series at 1 - Project Euclid

#2008 The Mathematical Society of Japan J. Math. Soc. Japan Vol. 60, No. 1 (2008) pp. 219–236 doi: 10.2969/jmsj/06010219

On Gauss’ formula for and finite expressions for the L-series at 1 Dedicated to Professor Gilles Lachaud on his sixtieth birthday By Masahiro H ASHIMOTO, Shigeru K ANEMITSU and Masayuki T ODA (Received Apr. 18, 2006) (Revised May 7, 2007)

Abstract. In this paper, we shall prove in Theorem 1 that Gauss’ famous closed formula for the values of the digamma function at rational arguments is equivalent to the well-known finite expression for the Lð1; Þ, which in turn gives rise to the finite expression for the class number of quadratic fields. We shall also prove several equivalent expressions for the arithmetic function NðqÞ introduced by Lehmer and reveal the relationships among them.

1.

Introduction and the main theorem.

Dirichlet’s celebrated class number pffiffiffi formula has two stages. To state them we introduce the notation. Let k ¼ Qð dÞ be a quadratic field with discriminant d and let h ¼ hk be its class number. Let ð d Þ be the Kronecker character associated to k, which is known to be a primitive Dirichlet character mod jdj. In general, let be a Dirichlet character to the modulus q and let Lðs; Þ be the Dirichlet L-function (L-series) associated to : Lðs; Þ ¼

1 X ðnÞ n¼1

ns

;

where the series on the right is absolutely convergent for :¼ Re s > 1 and is conditionally convergent for > 0 for non-principal . The value Lð1; Þ for nonprincipal is therefore meaningful and a fortiori for the Kronecker character ð d Þ. Let k ðsÞ be the Dedekind zeta-function of k. Then, decomposing k ðsÞ into h equivalent classes, we are led to considering the corresponding Epstein zeta2000 Mathematics Subject Classification. Primary 11R29, 33B15; Secondary 11R11. Key Words and Phrases. Gauss formula for the digamma function, Dirichlet class number formula, Hurwitz zeta-function, Lehmer’s arithmetic function, orthogonality of characters. This research was supported in part by Grant-in-Aid for Scientific Research (No. 17540050).

220

M. HASHIMOTO, S. KANEMITSU and M. TODA

function, and the residue is known to be ([10, p.182]) 2h pffiffiffiffiffiffi w jdj

for d < 0

2h log " pffiffiffi d

and

for

d > 0;

where w and " signify the number of roots of unity contained in the field and the funcdamental unit, usually so denoted. On the other hand, k ðsÞ has the product decomposition d ðsÞL s; ; which gives the residue Lð1; defined by

d

Þ, where ðsÞ is the Riemann zeta-function

ðsÞ ¼

1 X 1 ; s n n¼1

ð1:1Þ

> 1;

and L s; d is the L-series. Equating these residues, we obtain the first stage of the Dirichlet class number formula. The second stage consists in finite expressions for Lð1; Þ, where the underlying philosophy is that since h is finite, so is Lð1; Þ to be (which is given in its inception as an infinite series). The most well-known finite expressions for Lð1; Þ are jdj1 d i X d a L 1; ¼ B1 ; jdj G d a¼1 a

d 0;

ð1:3Þ

where B1 ðxÞ ¼ x 12 signifies the first (periodic) polynomial and GðÞ is Bernoulli the normalized Gauss sum defined by ðÞ ¼ d GðÞ ¼

X a mod jdj

ðaÞe

a 2i jdj

:

ð1:4Þ

Gauss formula and Lð1; Þ

221

(1.2) and (1.3) depend on the relation q1 1 X a ; Lðs; Þ ¼ ðaÞls GðÞ a¼1 q

ð1:5Þ

where is a primitive Dirichlet character mod q and ls ðxÞ is the polylogarithm (function) of the complex exponential argument ls ðxÞ ¼

1 X e2inx n¼1

ns

>1

;

