### 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*= 1:

(0.1) *ζ**K*(s) = *c**−1*(s*−*1)* ^{−1}*+

*c*0+

*c*1(s

*−*1) +

*· · ·*(c

*−1*

*6= 0).*

In this paper, we shall present a systematic study of the real number

(0.2) *γ** _{K}* =

*c*

_{0}

*/c*

_{−1}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→∞*

µ 1 + 1

2 +*· · ·*+ 1

*n* *−*log*n*

¶

= 0.57721566*· · ·* *,*

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*

*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*

_{K}*γ*

*more as an*

_{K}*invariant*of

*K*.

Before explaining our motivation for systematic study, let us briefly look at the FF-
case. When *K* is the function field of a curve *X* over a finite field F* _{q}* with genus

*g, so*that

*ζ*

*(s) is a rational function of*

_{K}*u*=

*q*

*of the form*

^{−s}(0.3) *ζ** _{K}*(s) =

Q_{g}

*ν=1*(1*−π**ν**u)(1−π*¯*ν**u)*

(1*−u)(1−qu)* *,* *π*_{ν}*π*¯* _{ν}* =

*q*(1

*≤ν*

*≤g),*then

*γ*

*is closely related to the*

_{K}*harmonic*mean of

*g*positive real numbers

(0.4) (1*−π** _{ν}*)(1

*−π*¯

*) (1*

_{ν}*≤ν*

*≤g),*

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,

*γ*_{K}

log*q* = (q*−*1)
X*g*

*ν=1*

1

(1*−π**ν*)(1*−π*¯*ν*) *−*(g*−*1)*−* *q*+ 1
2(q*−*1)
(0.5)

=
X*∞*

*m=1*

µ*q** ^{m}*+ 1

*−N*

_{m}*q*

^{m}¶

+ 1*−* *q*+ 1
2(q*−*1)*,*

where*N**m* denotes the number of F*q** ^{m}*-rational points of

*X*(see

*§1.4). The first expression*shows that

*γ*

*is a rational multiple of log*

_{K}*q, while the second shows that when*

*X*has many F

_{q}*-rational points for small*

^{m}*m*(esp.

*m*= 1),

*γ*

*tends to be*

_{K}*negative.*

Our first basic observation is that, including the NF-case, *γ**K* can sometimes be ”con-
spicuously 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.). More- over, the related problems often have their own arithmetic significance (e.g. the fields

*K*

*described below). We wish to know how negative*

_{p}*γ*

*can be also in the NF-case.*

_{K}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 1, 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

*§1.6). Among them, Theorem 1 gives a general upper bound for* *γ** _{K}*. The main term of
this upper bound is

(0.6)

½ 2 log logp

*|d|* (NF, under *GRH)*

2 log((g*−*1) log*q) + logq* (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

(0.7)

½ *−*logp

*|d|* (NF, unconditionally)

*−(g−*1) log*q* (F F).

Secondly, when we *fix* *q, the latter will be improved to be*

(0.8) *−* 1

*√q*+ 1(g*−*1) log*q* (F F)

(Theorem 2). In other words,

(0.9) *C(q) := lim inf* *γ*_{K}

(g_{K}*−*1) log*q* *≥ −* 1

*√q*+ 1*.*

This is based on a result of Tsfasman [Ts_{1}], and is somewhat stronger than what we
can prove only by using the Drinfeld-Vladut asymptotic bound [D-V] for *N*_{1} . 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

(0.10) *C* = lim inf *γ*_{K}

logp

*|d*_{K}*|.*

Clearly, (0.7) gives *C* *≥ −1 (unconditionally), but quite recently, Tsfasman proved, as a*
beautiful application of [T-V], that

(0.11) *C* *≥ −0.26049...* (under*GRH*)

([Ts2] 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* *≤ −0.1635 (under GRH; see* *§1.6), but [Ts*_{2}] 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* overQ resp. F*q*(t) is fixed (N >1), or grows slowly
enough, (0.7) will be improved to be

(0.12)

*−2(N* *−*1) log
µ

log*√*

*|d|*

(N−1)

¶

(NF, under *GRH),*

*−2(N* *−*1) log³

(g−1) log*q*
*N**−1*

´

(F F)

(Theorem 3), which is nearly as strong as the upper bound, and exactly so (with opposite
signs) when *N* = 2. Granville-Stark [G-S] (§3.1) gave an equivalent statement when
*N* = 2 (NF-case) , and our Theorem 3 was inspired by this work. The bound (0.12) is
quite sharp. In fact, some families of*K* having many primes with small norms insist that
(0.12) 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 >*2 (FF)) as satisfying

*f*(N)

*→ ∞*(resp.

*f*

*q*(N)

*→ ∞).*

*§1.7 is for supplementary remarks related to computations ofγ** _{K}*.

In Part 2, 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

(0.13) *C(q)≤ −c*_{0}log*q*

*q−*1

(§2.1), 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* *À* *q* (§2.1). In
this case, as a reflection of the fact that such a curve has so many F* _{q}* -rational points, we
can prove

(0.14) *γ*_{K}*≤ −* 1

*√q*+ 1(g_{K}*−*1) log*q*+*ε.*

Therefore, combining this with (0.9), we obtain

(0.15) *C(q) =* *−* 1

*√q*+ 1 (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}(0.16) *h** _{K}*log

*|d*

_{K}*|*

p*|d*_{K}*|* *>* *π*
3 *−ε,*

with an *explicit* description of the *ε-part (under GRH)(Theorem 5* *§2.2). 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

*-extension overQ (§2.3). It is the unique cyclic extension overQ of degree*

_{p}*p*contained in the field of

*p*

^{2}-th roots of unity. By classfield theory, a prime

*`*decomposes completely in

*K*

*if and only if*

_{p}(0.17) *`* ^{p−1}*≡*1 (mod *p*^{2}).

