On the Euler-Kronecker constants of global fields and primes with small norms

Transcription

1 On the Euler-Kronecker constants of global fields and primes with small norms Yasutaka Ihara Dedicated to V.Drinfeld Introduction Let K be a global field, i.e., either an algebraic number field of finite degree (abbrev. NF, or an algebraic function field of one variable over a finite field (FF. Let ζ K (s be the Dedekind zeta function of K, with the Laurent expansion at s = : (0. ζ K (s = c (s + c 0 + c (s + (c 0. In this paper, we shall present a systematic study of the real number (0. γ K = c 0 /c attached to each K, which we call the Euler-Kronecker constant (or invariant of K. When K = Q (the rational number field, it is nothing but the Euler-Mascheroni constant γ Q = lim ( n log n = , n and when K is imaginary quadratic, the well-known Kronecker limit formula expresses γ K in terms of special values of the Dedekind η function. This constant γ K appears here and there in several articles in analytic number theory, but as far as the author knows, it has not played a main role nor has it been studied so systematically. We shall consider γ K more as an invariant of K. Before explaining our motivation for systematic study, let us briefly look at the FFcase. When K is the function field of a curve X over a finite field F q with genus g, so that ζ K (s is a rational function of u = q s of the form (0.3 ζ K (s = g ν= ( π νu( π ν u, π ν π ν = q ( ν g, ( u( qu then γ K is closely related to the harmonic mean of g positive real numbers (0.4 ( π ν ( π ν ( ν g,

2 in contrast to the facts that their arithmetic (resp. geometric means are related to the number of F q -rational points of X (resp. its Jacobian J X. More explicitly, (0.5 γ K log q g = (q ( π ν= ν ( π ν ( q m + N m = m= q m (g q + (q + q + (q, where N m denotes the number of F q m-rational points of X (see.4. The first expression shows that γ K is a rational multiple of log q, while the second shows that when X has many F q m-rational points for small m (esp. m =, γ K tends to be negative. Our first basic observation is that, including the NF-case, γ K can sometimes be conspicuously negative, and that this occurs when K has many primes with small norms. In the FF-case, there are known interesting towers of curves over F q with many rational points, and we ask how negative γ K can be, in general and for such a tower. In the NF-case, there is no notion of rational points, but those K having many primes with small norms would be equally interesting for applications (to coding theory, etc.. Moreover, the related problems often have their own arithmetic significance (e.g. the fields K p described below. We wish to know how negative γ K can be also in the NF-case. A careful comparison of the two cases is very interesting. Thus we are led to studying γ K in both cases under a unified treatment, basically assuming the generalized Riemann hypothesis (GRH in the NF-case. We shall give a method for systematic computation of γ K, give some general upper and lower bounds, and study three special cases more closely, including that of curves with many rational points, for comparisons and applications. In Part, after basic preliminaries, we shall give some explicit estimations of γ K, and also discuss possibilities of improvements when we specialize to smaller families of K (see.6. Among them, Theorem gives a general upper bound for γ K. The main term of this upper bound is (0.6 { log log d (NF, under GRH log((g log q + log q (F F, d = d K being the discriminant. The lower bound is, as we shall see, necessarily much weaker. First, the main term of our general lower bound (Proposition 3 reads as { log d (NF, unconditionally (0.7 (g log q (F F. Secondly, when we fix q, the latter will be improved to be (0.8 q + (g log q (F F

3 (Theorem. In other words, (0.9 C(q := lim inf γ K (g K log q. q + This is based on a result of Tsfasman [Ts ], and is somewhat stronger than what we can prove only by using the Drinfeld-Vladut asymptotic bound [D-V] for N. We shall moreover see that the equality holds in (0.9 when q is a square (see below. In the NF-case, our attention will be focused on the absolute constant γ K (0.0 C = lim inf log d K. Clearly, (0.7 gives C (unconditionally, but quite recently, Tsfasman proved, as a beautiful application of [T-V], that (0. C (under GRH ([Ts ] in this Volume. The estimation of C(q or C from above is related to finding a sequence of K having many primes with small norms. As for C(q, see below. As for C, the author obtained C (under GRH; see.6, but [Ts ] contains a sharper unconditional estimation. At any rate, in each of the FF- and the NF-case, we see that the general (negative lower bound for γ K cannot be so close to 0 as the (positive upper bound. Thirdly, when the degree N of K over Q resp. F q (t is fixed (N >, or grows slowly enough, (0.7 will be improved to be ( log d (N log (NF, under GRH, (N (0. (N log (F F ( (g log q N (Theorem 3, which is nearly as strong as the upper bound, and exactly so (with opposite signs when N =. Granville-Stark [G-S] ( 3. gave an equivalent statement when N = (NF-case, and our Theorem 3 was inspired by this work. The bound (0. is quite sharp. In fact, some families of K having many primes with small norms insist that (0. cannot be replaced by its quotient even by log log N. To be precise, it cannot be replaced by its quotient by any such f(n (NF (resp. f q (N for a fixed q > (FF as satisfying f(n (resp. f q (N..7 is for supplementary remarks related to computations of γ K. In Part, we shall study some special cases. First, let q be any fixed prime power. Then, as an application of a result in [E-], we obtain (0.3 C(q c 0 log q q 3

4 (., where c 0 is a certain positive absolute constant. Then we treat the case where K is the function field over F q of a Shimura curve, with q a square, and g K q (.. In this case, as a reflection of the fact that such a curve has so many F q -rational points, we can prove (0.4 γ K q + (g K log q + ε. Therefore, combining this with (0.9, we obtain (0.5 C(q = q + (q : a square. Secondly, when K is imaginary quadratic, we combine our upper bound for γ K with the Kronecker limit formula, to give a lower bound for its class number h K ; (0.6 h K log d K dk > π 3 ε, with an explicit description of the ε-part (under GRH(Theorem 5.. As asymptotic formula, this is weaker than Littlewood s [Li] and almost equivalent with Granville-Stark s [G-S] (both conditional formulas; its merit is explicitness. Thirdly, we consider the case where K = K p is the first layer of the cyclotomic Z p -extension over Q (.3. It is the unique cyclic extension over Q of degree p contained in the field of p -th roots of unity. By classfield theory, a prime l decomposes completely in K p if and only if (0.7 l p (mod p. We shall apply our estimations of γ K to this case K = K p (Theorem 6 and its Corollaries. Among them, Corollary gives information on small l s satisfying (0.7 for a fixed large p, while Corollary 3 relates the question on the existence of many p satisfying (0.7 for a fixed l to that on lim inf(γ Kp /p. (Incidentally, lim(γ Kp / log d Kp = lim(γ Kp /(p log p = 0 under GRH. From Table.3A, see how the existence of a very small l satisfying (0.7 pushes the value of γ Kp drastically towards left on the negative real axis. For example, (0.7 is satisfied for l = and p = 093, and accordingly, γ K093 is so negative as about -747, while for several neighboring primes p, the absolute values of γ Kp are at most 0. Finally in.4.5, we shall give some application to the field index of K p. Our main tool is the explicit formula for the prime counting function (0.8 Φ K (x = ( x x N(P log N(P (x >, k N(P k x 4

