Q-linear relations of special values of the Estermann zeta function

LXXXVI.3 (1998)

Q-linear relations of special values of the Estermann zeta function

by

Makoto Ishibashi (Kagoshima)

1. Introduction. Let σ

(n) = P

d|n

d

, the sum of uth powers of divi- sors of n, and let e(x) = e

. For given integers a and q with (a, q) = 1, q ≥ 1, the Estermann zeta function E

= E

(·, a/q) is defined by the Dirich- let series

E

 s, a

q



= X

σ

(n)e

 an q



n

, Re(s) > max{0, Re(u) + 1}, and has an analytic continuation to the whole s-plane with possible poles at s = 1, u + 1. This function has its origin in Estermann’s paper [1] and plays an important role in recent theory of divisor functions and allied problems ([6], [7], [9]).

In this paper we determine the linear relations among the values of E

at negative integral arguments

{E

(−j, a/q) : 1 ≤ a ≤ q, (a, q) = 1}

over the rational number field Q, where j ≥ 1 and u ≥ 0 are rational integers, which extends our previous result [4]. Our main result is

Theorem. The numbers E

(−j, a/q), 1 ≤ a ≤ q, (a, q) = 1, belong to the qth cyclotomic field, in particular they vanish for any odd number u, and if q is a prime power , we have

X

(a,q)=1

c

E

(−j, a/q) = 0 if and only if c

= (−1)

c

for c

∈ Q, u even.

From this Theorem, we easily deduce

1991 Mathematics Subject Classification: Primary 11M41; Secondary 11J99.

Key words and phrases: Estermann zeta function, Q-linear relation, Leopoldt’s char- acter coordinate.

[239]

Corollary. If q is a prime power and u is even, then the numbers E

(−j, a/q), 1 ≤ a ≤ q/2, (a, q) = 1, are linearly independent over Q.

In Section 2, we prove the first part of the Theorem by evaluating these special values in terms of the cotangent function, and in Section 3, Q-linear relations are determined by the method of K. Girstmair [2].

2. Special values. Let B

and B

(x) be the mth Bernoulli number and the mth Bernoulli polynomial respectively, and let cot

(πx) be the mth derivative of cot(πx).

Proposition 1. Let q ≥ 2, 1 ≤ a ≤ q, (a, q) = 1, i = √

−1. Then

E



−j, a q



= q

j + 1



− i 2



X

l=1

B

 l q



cot

 πal q

 (1)

+ q

(1 + q)

B

j + 1 · B

j + u + 1 , for u ≥ 0, j ≥ 1. For q = 1,

E

(−j, 1) = B

j + 1 · B

j + u + 1 .

In particular , for q ≥ 2 the right hand side of (1) is 0 for u odd.

P r o o f. We can express the function E

(s, a/q) in terms of the Hurwitz zeta function

ζ(s, x) = X

(n + x)

, 0 < x ≤ 1, with ζ(s, 1) = ζ(s) = P

n=1

n

the Riemann zeta function, as follows:

E

 s, a

q



= q

X

e

 akl q

 ζ



s − u, k q

 ζ

 s, l

q



, Re(s) > Re(u) + 1 (cf. [4]). Since ζ(s, x) can be analytically continued to a meromorphic func- tion with simple pole at s = 1, this equation gives an analytic continuation of E

(s, a/q). To evaluate the values at negative integer points in terms of the cotangent function, we need

ζ(−j, a/q) = − 1

j + 1 B

(a/q), j ≥ 0, and

(j + 1)

 i 2



cot

 πa q



= q

X

e



− ak q

 B

 k q



(see [2]).

Substituting these formulas for E

(−j, a/q), we have

E



−j, a q



= q

j + 1



− i 2



X

l=1

B

 l q



cot

 πal q



+ q

B

(j + 1)(j + u + 1)

q−1

X

k=1

B

 k q



+ q

B

(j + 1)(j + u + 1)

X

B

 l q



= S

+ S

+ S

, say.

Using the Fourier series of cot

(πx/q) and changing the order of summation, we see that the symmetric terms appearing in the innermost sum over the range from 1 to q will cancel out each other for odd u, so that S

= 0. S

+ S

is evaluated by the distribution relation of the Bernoulli polynomial:

B

(x) = m

m−1

X

j=0

B

 x + j m

 ,

and we also have S

+ S

= 0 for odd u by the properties of Bernoulli numbers.

For q = 1, the formula follows from E

(s, 1) =

X

σ

(n)

n

= ζ(s)ζ(s − u).

Thus the proposition is proved.