We shall apply our estimations of*γ** _{K}* to this case

*K*=

*K*

*(Theorem 6 and its Corollaries).*

_{p}Among them, Corollary 1 gives information on small*`’s satisfying (0.17) for a fixed large*
*p, while Corollary 3 relates the question on the existence of ”many”* *p* satisfying (0.17)
for a fixed *`* to that on lim inf(γ_{K}_{p}*/p). (Incidentally, lim(γ*_{K}_{p}*/*logp

*d*_{K}* _{p}*) = lim(γ

_{K}

_{p}*/(p−*1) log

*p) = 0 under GRH.) From Table 2.3A, see how the existence of a*

*very*small

*`*satisfying (0.17) pushes the value of

*γ*

_{K}*drastically towards left on the negative real axis.*

_{p}For example, (0.17) is satisfied for *`* = 2 and *p* = 1093, and accordingly, *γ*_{K}_{1093} is so
negative as about -747, while for several neighboring primes*p, the absolute values of* *γ**K**p*

are at most 10. Finally in *§2.4∼*2.5, we shall give some application to the ”field index”

of *K** _{p}*.

Our main tool is ”the explicit formula” for the prime counting function

(0.18) Φ* _{K}*(x) = 1

*x−*1
X

*N(P*)^{k}*≤x*

µ *x*

*N*(P)^{k}*−*1

¶

log*N*(P) (x >1),

where (P, k) runs over the pairs of (non-archimedean) primes*P* of*K* and positive integers
*k* such that *N*(P)^{k}*≤x* (§2.2*∼*2.3). This function Φ*K*(x) is quite close to log*x* when *x*
is large, and the connection with our constant *γ** _{K}* is

*x→∞*lim (log*x−*Φ* _{K}*(x)) =

*γ*

*+ 1 (NF, unconditionally), (0.19)*

_{K}lim

*x∈q*^{Z}
*x→∞*

¡log*x−*Φ* _{K}*(x)¢

= *γ** _{K}*+

*q*+ 1

2(q*−*1)log*q* (F F).

(0.20)

It is a simple combination of two well-known prime counting functions, but two charac-
teristic 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

(0.21) *−* 1

2(x*−*1) lim

*T**→∞*

X

*|ρ|<T*

(x^{ρ}*−*1)(x^{1−ρ}*−*1)
*ρ(1−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+1)/(√*

*x−1))** ^{±1}*,
of the negative real constant

(0.22) *−* 1

2 X

*ρ*

1
*ρ(1−ρ).*

And*−γ** _{K}* is a translate of (0.22) 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 (0.19)(0.20) may be more canonical than
*γ** _{K}* as an invariant of

*K. Note that (0.20) with*

*q*= 1 ”corresponds to” (0.19), and that (0.5) will be simplified if we use the RHS of (0.20) instead of

*γ*

*K*itself (see

*§1.4).*

(ii) One can of course generalize the definition of *γ** _{K}* to the case of

*L-functions, al-*though 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 computedγ*

*K*in both cases

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*

*, seems to take both signs ”almost equally”.*

_{K}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 2004 Workshop on Cryptography and Related Mathematics”, Chuo University. The 2003-Worshop Proceedings contains a short summary of the present paper.

Part 1 The ”explicit formula” for Φ*K*(x), and estimations of *γ**K*

1.1. 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

(1.1.1) Φ* _{K}*(x) = 1

*x−*1
X

*N(P*)^{k}*≤x*

µ *x*

*N*(P)^{k}*−*1

¶

log*N*(P) (x >1).

Here, (P, k) runs over all pairs with *k* *≥* 1 and *N*(P)^{k}*≤* *x* (or what amounts to the
same effect, *N*(P)^{k}*< x). Call a point on the real axiscritical* if it is of the form*N*(P)* ^{k}*.
Then Φ

*(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*

_{K}*a < b*is

*c(x−*1)

*, where*

^{−2}*c*= P

*N*(P)^{k}*≤a*

³

1*−*_{N}_{(P)}^{1} *k*

´

log*N(P*)*>*0.

So, the slope near*x*is close to

X

*N*(P)^{k}*<x*

log*N*(P)

*x*^{−2}*∼x** ^{−1}*. 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 logx*

*−*Φ

*(x) ”at infinity” is closely related to*

_{K}*γ*

*, as we shall see later.*

_{K}1.2. The explicit formula for Φ* _{K}*(x)

From Weil’s general explicit formula [W_{1}],[W_{2}], we obtain, as will be indicated in*§1.3,*
the following formula for Φ* _{K}*(x);

(1.2.1) Φ* _{K}*(x) = log

*x*+ (α

*+*

_{K}*β*

*) +*

_{K}*`*

*(x) +*

_{K}*r*

*(x) (x >1).*

_{K}Here,

(1.2.2) *α** _{K}* =

^{1}

_{2}log

*|d|*(NF)