5 where (P, k runs over the pairs of (non-archimedean primes P of K and positive integers k such that N(P k x (..3. This function Φ K (x is quite close to log x when x is large, and the connection with our constant γ K is (0.9 (0.0 lim (log x Φ K(x = γ K + (NF, unconditionally, x ( log x ΦK (x = γ K + q + log q (F F. (q lim x q Z x It is a simple combination of two well-known prime counting functions, but two characteristic features of Φ K (x are : (i it is continuous, and (ii the oscillating term in the explicit formula for Φ K (x has the form (0. (x lim T ρ <T (x ρ (x ρ, ρ( ρ where ρ runs over the non-trivial zeros of ζ K (s, which, under GRH, is very easy to evaluate. In fact, then it is sandwiched in-between two multiples, by (( x+/( x ±, of the negative real constant (0. ρ ρ( ρ. And γ K is a translate of (0. by a more elementary constant associated to K. This is why (under GRH in the NF-case we can obtain results always with explicit error terms, and only by simple elementary arguments. Usually, one uses the truncated explicit formula where the summation over ρ is restricted to ρ < T and instead contains an error term R(x, T which is not easy to evaluate systematically. We add here three more observations. (i In some sense, the quantity on the RHS of (0.9(0.0 may be more canonical than γ K as an invariant of K. Note that (0.0 with q = corresponds to (0.9, and that (0.5 will be simplified if we use the RHS of (0.0 instead of γ K itself (see.4. (ii One can of course generalize the definition of γ K to the case of L-functions, although then they will not usually be real numbers. Multiplicative relations among the L-functions give rise to additive relations among these constants. In particular, when H runs over the subgroups of a given finite group G, any linear relation among those characters of G induced from the trivial character of H gives rise to the corresponding linear relation among the γ K, where K runs over the intermediate extensions of a given G-extension. (iii When K is either the cyclotomic field Q(µ m or its maximal real subfield Q(µ m +, it seems fairly likely that γ K is always positive!. The author has computed γ K in both cases 5

6 up to m = 600, and Mahoro Shimura more recently checked the first case K = Q(µ m for m as far as up to 8000, and we have found no counterexamples. On the other hand, their difference, the relative γ K, seems to take both signs almost equally. Studies of γ K for various families of global fields K including these cases will be left to future publications. Some open problems and numerical data can be found in my article in the (informal Proceedings of the 004 Workshop on Cryptography and Related Mathematics, Chuo University. The 003-Worshop Proceedings contains a short summary of the present paper. Part The explicit formula for Φ K (x, and estimations of γ K.. The function Φ K (x Let K be a global field. We denote by P any (non-archimedean prime divisor of K, and by N(P its norm. As mentioned in the Introduction, we shall consider the prime counting function (.. Φ K (x = ( x x N(P log N(P (x >. k N(P k x Here, (P, k runs over all pairs with k and N(P k x (or what amounts to the same effect, N(P k < x. Call a point on the real axis critical if it is of the form N(P k. Then Φ K (x remains to be 0 until the first critical point, then monotone increasing, and is everywhere continuous. In fact, at each critical point Φ K (x acquires new summands but their values are 0 at this point, so the visible increase at each critical point is that of the slope. The slope of Φ K (x between two adjacent critical points a < b is c(x, where c = ( log N(P > 0. N(P k So, the slope near x is close to N(P k a N(P k <x log N(P x x. Thus, Φ K (x is an arithmetic approximation of log x. If the field K has many primes P with small N(P, then Φ K (x increases faster than log x, at least for some while. The difference log x Φ K (x at infinity is closely related to γ K, as we shall see later. 6

7 .. The explicit formula for Φ K (x From Weil s general explicit formula [W ],[W ], we obtain, as will be indicated in.3, the following formula for Φ K (x; (.. Φ K (x = log x + (α K + β K + l K (x + r K (x (x >. Here, (.. α K = log d (NF = (g log q (F F, (d = d K : the discriminant, g = g K : the genus, F q : the exact constant field, (..3 β K = { r (γ + log 4π + r (γ + log π } (NF = 0 (F F, (r, r : the number of real, imaginary places of K, respectively, γ = γ Q : the Euler- Mascheroni constant = , (..4 ( l K (x = r log x+ + ( x+ log x x + r log x + log x (NF x x = φ(q, x (F F, where φ(q, x is a certain continuous function of x parametrized by q, satisfying (..5 0 φ(q, x < log q φ(q, x = 0 x = q m with some m N (see below. Finally, (..6 r K (x = (x ρ (x ρ (x ρ, ρ( ρ where ρ runs over all non-trivial zeros of ζ K (s, counted with multiplicities, and (..7 = lim. ρ T ρ <T By the functional equation for ζ K (s, if ρ is a non-trivial zero of ζ K (s then so is ρ, with the same multiplicity. 7

8 In the FF-case, when x = q m (m N, Φ K (x/ log q = q m (..8 where kdegp m log x/ log q = m, α K / log q = g, β K / log q = l K (x/ log q = 0, ( r K (x/ log q = q q m g ν= ( q m kdegp degp, (π m ν ( πm ν (π ν ( π ν, (..9 ζ K (s = g ν= ( π νu( π ν u, u = q s, π ν π ν = q ( ν g. ( u( qu (To derive the last formula for r K (q m / log q from the definition (..6 of r K (x, take any α C and q >, and substitute e z = α q s in the partial fraction expansion formula (..0 (e z + / = lim T n= T T (z πin, which gives (.. log q α q s + log q = lim T (s ρ. q ρ =α ρ T Now let q = αᾱ, s = 0 and take the real part of (.. to obtain (.. q log q = lim (α (ᾱ T The desired formula follows immediately from this. q ρ =α ρ T ( ρ + ρ Note that each reciprocal zero π ν (resp. π ν of ζ K (s in u = q s corresponds to infinitely many zeros in s, which are translations of one of them by πin/ log q (n Z. It also has poles at all translations of 0, by πin/ log q (n Z. The function φ(q, x arises from the poles θ 0, ; (..3 φ(q, x = (x poles θ 0,. (x θ (x θ, θ( θ where (..4 = lim θ. T θ <T 8

