (Revised by H.Montgomery), Graduate Tets in Mathematics, Springer-Verlag. [] M. Huley, Large values of Dirichlet polynomials III, Acta. Arith., 6(974/7), [3] M.N.Huley,M.Jutila, Large values of Dirichlet polynomials IV, Acta. Arith., 3(977), [4] Y.Ihara, On the Euler-Kronecker constants of global fields and primes with small norms, in: Algebraic Geometry and Number Theory, In Honor of Vladimir Drinfel d 0th Birthday, (V.Ginzburg ed.), Progress in Mathematics 3 (006), pp 407-4, Birkhauser. [] Y.Ihara, The Euler-Kronecker invariants in various families of global fields, to appear in Proceedings of AGCT 00 (Algebraic Geometry and Coding Theory 0) (F.Rodier et al. ed.), séminaires et congrès, Soc. math. de France. [6] Y.Ihara, On M-functions closely related to the distribution of L /L-values, RIMS-74 (Kyoto University RIMS preprint series), December 006. [7] J.C.Lagarias, A.M.Odlyzko, Effective versions of the Chebotarev density theorem, in: Algebraic Number Fields, (A.Fröhlich, Ed.) Proc. of the 97 Durham Symposium, Academic Press, London & New York, (977), [8] H.L.Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math.7, Springer Verlag 97. [9] M. Ram Murty and V. Kumar Murty, Non-vanishing of L-functions and applications, Birkhauser, Basel, 998. [0] A.Weil, [9b] Sur les formules eplicites de la théorie des nombres premiers, Comm. Sém. Math. Lund, (9), -6. (Collected Works, Vol, p48-6). (Yasutaka Ihara) COE, Chuo University, Kasuga -3-7, Bunkyo-ku, Tokyo - 8, Japan (through March, 07);RIMS, Kyoto University, Sakyo-ku, Kyoto , Japan (After April, 07) (V. Kumar Murty) Department of Mathematics, University of Toronto, Toronto, CANADA MS 3G3 3

24 (Mahoro Shimura) th Labo. Department of Mathematical Sciences, School of Science, Tokai University, 7, Kita-kaname, Hiratsuka, Kanagawa-Pref. 9-9, Japan address, Yasutaka Ihara: address, V. Kumar Murty: address, Mahoro Shimura: 4