= (g*−*1) log*q* (F F),

(d=*d** _{K}*: the discriminant,

*g*=

*g*

*: the genus, F*

_{K}*: the exact constant field),*

_{q}(1.2.3) *β** _{K}* =

*−*©

_{r}1

2(γ+ log 4π) +*r*_{2}(γ+ log 2π)ª

(NF)

= 0 (F F),

(r_{1}*, r*_{2}: the number of real, imaginary places of *K, respectively,* *γ* = *γ*_{Q}: the Euler-
Mascheroni constant = 0.57721566*· · ·*),

(1.2.4) *`** _{K}*(x) =

^{r}_{2}

^{1}¡

log ^{x+1}* _{x−1}* +

_{x−1}^{2}log

^{x+1}_{2}¢ +

*r*

_{2}¡

log _{x−1}* ^{x}* +

_{x−1}^{1}log

*x*¢

(NF)

= *φ(q, x)* (F F),

where *φ(q, x) is a certain continuous function of* *x* parametrized by *q, satisfying*
(1.2.5) 0*≤* *φ(q, x)<*log*q*

*φ(q, x) = 0←→x*=*q** ^{m}* with some

*m∈*N (see below). Finally,

(1.2.6) *r** _{K}*(x) =

*−*1

2(x*−*1)
X

*ρ*

(x^{ρ}*−*1)(x^{1−ρ}*−*1)
*ρ(1−ρ)* *,*

where *ρ* runs over all non-trivial zeros of*ζ** _{K}*(s), counted with multiplicities, and

(1.2.7) X

*ρ*

= lim

*T**→∞*

X

*|ρ|<T*

*.*

By the functional equation for *ζ** _{K}*(s), if

*ρ*is a non-trivial zero of

*ζ*

*(s) then so is 1*

_{K}*−ρ,*with the same multiplicity.

In the FF-case, when *x*=*q** ^{m}* (m

*∈*N),

(1.2.8)

Φ* _{K}*(x)/log

*q*=

_{q}*m*

^{1}

*−1*

P

*kdegP**≤m*

¡*q*^{m−kdegP}*−*1¢
degP,
log*x/*log*q* = *m,*

*α**K**/*log*q* = *g−*1,

*β*_{K}*/*log*q* = *`** _{K}*(x)/log

*q*= 0,

*r*

*K*(x)/log

*q*=

*−*

³ *q−1*
*q*^{m}*−1*

´P*g*

*ν=1*

(π_{ν}^{m}*−1)(¯**π*^{m}_{ν}*−1)*
(π*ν**−1)(¯**π**ν**−1)* *,*
where

(1.2.9) *ζ**K*(s) =
Q_{g}

*ν=1*(1*−π*_{ν}*u)(1−π*¯_{ν}*u)*

(1*−u)(1−qu)* *,* *u*=*q*^{−s}*,* *π**ν**π*¯*ν* =*q* (1*≤ν* *≤g).*

(To derive the last formula for *r**K*(q* ^{m}*)/log

*q*from the definition (1.2.6) of

*r*

*K*(x), take any

*α*

*∈*C

*and*

^{×}*q >*1, and substitute

*e*

*=*

^{z}*α*

^{−1}*q*

*in the partial fraction expansion formula*

^{s}(1.2.10) (e^{z}*−*1)* ^{−1}*+ 1/2 = lim

*T**→∞*

X*T*

*n=−T*

(z*−*2πin)^{−1}*,*

which gives

(1.2.11) log*q*

*α*^{−1}*q*^{s}*−*1 +log*q*
2 = lim

*T**→∞*

X

*q** ^{ρ}*=α

*|ρ|≤T*

(s*−**ρ)*^{−1}*.*

Now let*q*=*α¯**α,**s*= 0 and take the real part of (1.2.11) to obtain

(1.2.12) *q**−*1

(α*−*1)(¯*α**−*1)log*q*= lim

*T**→∞*

X

*q** ^{ρ}*=α

*|ρ|≤T*

µ1
*ρ*+1

¯
*ρ*

¶
*.*

The desired formula follows immediately from this.)

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 2πin/*log

*q*(n

*∈*Z).

It also has poles at all translations of 0,1 by 2πin/log*q* (n *∈* Z). The function *φ(q, x)*
arises from the poles *θ* *6= 0,*1;

(1.2.13) *φ(q, x) =* 1

2(x*−*1)

X

*poles θ* *6=0,1*

(x^{θ}*−*1)(x^{1−θ}*−*1)
*θ(1−θ)* *,*
where

(1.2.14) X

*θ*

= lim

*T**→∞*

X

*|θ|<T*

*.*

Since either *q** ^{θ}* = 1 or

*q*

^{1−θ}= 1, it is clear that

*φ(q, x) = 0 when*

*x*=

*q*

*(m*

^{m}*∈*N). In a finite form,

(1.2.15) *φ(q, x) = log*
µ*q*^{m}

*x*

¶

*−* (q^{m−1}*−*1)(q^{m}*−x)*
(x*−*1)(q^{m}*−q** ^{m−1}*)log

*q*

for *q*^{m−1}*≤x≤q** ^{m}* (m

*∈*Z, m

*≥*1, x

*6= 1). This follows immediately from the following*Proposition 1 (i). The functions

*`*

*(x)*

_{K}*and*

*r*

*(x)*

_{K}*are continuous.*

(ii). (FF) (x−1)(`* _{K}*(x)+log

*x)and*(x−1)r

*(x)*

_{K}*are linear on each interval*

*q*

^{m−1}*≤*

*x≤q*

*(m*

^{m}*≥*1).