9 Since either q θ = or q θ =, it is clear that φ(q, x = 0 when x = q m (m N. In a finite form, ( q m (..5 φ(q, x = log (qm (q m x x (x (q m q m log q for q m x q m Proposition (m Z, m, x. This follows immediately from the following (i. The functions l K (x and r K (x are continuous. (ii. (FF (x (l K (x+log x and (x r K (x are linear on each interval q m x q m (m. Proof (NF l K (x is continuous by definition. Since Φ K (x and l K (x are both continuous, r K (x is also continuous by (... (FF In this case, l K (x = φ(q, x is a function of x determined only by q. By (.. applied to the case g = 0, we have (..6 φ(q, x = Φ Fq(t(x log x + log q ; hence φ(q, x is continuous. Now, when q m x q m, ( x (..7 (x Φ K (x = N(P log N(P k N(P k q m is linear. Hence by (..6, (x (φ(q, x+log x is also linear on this interval. Moreover, the function (x r K (x = (x Φ K (x (x (φ(q, x + log x (x (α K + β K is also linear in the same interval. Remarks (i In the NF-case, β K and l K (x both come from the archimedean places. Among them, β K is the value at s = of the logarithmic derivative of the standard Γ- factor of ζ K (s (see.3 below, and l K (x comes from the trivial zeros of ζ K (s. Thus, l K (s for the (FF and the (NF cases have quite different origins poles 0,, vs. trivial zeros. We have given them the same name here only to save notation. (ii In the NF-case, β K + l K (x is the term coming from the archimedean places, and our separation into β K and l K (x can also be characterized by (cf. Lemma below (.5. (iii We note also that in both cases (cf. Lemma,.5. lim l K(x = 0 x l K (x 0 (x > 9

10 .3. The explicit formula for Φ K (x (continued The above explicit formula (.. for Φ K (x, at least in the NF-case, is a special case of Weil s general explicit formula. To be precise, use t for x of [W ], keeping x for our x, and put x (xe t/ e t/ 0 < t < log x, F (t = t = 0, 0 otherwise, in the formula ( of [W ]. Then we obtain (.. by straightforward computations. The FF-case is not fully treated in [W ] (nor in [W ] except when t is an integral multiple of log q, but this case is easier. In this.3, we shall give a brief account of some basic materials for, and a sketch of,the proof of (.. valid in both cases, which, hopefully is enough for the readers to convince themselves of the validity also in the FF-case, and to see why the term φ(q, x should appear. The formula (.3. obtained in this process will anyway be needed later. The advanced readers can skip this section. The explicit formula itself, and its connection with γ K, both follow from the partial fraction decomposition of the logarithmic derivative of ζ K (s. Put (.3. Z K (s = ζ K (s ζ K (s. Then from the Euler product expansion (.3. ζ K (s = P of ζ K (s follows the Dirichlet series expansion (.3.3 Z K (s = ( N(P s ( Re(s > P,k log N(P N(P ks ( Re(s > for Z K (s. In terms of Z K (s, the Euler-Kronecker constant γ K has the expression ( (.3.4 γ K = lim Z K (s. s s This Z K (s has the following partial fraction expansion ( Stark s lemma ; (.3.5 Z K (s = s + s ρ s ρ + α K + β K + ξ K (s, 0

11 with (.3.6 ( ξ K (s = r g( s g( ( + r g(s g( ( = r s + ( ( r s s s+n +n + ( (NF s s+n +n n= n= = (F F, s θ θ 0, where ρ runs over the non-trivial zeros of ζ K (s, θ runs over all poles 0, of ζ K (s (FF-case, (.3.7 = lim, = lim, and ρ T ρ <T (.3.8 g(s = Γ (s Γ(s. (Note that g( = γ Q, g( = γ Q log 4. θ T θ <T ( In the NF-case, (.3.5 is Stark s lemma [St](9 itself. The FF-case follows directly from the rational expression (.3.9 ζ K (s = α A( αq s λα (A : a finite subset of C, λ α = ± of ζ K (s ; (.3.0 Z K (s = d log ζ K(s = ds α A = λ α log q α A = (g log q ρ λ α q β =α log q α q s s β (by(.. s ρ + s + s + Now by combining (.3.4 with (.3.5, we obtain easily θ 0, poles s θ. (.3. γ K = α ρ K β K c K ρ = α ρ( ρ K β K c K, ρ where α K, β K are as defined by (..,(..3 respectively, and (.3. c K = (NF, = c q = q+ log q (q (F F.

12 The last formula for c K in the FF-case follows directly from (.. for s = α =, because ξ K ( + = ( θ = θ q =(. θ q θ =,q θ Remark If we define c q for each q R, q > by (.3., then c q > and lim q c q =. This matches with the well-known belief that the constant field of a number field should be F. The explicit formula (.. follows from the evaluation of the integral (.3.3 Φ (µ K (x = c+i x s µ πi c i s µ Z K(sds (c 0 for µ = 0 and in two ways, based on the classical formula c+i y s πi c i s ds = 0 0 < y <, / y =, < y. The Dirichlet series expansion (.3.3 of Z K (s gives the connection (.3.4 xφ ( K (x Φ(0 K (x = (x Φ K(x, while the partial fraction decomposition (.3.5 of Z K (s gives, via standard residue calculations, (.3.5 xφ ( K (x Φ(0 K (x = (x {log x + (α K + β K + l K (x + r K (x}. The terms log x, l K (x and r K (x inside { }, correspond to s + s, ξ K(s, and ρ s ρ in (.3.5, respectively. Remarks (i A word about the constant β K (NF-case. If ( (.3.6 Γ R (s = π s s Γ, resp. Γ C (s = (π s Γ(s

13 denote the standard Γ-factors at the real (resp. imaginary places, so that (.3.7 Λ K (s = Γ R (s r Γ C (s r ζ K (s satisfies the functional equation ( s (.3.8 Λ K (s = dk ΛK ( s, then one has (.3.9 ( d log ΓR (s ds s= = (γ Q + log 4π, (.3.0 Therefore, (.3. lim s ( d log ΓC (s ds (( d log ΛK (s Thus, β K is the archimedean counterpart of γ K. ds s= = (γ Q + log π. + = γ K + β K. s (ii Incidentally, the functional equation in the function field case for Λ K (s = ζ K (s is (.3. Λ K (s = (q g s Λ K ( s, and the comparison of (.3.8 and (.3. leads to our common recognition that the FF-analogue of log d should be (g log q. In both cases, the constant term in the partial fraction decomposition of Z K (s is determined from the functional equation..4. Some elementary formulas related to γ K We shall give a few more remarks related to the quantity (.4. γ K = ρ ρ α K β K c K. When α K is large, each of ρ ρ and α K + β K is usually much larger than the absolute value of γ K. (Only for some special families of K, they have the same order of magnitude; see.6. So, γ K is a finer object for study than ρ ρ. In the FF-case, in terms of the reciprocal roots π ν, π ν ( ν g of ζ K (s in u = q s, we have (as is obvious by (.. (.4. ρ g ρ = (q log q; (π ν ( π ν ν= 3