Since i

cot

(πal/q) belong to the qth cyclotomic field, the above proposition implies the first part of our Theorem.

3. Q-linear relations. Let Q

= Q(ζ) be the qth cyclotomic field with ζ = e(1/q) and let G = Gal(Q

/Q) be its Galois group. The Q-linear rela- tions of the conjugate numbers {σ(b) : b ∈ Q

, σ ∈ G} are determined by the annihilator ideal W

[b] in the group ring QG defined by

W

[b] = {α ∈ QG : α ◦ b = 0}, where the QG action on Q

is defined by

α ◦ b = X

σ∈G

a

σ(b) for α = X

σ∈G

a

σ ∈ QG.

In [2], K. Girstmair proves that W

[b] is generated by the idempotent element ε

= P

χ∈X

ε

, with ε

= |G|

P

σ∈G

χ(σ

)σ, attached to a

certain subset X of the character group b G of G determined by X = {χ ∈

G : y(χ|b) = 0}. Here, y(χ|b) are Leopoldt’s character coordinates defined b

by y(χ|b)τ (χ

|1) = P

σ∈G

χ(σ

)σ(b), where f is the conductor of χ, χ

is the primitive character modulo f attached to χ and τ (χ|k) is the kth Gauss sum.

He also proves, for q ≥ 2, (2) y(χ|i

cot

(π/q))

=

 



 

0, χ principal, j = 0,

 2q f



Y

p|q



1 − χ

(p) p

 B

j + 1 , otherwise,

where B

is the generalized Bernoulli number attached to the character χ

. Thus W

[i

cot

(π/q)] = h1+(−1)

σ

i, where σ

∈ G are such that σ

(ζ) = ζ

, (k, q) = 1.

In our case E

(−j, a/q) = σ

(E

(−j, 1/q)), and so we also have

Proposition 2. W

[E

(−j, 1/q)] = h1 + (−1)

σ

i for prime power q, j ≥ 1, and u even.

P r o o f. Let l = kd, d = (l, q) in the formula for E

(−j, 1/q), which gives E

(−j, 1/q)

= q

j + 1



− 1 2



X

d|q d−1

X

(k,d)=1k=1

B

 k d



i

cot

 πk d



= q

j + 1



− 1 2



C

, say.

By (2) and the QG-linearity of y(χ|−) with the reduction formula y(χ|b) =

 (ϕ(q)/ϕ(d)) · y(χ

|b), f | d,

0, otherwise,

for b ∈ Q

⊂ Q

, where χ

is the character mod d attached to χ (see [8]), we have

(3) y(χ|C

) = ϕ(q) j + u + 1

 2 f



X

f |d|qd

d

ϕ(d)

× Y

p|d



1 − χ

(p) p

 Y

p|d

(1 − χ

(p)p

)B

B

=

 

 

 

 

 

 



 

 

 

 

 

 

ϕ(q) j + u + 1

 2 f



X

f |d|qd

d

ϕ(d) B

B

, χ 6= 1, ϕ(q)

j + u + 1

 2 f



X

f |d|qd

d

ϕ(d)

Y

p|d



1 − 1 p



× Y

p|d

(1 − p

)B

B

, χ = 1.

Here

B

= d

X

χ

(k)B

(k/d), and we have the formula

B

= Y

p|d

(1 − χ

(p)p

) · B

,

which is a generalization of Hasse’s formula [3, p. 18], and can be proved in the same way, or instantly obtained by comparing both sides of the equality

L(s, χ

) = Y

p|d

(1 − χ

(p)p

)L(s, χ

)

at negative integral arguments, where L(s, χ

) denotes the Dirichlet L- function.

In the case of a primitive character it is known that

 B

6= 0, n 6≡ δ

mod 2, B

= 0, n ≡ δ

mod 2,

for n ≥ 1, where δ

= 0 for even χ and 1 for odd χ ([5]). Further, for principal χ, we see that B

= B

= 0 for even n ≥ 2, and B

6= 0 for n odd.

Hence we get X = {χ ∈ b G : χ(σ

) = (−1)

} from (3), so that ε

= 1 + (−1)

σ

generates W

[E

(−j, 1/q)].

This proposition implies the latter half of our Theorem.

Acknowledgements. The author is extremely grateful to the referee for some comments which helped the author to correct errors in the proof of the Theorem.

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Department of Liberal Arts

Kagoshima National College of Technology 1460-1 Shinko, Hayato-cho, Aira-gun Kagoshima 899-51, Japan

E-mail: isibasi@kctmgw.kagoshima-ct.ac.jp

Received on 13.10.1997

and in revised form on 29.4.1998 (3278)