Proof (NF) *`** _{K}*(x) is continuous by definition. Since Φ

*(x) and*

_{K}*`*

*(x) are both contin- uous,*

_{K}*r*

*(x) is also continuous by (1.2.1).*

_{K}(FF) In this case, *`**K*(x) =*φ(q, x) is a function ofx* determined only by *q. By (1.2.1)*
applied to the case *g* = 0, we have

(1.2.16) *φ(q, x) = Φ*_{F}_{q}_{(t)}(x)*−*log*x*+ log*q*;
hence *φ(q, x) is continuous. Now, when* *q*^{m−1}*≤x≤q** ^{m}*,
(1.2.17) (x

*−*1)Φ

*(x) = X*

_{K}*N(P*)^{k}*≤q*^{m−1}

µ *x*

*N*(P)^{k}*−*1

¶

log*N*(P)

is linear. Hence by (1.2.16), (x−1)(φ(q, x)+log*x) is also linear on this interval. Moreover,*
the function

(x*−*1)r*K*(x) = (x*−*1)Φ*K*(x)*−*(x*−*1) (φ(q, x) + log*x)−*(x*−*1)(α*K* +*β**K*)

is also linear in the same interval. *2*

Remarks (i) In the NF-case, *β**K* and *`**K*(x) both come from the archimedean places.

Among them, *β** _{K}* is the value at

*s*= 1 of the logarithmic derivative of the ”standard Γ- factor” of

*ζ*

*(s) (see*

_{K}*§1.3 below), and*

*`*

*(x) comes from the trivial zeros of*

_{K}*ζ*

*(s). Thus,*

_{K}*`**K*(s) for the (FF) and the (NF) cases have quite different origins *· · ·* poles *6= 0,*1, vs.

trivial zeros. We have given them the same name here only to save notation.

(ii) In the NF-case, *β** _{K}*+

*`*

*(x) is the term coming from the archimedean places, and our separation into*

_{K}*β*

*K*and

*`*

*K*(x) can also be characterized by

*x→∞*lim *`** _{K}*(x) = 0
(cf. Lemma 2 below (§1.5)).

(iii) We note also that

*`** _{K}*(x)

*≥*0 (x >1) in both cases (cf. Lemma 2,

*§1.5).*

1.3. The explicit formula for Φ* _{K}*(x) (continued)

The above explicit formula (1.2.1) 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

_{1}], keeping

*x*for our

*x,*and put

*F*(t) =

1

*x−1*(xe^{−t/2}*−e** ^{t/2}*)

*· · ·*0

*< t <*log

*x,*

1

2 *· · ·t*= 0,
0 *· · ·*otherwise,

in the formula (11) of [W_{1}]. Then we obtain (1.2.1) by straightforward computations.

The FF-case is not fully treated in [W_{1}] (nor in [W_{2}] except when*t* is an integral multiple
of log*q), but this case is easier.*

In this *§1.3, we shall give a brief account of some basic materials for, and a sketch*
of,the proof of (1.2.1) 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 (1.3.11) 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*

*(s). Put*

_{K}(1.3.1) *Z**K*(s) =*−* *ζ*_{K}* ^{0}* (s)

*ζ** _{K}*(s)

*.*Then from the Euler product expansion

(1.3.2) *ζ** _{K}*(s) = Q

*P*

¡1*−N*(P)* ^{−s}*¢

*¡*

_{−1}Re(s)*>*1¢
of *ζ**K*(s) follows the Dirichlet series expansion

*Z** _{K}*(s) = X

*P,k≥1*

log*N*(P)
*N*(P)^{ks}

¡Re(s)*>*1¢
(1.3.3)

for *Z** _{K}*(s). In terms of

*Z*

*(s), the Euler-Kronecker constant*

_{K}*γ*

*has the expression*

_{K}(1.3.4) *γ** _{K}* =

*−*lim

*s→1*

µ

*Z** _{K}*(s)

*−*1

*s−*1

¶
*.*

This *Z** _{K}*(s) has the following partial fraction expansion (”Stark’s lemma”);

(1.3.5) *Z**K*(s) = 1
*s* + 1

*s−*1 *−*X

*ρ*

1

*s−ρ*+*α**K* +*β**K*+*ξ**K*(s),

with

(1.3.6)

*ξ**K*(s) = ^{r}_{2}^{1}¡

*g(*^{s}_{2})*−g(*^{1}_{2})¢
+*r*2

¡*g(s)−g(1)*¢

= *−r*_{1}¡_{1−s}

*s* + P^{∞}

*n=1*

(_{s+2n}^{1} *−*_{1+2n}^{1} )¢

*−r*_{2}¡_{1−s}

*s* + P^{∞}

*n=1*

(_{s+n}^{1} *−* _{1+n}^{1} )¢

(NF)

= P

*θ6=0,1*
1

*s−θ* (F F),

where *ρ* runs over the non-trivial zeros of *ζ** _{K}*(s),

*θ*runs over all poles

*6= 0,*1 of

*ζ*

*(s) (FF-case),*

_{K}(1.3.7) X

*ρ*

= lim

*T**→∞*

X

*|ρ|<T*

*,* X

*θ*

= lim

*T**→∞*

X

*|θ|<T*

*,*
and

(1.3.8) *g(s) =* Γ* ^{0}*(s)

Γ(s)*.*
(Note that *g(1) =−γ*_{Q},*g(*^{1}_{2}) = *−γ*_{Q}*−*log 4.)