14 hence (.4.3 γ K = = = = { (q g ν= } (π ν ( π ν (g log q c q log q + (log q c q π ν g ( π ν + g (πν m + π ν m q m log q + (log q c q ν= m= ν= (q m + N m q m log q + (log q c q. m= Consider the arithmetic, geometric, and harmonic means of g positive real numbers (.4.4 (π ν ( π ν ( ν g. Then if X denotes the proper smooth curve over F q corresponding to K, and J its Jacobian, the above three means of (.4.4 are given respectively by a.m. = ( g #X(F q + (q + g (.4.5 g.m. = (#J ( F q /g h.m. = g(q log q γ K + α K + c q. The three properties #X(F q large, #J(F q large, and γ K large, are different but correlated, and are in a sense in the same direction. (The denominator of h.m. is ρ ρ > 0. By the Riemann hypothesis for curves, we have ( q (π ν ( π ν ( q + ; hence, by (.4.3, we obtain immediately ( g ( log q γ K + c q q + ( g + log q. q Later, we shall obtain much better bounds (.6. In particular, when g is fixed and q (e.g. the constant field extensions, we have the limit formula γ K (.4.7 lim q log q =. 4

15 When g = 0, γ K is given by the equality (.4.8 γ K + c q = log q. (See Remark (i below. Remarks (i We defined γ K as a natural generalization of the Euler-Mascheroni constant γ Q. But, in a sense, the quantity γ K +c K may be more canonical, as some of the preceding formulas indicate! The quantities obtained by (further adding β K (NF or log q (FF can sometimes be better. (ii In the FF-case, if we use other poles of ζ K (s, instead of s =, to define γ K similarly, then what we obtain is still γ K if the pole is congruent to modulo πi/ log q, and is α K γ K if the pole is 0 mod (πi/ log q. In the NF-case, the order of zero at s = 0 of ζ K (s is r + r, and (.4.9 γ K = γ K log d K + [K : Q](γ Q + log(π, where γ K is the coefficient of sr +r divided by that of s r +r in the Taylor expansion of ζ K (s at s = Estimations of r K (x and l K (x (.5. Now we return to the explicit formula (... By (.3., one may rewrite it as log x Φ K (x = (α K + β K r K (x l K (x = γ K + c K (r K (x + ρ l K (x ( ρ = γ K + c K r K (x + l ρ( ρ K (x. ρ We are going to estimate the non-constant terms on the right side of (.5.. In most of what follows, we shall assume GRH (which is satisfied in the FF-case. Main Lemma ((FF; and (NF under (GRH For any x > we have ( x ( x + (.5. r K (x. x + ρ( ρ x ρ( ρ ρ Proof Since (.5.3 r K (x = { } (x ρ (x ρ (x ρ( ρ ρ and ρ = + iγ (γ R, it follows that ρ( ρ = 4 + γ > 0, and that (.5.4 (x ρ (x ρ = x + x cos(γ log x lies in-between ( x and ( x + ; whence our inequalities. 5 ρ

16 The graphs of the three functions of x appearing in (.5. in the Main lemma, for the cases K = Q, Q( 48, are as shown in Figures.5A, A. (As (.5.3 indicates, each ρ with small γ contributes to a high wave calm on the surface, whereas a larger γ, to a lower ripple. The effect of the first few ρ is not particularly large, but sometimes determines the main shape of the graph (for x not too large. Thus, these graphs seem to indicate that the smallest γ for Q( 48 would be much smaller than that of K = Q (i.e., Figure.5A : K = Q Figure.5A : K = Q( 48 By this lemma and (.3. we obtain (.5.5 (γ K + α K + β K + c K r K (x x + ρ( ρ (γ K + α K + β K + c K, ρ x and hence by (.5. (.5.6 x x+ (γ K + c K x+ (α K + β K l K (x log x Φ K (x (under GRH. As for l K (x, we have Lemma (i. (NF-case l K (x is monotone decreasing, x+ x (γ K + c K + x (α K + β K l K (x and lim l K(x = +, x lim l K (x = 0, x ( log x + 0 < l K (x < [K : Q] x (x >. 6

17 (ii. (FF-case l K (x = 0 if and only if x = q m (m N; for other x, l K (x always remains within the open interval (0, log q, but does not tend to 0 as x. Proof (.5.7 (i(nf-case. In this case, l K (x = r F (x + r F (x, F (x = log x+ x + x F (x = log x + log x. x x x+ log, with First, since F (x = (x log((x + / < 0, F (x is monotone decreasing. Secondly, since F (x = F (x, F (x is also monotone decreasing and F (x < F (x. Thirdly, since log ( x+ x < (x and log ( x+ < log x, we obtain (.5.8 F (x < (log x + (x, and it is clear that F (x > 0. The desired inequalities follow immediately from these. The assertions for the limits at x, of l K (x are also obvious. (The following inequality will be used later (.4 ; [ ( (.5.9 (x F (x = log(x + + log + x ] log(x + (x 3. x (ii(ff-case. We already know that φ(q, x = 0 if x = q m (m N. So, put x = q m +y, with m, 0 < y <. Then by (..5, ( φ(q, x = y (qm (q q y (.5.0 log q (q m +y (q ( (q m (q y = (q m +y (q y log q. It is easy to see that if we fix y, then this is monotone decreasing as a function of m, and tends uniformly to ( q y (.5. s q (y = q y log q (> 0 as m. Therefore, (.5. 0 < q y φ(q, x y < q log q y <, which proves all the assertions stated in Lemma (ii. 7

18 Remark Note that s q (0 = s q ( = 0, s q (y > 0 for 0 < y <. The maximal value of s q (y for 0 < y < is (.5.3 which is attained at log q q (log log q log( q +, (.5.4 y = log log q log( q. log q The graphs of F (x > F (x will be shown in Figure.5B, and that of φ(q, q z for q = 5, in Figure.5B. The horizontal line in the latter gives the value of (.5.3 for q = Figure.5B : Figure.5B :.6. Estimations related to γ K From (.5.6 we obtain immediately: Proposition (Under (GRH in the (NF case For any x > we have (i (ii γ K γ K x + ( log x ΦK (x + l K (x + (α K + β K c K, x x x ( log x ΦK (x + l K (x (α K + β K c K. x + x + Since by (.5.6 the difference between the upper and the lower bounds tends to 0 as x, this gives a method for computing the constant γ K (under GRH to as much accuracy as one desires. Although the convergence is slow, one can usually determine the approximate size of γ K (e.g. its sign even by hand calculations. 8