ð1:6Þ

(also referred to as the Lerch zeta-function [14]). Its limiting case s ¼ 1, x 2 =Z l1 ðxÞ ¼ A1 ðxÞ iB1 ðxÞ;

0 0;

ð1:8Þ

which for x ¼ 1 reduces to the Riemann zeta-function defined by (1.1). The polylogarithm function and the Hurwitz zeta-function are interrelated by the functional equation (sometimes referred to as Lerch’s formula) ð1 s; xÞ ¼

i ðsÞ i s 2 ls ðxÞ þ e 2 s ls ð1 xÞ : s e ð2Þ

ð1:9Þ

As suggested by (1.9), there is a counterpart of (1.5), which is a decomposition into residue classes mod q: q1 1X a Lðs; Þ ¼ s ðaÞ s; q a¼1 q being valid for any Dirichlet character mod q, not necessarily primitive. Recall the Laurent expansion for ðs; xÞ,

ð1:10Þ

222


ðs; xÞ ¼

1 ðxÞ þ Oðs 1Þ; s1

s ! 1;

ð1:11Þ

where ðxÞ signifies the Euler digamma function ðxÞ ¼

0 ðxÞ ¼ ðlog ðxÞÞ0 :

ð1:12Þ

Also recall the orthogonality of characters q1 X

ðaÞ ¼

a¼1

0;

if 6¼ 0 ,

’ðqÞ;

if ¼ 0 ,

ð1:13Þ

where 0 and ’ðqÞ stand for the principal character mod q and the Euler function P defined by ’ðqÞ ¼ 1aq;ða;qÞ¼1 1, respectively. From (1.10), (1.11) and (1.13) we obtain q1 1X ðaÞ Lðs; Þ ¼ q a¼1

a þ Oðs 1Þ; q

s!1

and a fortiori Lð1; Þ ¼ For the values of

p q

q1 1X ðaÞ q a¼1

a : q

ð1:14Þ

, there is a remarkable formula of Gauss (1 p < q):

q1 X p p 2pk k ¼ log q cot þ log 2 sin cos q 2 q q q k¼1 X p 2pk k ¼ log q cot þ 2 cos log 2 sin ; 2 q q q q

ð1:15Þ

k 2

where is the Euler constant ( ð1Þ ¼ ) and where the second equality is a consequence of Lemma 1 below ([2], [4], [9]). It was D. H. Lehmer [15] who first used (1.15) in his study of the generalized Euler constant ðp; qÞ for an arithmetic progression p mod q. Using [15, (11)] and


223

p the relation [15, Theorem 7] between ðp; qÞ and q , he deduced (1.15), and stated ([15, p.135]) ‘‘Our proof via finite Fourier series indicates that Gauss’ remarkable result has a completely elementary basis.’’ Our main purpose is to elaborate on this statement of Lehmer and, on streamlining the argument, to show that (1.15) has a purely number-theoretic basis and that is a number-theoretic function. As a converse to this, we shall also put into practice the statement of Deninger [6, p.180], to the effect that (1.15) can be used to evaluate Lð1; Þ. Indeed, Funakura was on these lines (cf. [8, (1)]) but he appealed to the integral representation of Legendre and applied Lehmer’s argument of using logð1 e2ix Þ, 0 < x < 1. We may now state our main theorem. THEOREM 1. Lð1; Þ:

Gauss’ formula (1.15) is equivalent to finite expressions for

Lð1; Þ ¼

q1 X a ðaÞ cot 2q a¼1 q

ð1:16Þ

for odd and q1 1 X a bðaÞ log 2 sin Lð1; Þ ¼ pffiffiffi q q a¼1

ð1:17Þ

1 X 2i kq a bðaÞ ¼ pffiffiffi ðkÞe q k mod q

ð1:18Þ

for even, where

is the finite Fourier transform of . The finite Fourier transform of is intimately related to the generalized Gauss sum Gða; Þ, see (2.3) below. COROLLARY 1.