( In the NF-case, (1.3.5) is Stark’s lemma [St](9) itself. The FF-case follows directly from the rational expression

(1.3.9) *ζ**K*(s) = Y

*α∈A*

(1*−**αq** ^{−s}*)

^{λ}*(A: a finite subset of C*

^{α}

^{×}*, λ*

*α*=

*±1)*of

*ζ*

*K*(s) ;

*Z**K*(s) = *−d*log*ζ**K*(s)

*ds* = X

*α∈A*

*λ**α* log*q*
1*−**α*^{−1}*q** ^{s}*
(1.3.10)

= X

*α∈A*

*λ*_{α}

log*q*

2 *−* X

*q** ^{β}*=α

1
*s**−**β*

(by(1.2.11))

= (g*−*1) log*q**−*X

*ρ*

1
*s**−**ρ*+

1
*s*+ 1

*s**−*1 + X

*θ6=0,1*
*poles*

1
*s**−**θ*

*.)*

Now by combining (1.3.4) with (1.3.5), we obtain easily (1.3.11)

*γ** _{K}* = P

*ρ*
1

*ρ* *−α*_{K}*−β*_{K}*−c*_{K}

= ^{1}_{2}P

*ρ*
1

*ρ(1−ρ)**−α*_{K}*−β*_{K}*−c*_{K}*,*
where *α** _{K}*,

*β*

*are as defined by (1.2.2),(1.2.3) respectively, and*

_{K}(1.3.12) *c** _{K}* = 1 (NF),

= *c** _{q}*=

_{2(q−1)}

*log*

^{q+1}*q*(F F).

The last formula for *c** _{K}* in the FF-case follows directly from (1.2.11) for

*s*=

*α*= 1, because

*ξ*

*K*(1) + 1 = X

*q** ^{θ}*=1,q

*θ6=1*

(1*−**θ)** ^{−1}*= X

*q** ^{θ}*=1

(1*−**θ)** ^{−1}*.

Remark If we define *c**q* for each*q* *∈*R,*q >*1 by (1.3.12), then*c**q* *>*1 and lim*q→1**c**q* = 1.

This matches with the well-known belief that ”the constant field of a number field should
be F_{1}”.

The explicit formula (1.2.1) follows from the evaluation of the integral
(1.3.13) Φ^{(µ)}* _{K}* (x) = 1

2πi

Z _{c+i∞}

*c−i∞*

*x*^{s−µ}

*s−µZ** _{K}*(s)ds (c

*À*0) for

*µ*= 0 and 1 in two ways, based on the classical formula

1 2πi

Z _{c+i∞}

*c−i∞*

*y*^{s}*s* *ds*=

0 *· · ·* 0*< y <* 1,
1/2 *· · ·* *y*= 1,
1 *· · ·* 1*< y.*

The Dirichlet series expansion (1.3.3) of *Z**K*(s) gives the connection
(1.3.14) *xΦ*^{(1)}* _{K}* (x)

*−*Φ

^{(0)}

*(x) = (x*

_{K}*−*1)Φ

*K*(x),

while the partial fraction decomposition (1.3.5) of *Z** _{K}*(s) gives, via standard residue cal-
culations,

(1.3.15) *xΦ*^{(1)}* _{K}* (x)

*−*Φ

^{(0)}

*(x) = (x*

_{K}*−*1){log

*x*+ (α

*+*

_{K}*β*

*) +*

_{K}*`*

*(x) +*

_{K}*r*

*(x)}.*

_{K}The terms log*x,* *`** _{K}*(x) and

*r*

*(x) inside*

_{K}*{ }*, correspond to 1

*s* + 1

*s−*1*,* *ξ**K*(s), and *−*X

*ρ*

1
*s−ρ*
in (1.3.5), respectively.

Remarks (i) A word about the constant*β**K* (NF-case). If
(1.3.16) ΓR(s) =*π*^{−}^{s}^{2}Γ

³*s*
2

´

*, resp.* ΓC(s) = (2π)* ^{−s}*Γ(s)

denote the standard Γ-factors at the real (resp. imaginary) places, so that
(1.3.17) Λ*K*(s) = ΓR(s)^{r}^{1}ΓC(s)^{r}^{2}*ζ**K*(s)
satisfies the functional equation

(1.3.18) Λ*K*(s) =³p

*|d**K**|*´1−2s

Λ*K*(1*−**s),*
then one has

(1.3.19)

µ*d*log ΓR(s)
*ds*

¶

*s=1*

=*−*1

2(γQ+ log 4π),

(1.3.20)

µ*d*log ΓC(s)
*ds*

¶

*s=1*

=*−(γ*Q+ log 2π).

Therefore,

(1.3.21) lim

*s→1*

µµ*d*log Λ*K*(s)
*ds*

¶

+ 1

*s**−*1

¶

=*γ**K*+*β**K**.*
Thus,*β**K* is the ”archimedean counterpart” of*γ**K*.

(ii) Incidentally, the functional equation in the function field case for Λ*K*(s) =*ζ**K*(s) is
(1.3.22) Λ* _{K}*(s) = (q

*)*

^{g−1}^{1−2s}Λ

*(1*

_{K}*−*

*s),*

and the comparison of (1.3.18) and (1.3.22) leads to our common recognition that the FF-analogue of

1

2log*|d|*should be (g*−*1) log*q. In both cases, the constant term in the partial fraction decomposition of*
*Z**K*(s) is determined from the functional equation.