19 Figure.6A : K = Q( Figure.6A : K = Q(cos π 9 Figures.6A,.6A show two examples for the graphs of the upper and the lower bounds given by Proposition (denoted respectively as upp K (x, low K (x. The horizontal lines indicate the expected values of γ K. Examples(by computer. Let K = Q(, and take x = 50, 000. Then the upper and the lower bounds for γ K given by Proposition (i(ii are , 0.843, respectively. The value of γ K computed by using the Kronecker formula (cf.. is Incidentally, in this case, the value of log x Φ K (x is which is close to the actual value, and lies in between the above upper and lower bounds. But in general, log x Φ K (x need not lie in between the two bounds of Proposition (see Remark (ii below. For other imaginary quadratic fields, 0 < γ K < holds for d K 43, but γ K < 0 for d K = 47, 56,. For example, 0.07 < γ Q( 47 < For real quadratic fields, 0 < γ K < for d K < 00, but 0.8 < γ Q( 48 < These are, of course, under (GRH. Some other examples will be given in.7 and.3. We shall give some applications. First, by letting x in (.5.6 we obtain Corollary (.6. γ K = lim x (log x Φ K (x (NF, 9

20 (.6. γ K = lim (log(q m Φ K (q m c q m m N (FF. A formula equivalent to (.6. can be found also in a recent preprint [HIKW](Th. B. Remarks (i For (.6., GRH is unnecessary. In fact, by using a standard zero-free region for ζ K (s, one can show, unconditionally, that { (.6.3 lim r K (x + } = 0, x ρ( ρ ρ from which (.6. follows directly by (.5. and Lemma. The proof of (.6.3 runs as follows. As is well-known (cf. e.g. [L-O] lemma 8., there exists a positive constant c (depending on K such that if ρ = β + iγ is a non-trivial zero of ζ K (s with γ sufficiently large, then (.6.4 β < c(log γ. We claim that (.6.5 lim x ( ρ x β γ = 0, where ρ runs over all imaginary zeros of ζ K (s. To show this, since β <, we may exclude finitely many ρ s and assume that (.6.4 is satisfied. Then, for x >, ρ x β γ < ρ c(log γ x γ = log γ <T + log γ T, where we choose T = log x. Then and log γ <T whence (.6.5. But since (.6.6 r K (x + γ log γ <T ρ log γ T ct x ( ρ log γ log x ρ( ρ = ρ( ρ ρ γ exp( c log x 0, γ 0, ( x ρ + x ρ x, and each term tends to 0 as x, by (.6.5, we obtain lim x r K(x + ρ( ρ lim x ρ ρ, γ 0 γ x β x = 0. 0

21 (ii Some readers may be interested in the comparison between A K (x := log x Φ K (x and the two bounds upp K (x, low K (x of Proposition. From (.. we obtain easily x x (upp K (x A K (x = r K (x + l K (x, x + x + (A K (x low K (x = r K (x l K (x, and we moreover have r K (x > 0 under GRH. Therefore, A K (x < upp K (x always holds under GRH. But A K (x > low K (x need not hold in general ; a counterexample being given by K = Q(, x = (say 800. [Examples of graphs of A K (x = log x Φ K (x ] Figure.6B : K = Q( 48 Figure.6B : K = Q(cos π 9 [Upper bounds] The second application of Proposition is to the problem of finding a reasonably good general upper bound for γ K in terms of more elementary invariants of K. It can be obtained from Proposition (i by the substitution of a suitable value of x. Since we do not know a priori the local behavior of Φ K (x, except that Φ K (x 0, what we do is try to minimize (.6.7 x + (log x + l K (x + (α K + β K c K. x x We leave the discussion of this delicate question until a little later (after the proof of Theorem, and first see what we can obtain by choosing the value x 0 of x which minimizes log x + α K x, i.e., x 0 = αk. We then obtain

22 Theorem (Under (GRH in the (NF-case ( α γ K < K + ( α (.6.8 K log αk + a Φ K (αk ( ( log α K + a, provided that α K + α K (.6.9 g >, or g = and q > (F F n >, or n = and d K > 8 (NF. Here, a = and n = [K : Q] (NF, a = + log q (FF. Proof The right hand side of Proposition (i for x = α K can be rewritten as (.6.0 { αk + ( log α α K K Φ K (αk + + α K K + l K (αk + β K (NF, α K + α K K Φ K (αk + + φ(q, α K + ( c q (F F. In the FF-case, (.6.9 implies α K >. And since φ(q, α K < log q and c q < 0, we are done. In the NF-case, we have the following inequalities; (.6. { ( f (x := (x + log { ( f (x := (x + log x + x x x + x log ( } (γ Q + log 4π < 0 for x >.6, } (γ Q + log π < 0 for x >.6. x + + x log(x They hold because f (x, f (x are both monotone decreasing for x >, and their values at.6 are both negative (being , ,respectively. Therefore, (.6. (α K + l K (α K + β K < 0 for α K >.6. But even the Minkowski lower bound for d K shows that α K >.6 holds for n > and for n = with d K > 0. (which is actually the same as d K > 8. [Discussions on minimizing (.6.7] Write x = t (t >, and put (.6.3 s = s K = α K + β K. Then (.6.7 can be expressed as (.6.4 g(t = g (t + g (t, with (.6.5 g (t := t ((t + log t + s c K,

23 (.6.6 g (t : = t + { r } t F (t + r F (t (> 0 (NF = t + t φ(q, t ( 0 (F F (cf.(.5.7. These are continuous functions of t > parametrized by s( R ; r, r 0, r + r > 0 (NF, or q (FF. We shall exclude the trivial case of genus 0 (FF. Then g(t always achieves its minimal value at some θ ( < θ <, because it is continuous and tends to + at both ends, i.e., t and t. [NF-case] In this case, θ is unique, as (t g (t is monotone increasing. Indeed, (.6.7 (t g (t = (t log t t s is monotone increasing, and so are the r - and the r -components of (t g (t. When s >, θ is close to ( (.6.8 s + + r + r log s. s We have included the (r +r s term, because when s K, the quantity (r +r s K, though bounded (by a standard unconditional lower bound for d K, does not tend to 0. [FF-case] The local differential structure of g(t is, in a sense, opposite to the NFcase. The graph of g(t looks like a bouncing ball, bouncing at each integral power of q, first coming down a slope and then going up another forever. (The slopes correspond to the graph of g (t. Indeed, (t g (t is a negative constant g log q (g: the genus on < t < q, and is monotone decreasing on every open interval q m < t < q m (m >. (The derivative of (t g (t on this interval is a(t, where a = (q m (q m (q m q m log q. Therefore, g (t < 0 wherever g (t = 0. Therefore, g(t can acquire its minimal value only at the bouncing points t ( q Z. Note that φ(q, t = 0 at bouncing points. Moreover, by (.6.7, we conclude that θ must be one of the (at most two integral powers of q which are adjacent to the unique root of the equation (.6.9 t log t t = s. Thus, again, θ is close to s + log s as long as s is large compared with q. If, on the other extreme, (.6.0 g < ( q, log q q 3