For primitive , (1.16) and (1.17) reduce, respectively, to

Lð1; odd Þ ¼

and

q1 i X a ðaÞB1 GðÞ a¼1 q

(1.16)0

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Lð1; even Þ ¼

q1 1 X a ðaÞ log 2 sin ; GðÞ a¼1 q

(1.17)0

and a fortiori, finite expressions (1.2) and (1.3) are consequences of Gauss’ formula (1.15). REMARK 1.

(i) On symmetry grounds, (1.16) may be stated as q1 i X a bðaÞB1 Lð1; odd Þ ¼ pffiffiffi ; q q a¼1

(1.16)00

which can be explicitly computed to be (1.16) (cf. e.g. [8]). We note that both Funakura [8] and Ishibashi-Kanemitsu [12] treated the case of periodic functions fðnÞ of period q, and obtained generalizations of the formulas (1.16)00 and (1.17), but they are already implicit in Yamamoto’s work [19], depending on (1.5) and (1.7). (ii) The last statement of Corollary 1 follows, on recalling that the Kronecker characters d are primitive odd or even characters mod jdj, according as d < 0 or d > 0, respectively. In the course of proof of Theorem 1, we shall encounter an interesting P number-theoretic function NðqÞ ¼ Nq defined by log Nq ¼ djq ððdÞ log dÞ dq which eventually cancels out in view of (1.20) below. We believe this function deserves wider attention and we state THEOREM 2. For q > 1, the number-theoretic function log NðqÞ ¼ log Nq admits the following expressions. log Nq ¼ q

X ðdÞ djq

d

ð1:19Þ

log d

11 q a ða;qÞ AA ¼ ’ðqÞ log@2 sin@ q q ’ ða;qÞ a¼1 X q ðdÞ’ ¼ d djq q1 X

¼ ’ðqÞ

0

X log p ; p1 pjq

0

ð1:20Þ

ð1:21Þ ð1:22Þ


225

where the last sum is extended over all prime divisors p of q, and where and signify the Mo ¨bius function and the von Mangoldt function, respectively. 2.

Proof of the theorems.

Let fðnÞ be an arithmetic periodic function of period q: f : Z ! C;

fðn þ qÞ ¼ fðnÞ;

n 2 Z:

We define the parity of f as follows: f is called even if fðnÞ ¼ fðnÞ and odd if fðnÞ ¼ fðnÞ. We prepare some lemmas, of which Lemma 1 is repeatedly used in what follows, without notice. LEMMA 1.

If f is odd, then q1 X

fðaÞ ¼ 0

a¼1

and if f is even, then q1 X

fðaÞ ¼ 2

X q

a¼1

a< 2

1 þ ð1Þq q : fðaÞ þ f 2 2

In particular, if f and mod q are of opposite parity, then q1 X

ðaÞfðaÞ ¼ 0

a¼1

while if f and are of the same parity and q > 2, then q1 X

ðaÞfðaÞ ¼ 2

X

ðaÞfðaÞ

q a 2

a¼1

¼2

X

ðaÞfðaÞ:

q

a< 2

LEMMA 2. The function satisfies Gauss’ multiplicative formula or the modified Kubert identity

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q1 1X ðxÞ ¼ log q þ q a¼0

LEMMA 3.

xþa : q

ð2:1Þ

Let denote a Dirichlet character mod q, q 3. Then 8 > > > > < ’ðqÞ X ðnÞ ¼ 2 > > odd > ’ðqÞ > > : 2

if n 6 1 (mod qÞ, if n 1 (mod qÞ, if n 1 (mod qÞ,

where the sum is extended over all even and odd characters, respectively. PROOF. For q 3, the set f1g forms a subgroup of the reduced residue class group G ¼ ðZ=qZ Þ of index 2. Hence the factor group G=f1g has order ’ðqÞ 2 . Since the group of all even characters may be identified with the character group of G=f1g, it follows, from the orthogonality of characters, that X

X

ðnÞ ¼

even

ðnÞ

2 G=f1g d

8