1.4. Some elementary formulas related to *γ*_{K}

We shall give a few more remarks related to the quantity

(1.4.1) *γ** _{K}* =X

*ρ*

1

*ρ−α*_{K}*−β*_{K}*−c*_{K}*.*
When *α** _{K}* is large, each ofP

*ρ**ρ** ^{−1}* and

*α*

*+*

_{K}*β*

*is*

_{K}*usually*much larger than the absolute value of

*γ*

*. (Only for some special families of*

_{K}*K, they have the same order of magnitude;*

see*§1.6.) So,* *γ** _{K}* is a

*finer*object for study thanP

*ρ**ρ** ^{−1}*.

In the FF-case, in terms of the reciprocal roots *π*_{ν}*,*¯*π** _{ν}* (1

*≤ν*

*≤g) ofζ*

*(s) in*

_{K}*u*=

*q*

*, we have (as is obvious by (1.2.12))*

^{−s}(1.4.2) X

*ρ*

1

*ρ* = (q*−*1)
X*g*

*ν=1*

1

(π_{ν}*−*1)(¯*π*_{ν}*−*1)log*q;*

hence

*γ** _{K}* =
n

(q*−*1)
X*g*

*ν=1*

1

(π*ν* *−*1)(¯*π**ν* *−*1)*−*(g*−*1)
o

log*q−c** _{q}*
(1.4.3)

=
X*g*

*ν=1*

µ 1

*π*_{ν}*−*1+ 1

¯
*π*_{ν}*−*1

¶

log*q*+ (log*q−c**q*)

=
X*∞*

*m=1*

X*g*

*ν=1*

(π_{ν}* ^{m}*+ ¯

*π*

_{ν}*)q*

^{m}*log*

^{−m}*q*+ (log

*q−c*

*)*

_{q}=
X*∞*

*m=1*

(q* ^{m}*+ 1

*−N*

*)q*

_{m}*log*

^{−m}*q*+ (log

*q−c*

*).*

_{q}Consider the arithmetic, geometric, and harmonic means of *g* positive real numbers
(1.4.4) (π*ν* *−*1)(¯*π**ν* *−*1) (1*≤ν* *≤g).*

Then if *X* denotes the proper smooth curve over F* _{q}* corresponding to

*K*, and

*J*its Jaco- bian, the above three means of (1.4.4) are given respectively by

*a.m.* = 1

*g*#X(F* _{q}*) +
µ

1*−* 1
*g*

¶

(q+ 1)

*≤*

*g.m.* = (#J¡

F* _{q}*)¢

_{1/g}(1.4.5)

*≤*

*h.m.* = *g(q−*1) log*q*
*γ** _{K}*+

*α*

*+*

_{K}*c*

_{q}*.*

The three properties #X(F* _{q}*) large, #J(F

*) large, and*

_{q}*−γ*

*large, are different but correlated, and are in a sense in the same direction. (The denominator of*

_{K}*h.m.*isP

*ρ**ρ*^{−1}*>*

0.)

By the Riemann hypothesis for curves, we have
(*√*

*q−*1)^{2} *≤*(π_{ν}*−*1)(¯*π*_{ν}*−*1)*≤*(*√*

*q*+ 1)^{2};
hence, by (1.4.3), we obtain immediately

(1.4.6)

µ *−2g*

*√q*+ 1 + 1

¶

log*q≤γ**K*+*c**q* *≤*

µ 2g

*√q−*1 + 1

¶
log*q.*

Later, we shall obtain much better bounds (§1.6). In particular, when *g* is fixed and
*q→ ∞* (e.g. the constant field extensions), we have the limit formula

(1.4.7) lim

*q→∞*

*γ**K*

log*q* = 1
2*.*

When *g* = 0, *γ** _{K}* is given by the equality

(1.4.8) *γ** _{K}* +

*c*

*= log*

_{q}*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

*may be more canonical, as some of the preceding formulas indicate! The quantities obtained by (further) adding*

_{K}*β*

*(NF) or*

_{K}*−*log

*q*(FF) can sometimes be better.

(ii) In the FF-case, if we use other poles of *ζ** _{K}*(s), instead of

*s*= 1, to define

*γ*

*similarly, then what we obtain is still*

_{K}*γ*

*if the pole is congruent to 1 modulo 2πi/log*

_{K}*q,*and is

*−α*

*K*

*−γ*

*K*if the pole is 0 mod (2πi/log

*q).*

In the NF-case, the order of zero at *s*= 0 of *ζ** _{K}*(s) is

*r*

_{1}+

*r*

_{2}

*−*1, and (1.4.9)

*γ*

*=*

_{K}*−γ*

_{K}

^{0}*−*log

*|d*

_{K}*|*+ [K :Q](γ

_{Q}+ log(2π)),

where*γ*_{K}* ^{0}* is the coefficient of

*s*

^{r}^{1}

^{+r}

^{2}divided by that of

*s*

^{r}^{1}

^{+r}

^{2}

*in the Taylor expansion of*

^{−1}*ζ*

*(s) at*

_{K}*s*= 0.

1.5. Estimations of *r** _{K}*(x) and

*`*

*(x)*

_{K}Now we return to the explicit formula (1.2.1). By (1.3.11), one may rewrite it as

(1.5.1)

log*x−*Φ* _{K}*(x) =

*−(α*

*+*

_{K}*β*

*)*

_{K}*−r*

*(x)*

_{K}*−`*

*(x)*

_{K}= *γ** _{K}* +

*c*

_{K}*−*(r

*(x) +P*

_{K}*ρ*

*ρ** ^{−1}*)

*−`*

*(x)*

_{K}= *γ** _{K}* +

*c*

_{K}*−*³

*r** _{K}*(x) +

^{1}

_{2}P

*ρ*
1
*ρ(1−ρ)*

´

*−`** _{K}*(x).