24 so that the root of (.6.9 is smaller than q, then θ is always equal to q, and this gives rise only to the trivial general upper bound (.4.6 for γ K. Each of Figures.6C,.6C gives the graphs of two functions g(t g (t when K = Q( 5003 (Figure.6C, and q =, g = 5 for t = q y (the horizontal axis is for y (Figure.6C Figure.6C : Figure.6C : Thus, the best possible approximation of θ would depend on the specific family of K and purpose of applications. For example, the β K -part of s K in the NF-case should not be neglected if (r + r s K is not small. But here, we are satisfied with having given a basic result expressed simply in terms of α K only, together with some indications for possible improvements. We note that choices of other θ can improve only minor terms in Theorem, unless we restrict ourselves to some special families of K. Further studies of the upperbounds of γ K for various families of K will be left to future publications. The author has yet no idea about the minimal possible size of Φ K (θ for each given r, r and given approximate range of s. This is of course related to the question how sharp an upper bound our method can possibly give. [Lower bounds] Proposition 3 This follows immediately from (.6. γ K + α K + β K + c K = ρ A general, unconditional (and trivial lower bound is: γ K > α K β K c K. 4 ρ = ρ ( ρ + ρ > 0.

25 (Under (GRH, one can deduce this also from Proposition (ii by letting x. We note that the absolute value of the negative lower bound given above is much larger than that of the upper bound given in Theorem. For example, in the NF-case, the former is log d while the latter is log log d. This is not just because we have not assumed GRH for the above lower bound as we did for the upper bound. Indeed, in the FF-case, γ K can be of the order of ( (.6. + ε (g K log q q + for Shimura curves over F q (q: a square, g K ; see.. But we can show Theorem (FF Fix q. Then (.6.3 lim inf γ K (g K log q. q + Proof This is obtained by using Corollary of Tsfasman [Ts ]. (If we combine the inequality h.m a.m in (.4.5 with the Drinfeld-Vladut asymptotic upper bound [D-V] for N, then what we obtain is a somewhat weaker statement, where the denominator q + on the RHS of (.6.3 is replaced by q. First, let us recall the basic materials from [Ts ] that will be needed. By a curve over F q, we shall always mean a complete, smooth, geometrically irreducible algebraic curve over F q. For a curve C over F q, let g = g(c denote the genus, and B m = B m (C denote the number of prime divisors (i.e., scheme theoretic closed points of C with degree m over F q. Thus, N m (C = db d (C d m is the number of F q m-rational points of C. If {C α } is a family of curves over F q (q: fixed with growing genus such that β m = lim α B m (C α g(c α exists for all m, we call the family {C α } asymptotically exact. Each sequence of curves over F q with growing genus contains a subsequence which is asymptotically exact ([Ts ]p.8. Moreover, for any asymptotically exact family of curves over F q, one has (.6.4 m= mβ m q m/ (Corollary in [Ts ]. 5

26 Now, to prove Theorem, let us write γ(c = γ K, where K is the function field of C. Put γ(c γ(c λ = lim inf = lim inf (g(c log q g(c log q. Then there exists a family of curves C over F q with g(c such that lim C γ(c g(c log q = λ, and we may assume that this family is asymptotically exact. Let C run over such a family. Then, by (.4.3, (.6.5 γ(c g(c log q = m= q m + N m (C q m g(c + ( c q. g(c log q Since the summand on the right-hand side of (.6.5 has absolute value at most equal to q m/, the sum is uniformly convergent w.r.t. C. We thus obtain (.6.6 λ = lim C m= = m= ( q m + N m (C q m g(c dβ d = q m d m ( dβ d q + q d/ d= d= dβ d q d q + by (.6.4,as desired. Remark The above proof shows also that the equality λ = q+ holds if and only if β = q holds, i.e., if and only if the Drinfeld-Vladut asymptotic upper bound for N (C g(c is attained by this family. In., we shall show that when q is a square, then the equality holds for (.6.3 (see (... It is also a very interesting problem to find out the precise value of the quantity (.6.7 C = lim inf γ K α K in the NF-case. As for this, what the author obtained are : 6

27 (i By Proposition 3(.6, we have (.6.8 C (unconditionally. (ii If there exists an infinite unramified Galois extension M/k over a number field k in which some prime ideals p, p m of k decompose completely, then, under (GRH, we can show easily that ( (.6.9 C m log N(p i, α k N(p i i= by applying Theorem (the first inequality in (.6.8 to finite intermediate extensions of M/k. For example, one may choose the examples given in Cor of [T-V], among which (Cor 9.5 is an old example due to the present author, but Cor 9.4 gives the best result. In this case, k = Q( d, d = d 73 79, where d is the product of all prime numbers q with 3 q 6. It has an infinite - classfield tower in which ten primes above,3,5,7, split completely. By taking p, p m to be these ten primes of k, we obtain by (.6.9, (.6.30 C (under GRH. Now, after the present article was submitted to this Volume, Tsfasman has kindly informed me that he can prove better results, namely, ( C (the LHS inequality under GRH, by using [T-V] for the LHS inequality, and an unconditional (.6.9 ([Ts ] Theorem 5 with a better classfield tower, for the RHS inequality (see [Ts ]. The LHS inequality was surprising to the author who had considered it plausible that C = / (the value obtained by putting q = on the RHS of (.6.3. On the other hand, (.6.3 lim sup γ K α K = 0 (under GRH by Theorems, and 3 below (for, say, N =. When the degree of K over Q (NF, or over a rational subfield of K (FF, is relatively small, there is a much better lower bound for γ K, as follows. 7

28 Theorem 3 N >. Put (.6.33 and assume α K Put k = Q (NF, = F q (t (FF, and let K be an extension of k of degree αk = α K/(N = log d N (NF, (g log q = N (F F, >. Then (.6.34 α K + α (γ K + c K K > (N (log αk + (NF, under GRH > (N (log αk + α K α K (F F. Remarks (i Granville-Stark [G-S] ( 3. gave an equivalent statement when [K : Q] = (in fact, L (, χ/l(, χ = γ K γ Q, whose argument applies also to abelian extensions over Q. Our Theorem was motivated by [G-S]. (ii The bound given by Theorem 3 is sharp in the following sense. The RHS of (.6.34 cannot be replaced by its quotient by such an f(n (NF (resp. f q (N (FF, for a fixed q > that tends to as N. This can be proved easily by using a family of K satisfying (.6.9 (NF resp. (..8(. (FF. The point is that, in each case, one can find a subsequence of K such that α K and that the following (finite limits lim αk > and lim γ K α K < 0 exist. Proof (.6.35 (.6.36 (.6.37 This will be based on the Main lemma and the following 4 inequalities; Φ K (x N Φ k (x, Φ k (x < log x, l K (x 0, each for all x >, and (.6.38 β K < [K : Q]. Among them, (.6.35 is trivial, (.6.36(FF and (.6.37 both follow directly from lemma of.5., and (.6.38 is because γ +log π =.45 >. The inequality Φ Q (x < log x can be proved easily as follows. Since Φ Q (x, log x are both monotone increasing, it is enough to show Φ Q (x log(x for integers x = n (note the shift x x on the right side. But by the prime factorization of n!, we have log n! = [ ] n log p ( n + (.6.39 log p p k p k p k n p k n = nφ Q (n +, 8

29 hence (.6.40 Φ Q (n + log(n! log n n for all n. Now we proceed to the proof of Theorem 3. By (.6.35, (.6.36, and then by (.4., (.5., (.6.37, we obtain N log x > Φ K (x = log x + α K + β K + r K (x + l K (x ( x + log x + α K + β K (γ K + α K + β K + c K. x Therefore, (N log x > x + (α K + β K (γ K + c K. x x Putting x = (α K, we obtain αk + αk (γ K + c K > (N log αk αk (α K + β K. The rest follows directly, by (.6.38 in the NF-case. Corollary If N and q (in the FF-case are fixed and α K, then (.6.4 γ K > (N + ε log(log d (NF, under GRH > (N + ε log ( (g log q (F F..7. Supplementary remarks related to computations of γ K An accurate computation of γ K for each individual K is not the main issue of this paper. Still, it should probably be pointed out that there are other ways of computing γ K that are better at least microscopically. One is the classical Landau formula generalizing the well-known Euler formula for K = Q. Let K be a number field, let ζ K (s = a n n s be the Dirichlet series expansion n= of ζ K (s on Re(s >, and put S K (x = n x a n, so that (.7. ζ K (s = s S K (xx s dx (Re(s >. Further, put κ K = lim s (s ζ K (s = lim x (S K (x/x, and (.7. B K (x = κ K a n n log x (x. n x 9

30 Then the Landau formula asserts that (.7.3 γ K = lim x B K (x. (Compare this with (.6., and note that log(x appears with opposite signs! Now, there exist positive constants ɛ, C and x 0 (all depending on K such that (.7.4 S K (x κ K x Cx ɛ (x x 0. So, we can change the order of integration and passage to the limit s, on the RHS of (.7. with S K (x κ K x in place of S K (x. This gives an expression of γ K in terms of the definite integral of (S K (x κ K xx dx from to. But since that from to x is nothing but κ K B K (x S K (xx, we obtain directly (.7.5 γ K B K (x κ K C( + ɛ x ɛ ( x x0. As for (.7.4, although certainly not the best possible bound (as the well-known case of K = Q(...counting lattice points in circles...indicates, a general method (cf.e.g.[la] VI shows that one can take ɛ = [K : Q] and can compute C by using geometry of numbers. For [K : Q] =, the exponent= / of x in (.7.5 is as strong as our conditional estimate given in.6. The constant C computed by following the method of loc.cit. is generally large compared with α K, but the actual convergence seems considerably faster than what we expect from such a bound. The second is for the case of quadratic fields. We have to rely on the notation of. below. When K is imaginary, we have the Kronecker formula (... When K is real, there is also Hecke s formula [H], which gives (.7.6 γ K = log(d K + γ Q + i C. h K Here, C runs over the narrow ideal classes of K, h K is the narrow class number, and i C is defined as follows. Pick any ideal from C with a Z-basis α,α satisfying α α -α α > 0 (α i is the conjugate of α i, and put (.7.7 ω(y = α y + iα α y + iα C ( i =, 0 < y < (its image is a semi-circle in the complex upper half plane vertical to the real axis.then (.7.8 i C = log ɛ ɛ ɛ log( η(ω Im(ω / dy/y, where ɛ is the fundamental unit > of K. (There is a small error in [H]. The formula for m ( lines below (4 of p.0 is actually that for 4m; the formulas in p. 03 will give / of the residue and of the constant term, if log 4 in (5 is replaced by log. 30

31 For example, when K = Q( 0, where ɛ = and h =, (.7.6 (rewritten as the average of log(d K + γ Q + i C gives γ K = ( = On the other hand, B K (0 5 = , and the GRH bounds given by Proposition of.6 for x = 0 5 give < γ K < Finally, A K (0 5 = , for (.7.9 A K (x = log(x Φ K (x. Part Some special families of K.. Curves over F q with many rational points When K corresponds to such a curve, γ K tends to be negative with a large absolute value. In fact, as a direct application of Theorem, we obtain Theorem 4 (FF Fix any prime power q and ε > 0. Then (.. γ K (g K log q < ε N (K (q (g K holds as long as the exact constant field of K is F q and the genus g K of K is sufficiently large. Here, N (K is the number of F q -rational points of the curve corresponding to K. Proof Let α K = (g K log q > 0. Then Theorem gives (.. γ K < ( αk + ( log αk + + log q Φ K (α α K K. Let g K be so large that α K > q, and take m N such that (..3 q q m α K < q m+. Then, by the definition of Φ K (x, Φ K (α K N (K α K ( ( αk q + + m log q; q m 3

32 hence ( αk + Φ K (α α K K N { (K α K ( q m (α K q Now let g K be so large that hence ( log αk α K > (q log q + +, (..4 α K > (q (m + +. } m log q. Then by (..3, (..4, we obtain α K q m + mq < (m + q < α K + m ; hence (..5 α K( q m m(q > (α K. Therefore, (..6 ( αk + Φ K (α α K K < N (K log q; q hence by (.., we obtain (..7 γ K α K < ( αk + α K ( log αk α K + + log q N (K α K (q (g K. Therefore, if g K is so large that the first term on the RHS is < ε, we have γ K α K < ε N (K (q (g K. We shall combine this with two typical results on curves with many rational points. First, we refer to Theorem (Elkies-Howe-Kresch-Poonen-Wetherell-Zieve[E-]( 3. There exists a positive absolute constant c 0 such that for any prime power q, and any g, there exists a curve X over F q with genus g such that (..8 #X(F q c 0 (log q(g. 3