We are going to estimate the non-constant terms on the right side of (1.5.1). 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 >*1 *we have*
(1.5.2)

*√x−*1

*√x*+ 1
Ã

1 2

X

*ρ*

1
*ρ(1−ρ)*

!

*≤ −r**K*(x)*≤*

*√x*+ 1

*√x−*1
Ã

1 2

X

*ρ*

1
*ρ(1−ρ)*

!
*.*
Proof Since

(1.5.3) *−r**K*(x) = 1

2 X

*ρ*

½(x^{ρ}*−*1)(x^{1−ρ}*−*1)

(x*−*1) *·* 1

*ρ(1−ρ)*

¾

and *ρ*= ^{1}_{2} +*iγ* (γ *∈*R), it follows that *ρ(1−ρ) =* ^{1}_{4} +*γ*^{2} *>*0, and that
(1.5.4) (x^{ρ}*−*1)(x^{1−ρ}*−*1) =*x*+ 1*−*2*√*

*x*cos(γlog*x)*
lies in-between (*√*

*x−*1)^{2} and (*√*

*x*+ 1)^{2}; whence our inequalities. *2*

The graphs of the three functions of *x*appearing in (1.5.2) in the Main lemma, for the
cases *K* =Q, Q(*√*

481), are as shown in Figures 1.5A1*, A*2.

(As (1.5.3) 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|*forQ(*√*

481) would be much smaller than that of*K*=Q(i.e., 14.1347...).)

20000 40000 60000 80000 100000 0.0229

0.0231 0.0232 0.0233

Figure 1.5A_{1}: *K* =Q

20000 40000 60000 80000 100000 0.795

0.805 0.81 0.815 0.82 0.825

Figure 1.5A_{2}: *K* =Q(*√*
481)
By this lemma and (1.3.11) we obtain

(1.5.5)

*√−2*

*x*+ 1(γ*K*+*α**K*+*β**K*+*c**K*)*≤ −r**K*(x)*−*1
2

X

*ρ*

1

*ρ(1−ρ)* *≤* 2

*√x−*1(γ*K*+*α**K*+*β**K*+*c**K*),

and hence by (1.5.1)
(1.5.6)_{√}

*√**x−1*

*x+1*(γ* _{K}*+

*c*

*)*

_{K}*−*

^{√}

_{x+1}^{2}(α

*+*

_{K}*β*

*)*

_{K}*−`*

*(x)*

_{K}*≤*log

*x−*Φ

*(x)*

_{K}*≤* ^{√}^{√}^{x+1}* _{x−1}*(γ

*+*

_{K}*c*

*) +*

_{K}

^{√}

_{x−1}^{2}(α

*+*

_{K}*β*

*)*

_{K}*−`*

*(x) (under GRH).*

_{K}As for *`** _{K}*(x), we have

Lemma 2 (i). (NF-case) *`** _{K}*(x)

*is monotone decreasing,*

*x→1*lim*`** _{K}*(x) = +∞, lim

*x→∞**`** _{K}*(x) = 0,

*and*

0*< `** _{K}*(x)

*<*[K :Q]

³log*x*+ 1
*x−*1

´

(x >1).

(ii). (FF-case) *`** _{K}*(x) = 0

*if and only if*

*x*=

*q*

*(m*

^{m}*∈*N);

*for other*

*x,*

*`*

*(x)*

_{K}*always*

*remains within the open interval*(0,log

*q), but does*not

*tend to*0

*as*

*x→ ∞.*

Proof (i)(NF-case). In this case,

(1.5.7)

*`** _{K}*(x) =

^{r}_{2}

^{1}

*F*

_{1}(x) +

*r*

_{2}

*F*

_{2}(x), with

*F*

_{1}(x) = log

^{x+1}*+*

_{x−1}

_{x−1}^{2}log

^{x+1}_{2}

*,*

*F*

_{2}(x) = log

_{x−1}*+*

^{x}

_{x−1}^{1}log

*x.*

First, since *F*_{1}* ^{0}*(x) =

*−2(x*

*−*1)

*log((x+ 1)/2)*

^{−2}*<*0,

*F*

_{1}(x) is monotone decreasing.

Secondly, since*F*_{2}(x) =*F*_{1}(2x*−*1), *F*_{2}(x) is also monotone decreasing and *F*_{2}(x)*< F*_{1}(x).

Thirdly, since log¡_{x+1}

*x−1*

¢*<*2(x*−*1)* ^{−1}* and log¡

_{x+1}2

¢*<*log*x, we obtain*
(1.5.8) *F*1(x)*<*2(log*x*+ 1)(x*−*1)^{−1}*,*

and it is clear that *F*_{2}(x) *>* 0. The desired inequalities follow immediately from these.

The assertions for the limits at *x* *→* 1, *∞* of *`**K*(x) are also obvious. (The following
inequality will be used later (§2.4) ;

(1.5.9) 1

2(x*−*1)F_{1}(x) = log(x+ 1) + log

"

1 2

µ

1 + 2
*x−*1

¶^{x−1}

2

#

*≥*log(x+ 1) (x*≥*3).)
(ii)(FF-case). We already know that *φ(q, x) = 0 if* *x*=*q** ^{m}* (m

*∈*N). So, put

*x*=

*q*

*, with*

^{m−1+y}*m≥*1, 0

*< y <*1. Then by (1.2.15),

*φ(q, x) =*
µ

1*−y−* (q^{m−1}*−*1)(q*−q** ^{y}*)
(q

^{m−1+y}*−*1)(q

*−*1)

¶
log*q*
(1.5.10)

=

µ (q^{m}*−*1)(q^{y}*−*1)
(q^{m−1+y}*−*1)(q*−*1) *−y*

¶
log*q.*

It is easy to see that if we fix *y, then this is monotone decreasing as a function ofm, and*
tends uniformly to

(1.5.11) *s** _{q}*(y) =

µ1*−q** ^{−y}*
1

*−q*

^{−1}*−y*

¶

log*q* (>0)
as *m→ ∞. Therefore,*

(1.5.12) 0*<* 1*−q*^{−y}

1*−q*^{−1}*−y <* *φ(q, x)*

log*q* *≤*1*−y <*1,

which proves all the assertions stated in Lemma 2 (ii). *2*

Remark Note that *s**q*(0) = *s**q*(1) = 0, *s**q*(y) *>* 0 for 0 *< y <* 1. The maximal value of *s**q*(y) for
0*< y <*1 is

(1.5.13) log*q*

1*−**q*^{−1}*−*(log log*q**−*log(1*−**q** ^{−1}*) + 1),
which is attained at

(1.5.14) *y*=log log*q**−*log(1*−**q** ^{−1}*)

log*q* *.*

The graphs of *F*_{1}(x) *> F*_{2}(x) will be shown in Figure 1.5B_{1}, and that of *φ(q, q** ^{z}*) for

*q*= 5, in Figure 1.5B

_{2}. The horizontal line in the latter gives the value of (1.5.13) for

*q*= 5.

10 20 30 40 50

0.2 0.4 0.6 0.8

Figure 1.5B_{1}:

2 4 6 8 10

0.1 0.2 0.3 0.4 0.5 0.6

Figure 1.5B_{2}:

1.6. Estimations related to *γ** _{K}*
From (1.5.6) we obtain immediately:

Proposition 2 (Under (GRH) in the (NF) case) *For any* *x >*1 *we have*

(i) *γ*_{K}*≤*

*√x*+ 1

*√x−*1

¡log*x−*Φ* _{K}*(x) +

*`*

*(x)¢*

_{K}+ 2

*√x−*1(α* _{K}*+

*β*

*)*

_{K}*−c*

_{K}*,*

(ii) *γ*_{K}*≥*

*√x−*1

*√x*+ 1

¡log*x−*Φ* _{K}*(x) +

*`*

*(x)¢*

_{K}*−* 2

*√x*+ 1(α* _{K}*+

*β*

*)*

_{K}*−c*

_{K}*.*Since by (1.5.6) the difference between the upper and the lower bounds tends to 0 as

*x*

*→ ∞, this gives a method for*

*computing the constant*

*γ*

*(under GRH) to as much accuracy as one desires. Although the convergence is slow, one can usually determine the approximate size of*

_{K}*γ*

*K*(e.g. its sign) even by hand calculations.

20000 40000 60000 80000 100000 -0.21

-0.19 -0.18 -0.17 -0.16 -0.15 -0.14

Figure 1.6A_{1}: *K* =Q(*√*
481)

10000 20000 30000 40000 50000 1.76

1.77 1.78 1.79

Figure 1.6A_{2}: *K* =Q(cos ^{2π}_{9} )
Figures 1.6A_{1}, 1.6A_{2} show two examples for the graphs of the upper and the lower
bounds given by Proposition 2 (denoted respectively as upp* _{K}*(x), low

*(x)). The horizon- tal lines indicate the expected values of*

_{K}*γ*

*.*

_{K}Examples(by computer). Let *K* = Q(*√*

*−1), and take* *x* = 50,000. Then the upper
and the lower bounds for *γ** _{K}* given by Proposition 2 (i)(ii) are 0.8239498, 0.8221413,
respectively. The value of

*γ*

*computed by using the Kronecker formula (cf.*

_{K}*§2.2) is*0.82282525. Incidentally, in this case, the value of log

*x−*Φ

*K*(x)

*−*1 is 0.82280515 which is close to the actual value, and lies in between the above upper and lower bounds. But in general, log

*x−*Φ

*(x)*

_{K}*−*1 need not lie in between the two bounds of Proposition 2 (see Remark (ii) below).

For other imaginary quadratic fields, 0*< γ*_{K}*<*1 holds for *|d*_{K}*| ≤* 43, but *γ*_{K}*<*0 for
*d** _{K}* =

*−47,−56,· · ·*. For example,

*−0.072< γ*_{Q(}^{√}_{−47)}*<−0.053.*

For real quadratic fields, 0*< γ*_{K}*<*2 for *d*_{K}*<*100, but

*−0.181* *< γ*_{Q(}^{√}_{481)} *<−0.167.*

These are, of course, under (GRH).

Some other examples will be given in *§1.7 and* *§2.3.*

We shall give some applications. First, by letting *x→ ∞* in (1.5.6) we obtain
Corollary 1

(1.6.1) *γ** _{K}* = lim

*x→∞*(log*x−*Φ* _{K}*(x)

*−*1) (NF),