33 Combining this with Theorem 4, we obtain Corollary For any fixed prime power q, we have (..9 C(q c 0 log q q. Corollary Fix any prime power q. Then, for any sufficiently large g, there exists K over F q with genus g such that γ K < 0. Secondly, let us recall Theorem ([I ] [I ], [TVZ] When q is a square, there exist Shimura curves X over F q with growing genus g such that (..0 #X(F q ( q (g. Therefore, by Theorems and 4, we obtain Corollary Let q be a square. Then (.. C(q = q +. Corollary Let q be a square, and K be the function field of a Shimura curve over F q corresponding to a ( p-adic discrete subgroup Γ in the sense of [I ] [I ]. Suppose that Γ is torsion-free, and { q > 3 (.. g > 3(q + /( q 3. Then γ K < 0. Proof (..3 In Proposition (i, take x = q. Then γ K q + ( log q ΦK (q + α K q q c q ( q + q log q N (K (q log q q + α K q c q. But N (K ( q (g K when K corresponds to such a Shimura curve. Therefore, γ K q + ( q α K q g K + q + q c q (..4 α { K = (3 q + 3 } q +, q g K from which the desired assertion follows at once. 33

34 .. Imaginary quadratic fields Let K be an imaginary quadratic field with discriminant d(< 0. Then the Kronecker limit formula, averaged over all the ideal classes of K, gives Theorem (Kronecker (.. γ K = log d + γ Q log + t C. h Here, h = h K is the class number of K, C runs over all ideal classes of K, (.. t C = log ( η(ω C Im(ω C /, C (..3 η(τ = q 4 ( q n, q = e πiτ (Im(τ > 0 n= is the Dedekind η-function, and ω C is defined as follows. Pick any ideal from C, with such a Z-basis [ω, ω ] that Im(ω /ω > 0. Then we put ω C = ω /ω. Since the function (..4 η(τ (Im(τ is SL (Z-invariant, t C is well-defined. Remark Each t C is positive. Indeed, the maximal value M of (..4 on the upper half plane is attained at τ = ( + 3i, and (..5 M = , log M = Since t C log M < 0, we see that t C is always positive. Now, one notes that the contribution of the principal ideal class C 0 to the formula (.. is extraordinarily large. Indeed (..6 t C0 6 π d, which is too large compared with our upper bound log log d for γ K under GRH. So, this too outstanding a contribution of the principal class should be pulled down by averaging over a large number of non-principal classes. We thus obtain, by combining with Theorem, the following 34

35 Theorem 5(under (GRH. If α K = log d K >.6. (i.e., d K, then (..7 h K > π 6 dk α K + b α K + log α K + b + c(α K, with small b, b, c(α K given by b = log M + log 4q 0 = q 0, q 0 = e π d K, (..8 b = log M γ Q + log + = c(t = 4 log t+ t. Proof (..9 Write ( γ K = t h C ξ, C ξ = α K γ Q + log. Take t 0 > t > 0 such that (..0 t 0 t C0, t t C (all C, and a majorant U for γ K ; (.. γ K U. Then (.. t 0 t h ( + t t C U + ξ ; h C hence U + ξ t is positive and (..3 h t 0 t U + ξ t. For t, we choose t = log M; and for t 0, we may choose (..4 t 0 = π d K dk ( 4q 0 log q 0 = e π d K 6 (see below. Finally, choose U as in Theorem, ( αk + (..5 U = ( log α K +, α K 35

36 and we obtain the Theorem by (..3. Here, it remains to check that t 0 given by (..4 satisfies both (i t 0 t C0 and (ii log M < t 0. (i For ω C0, we may choose ( d K i/, resp. ( + d K i/, according to whether d K 0 (mod 4, resp. (mod 4; hence (..6 t C0 = log dk + π d K 6 4 log ( (εq 0 n, with ε = resp.. But n= ( (εq 0 n < resp. < + q 0 ; hence (..7 t C0 > log dk + π d K 6 n= { 0 d K 0 (mod 4, 4q 0 d K (mod 4, which settles (i. (ii For α K >.6, we have t 0 >.03 > = log M. Remark As the above proof shows, the term 4q 0 in the formula for b in Theorem 5 is unnecessary when d K 0 (mod 4. A similar result has already been obtained by Granville-Stark[G-S] (Th ; note that the unconditional Th still contains L (, χ/l(, χ = γ K γ Q. S.Louboutin kindly informed the author that, as an asymptotic formula, there is an essentially stronger (and much older result due to Littlewood [Li]; (..8 h log log d > π. exp( γ Q o( d (under GRH. (As asymptotic formula, this is better than Theorem 5 when log d / log log d 4 exp(γ Q ; hence when d has 0 or more digits..3. The field K p For each odd prime p, let K p denote the unique cyclic extension of degree p over Q contained in the field Q(µ p of p -th roots of unity. It is totally real, with discriminant d = d p = p p ; whence (.3. log d = (p log p. 36

37 Let l be any prime number p. Then, by classfield theory, We shall study the invariant l decomposes completely in K p l p (mod p. (.3. γ p = γ Kp in connection with the following set of primes (.3.3 W (p = {l; primes < p, l p (mod p }. For example, the list of non-empty W (p with p < 00 is W ( = {3}, W (43 = {9}, W (59 = {53}, W (7 = {}, W (79 = {3}, W (97 = {53}. Among the 4 primes p with 900 < p < 000, only 3 primes p satisfy W (p (namely, p = 907, 99, 983. The known primes p such that W (p contains (resp. 3 are p = 093, 35 (resp. p =, Theorem 6 (i. (ii. Under (GRH for K p, we have ( γ p < i p{ log(p + log log p + } p l k <p l W (p γ p > i p {(p (log log p + }. log l l k, Here, (.3.4 i p = + (p log p, i p = log p+. The first inequality is a direct consequence of Theorem (.6. Indeed, each l W (p has p distinct primes of K p above l, so if we write α p = α Kp = (p log p, then (since αp p (α p for p 3 we have (.3.5 α p + α p Φ K(αp p αp l k log l p log l (α p l k l. k l k <p l k <p l W (p l W (p The second inequality is a special case of Theorem 3. Note that α K = log p. 37