Errata to the paper

ACTA ARITHMETICA LXXXII.1 (1997)

Errata to the paper

“On a functional equation satisfied by certain Dirichlet series”

(Acta Arith. 71 (1995), 265–272) by

E. Carletti and G. Monti Bragadin (Genova)

We have to point out that formula (5) in [1] is wrong, as well as the formula for Φ

L

(s) given in the statement of the Theorem in [1]. The following lemma will take the place of formula (5) in [1].

Lemma 0.1. The following formula for derivatives of higher order of z

^ν

I

ν

(z) holds:

(0.1) d

^p

dz

^p

(z

^ν

I

ν

(z)) =

[p/2]

X

l=0

(2l − 1)!!  p 2l



z

^ν−l

I

ν−(p−l)

(z) if we put (−1)!! = 1.

P r o o f. From the well known formula (see [2])

(0.2) d

dz (z

^ν

I

ν

(z)) = z

^ν

I

ν−1

(z) we derive, by induction, that if p ≥ 1 then

(0.3) d

^p

dz

^p

(z

^ν

I

ν

(z)) =

[p/2]

X

l=0

β

p,l

z

^ν−l

I

_ν−(p−l)

(z).

By a direct computation we get β

p,0

= 1 for all p ≥ 1. Comparing d

^p+1

dz

^p+1

(z

^ν

I

ν

(z)) =

[(p+1)/2]

X

t=0

β

p+1,t

z

^ν−t

I

ν−(p+1−t)

(z) with

d dz

 d

^p

dz

^p

(z

^ν

I

ν

(z))



[99]

100 E. C a r l e t t i a n d G. M o n t i B r a g a d i n

developed by (0.2) from (0.3), we obtain the following recurrence formula:

(0.4) β

p+1,t

= (p − 2t + 2)β

p,t−1

+ β

p,t

,

where p ≥ 1, 0 ≤ t ≤ [(p + 1)/2] and β

p,i

= 0 if i > [p/2] or i < 0. From (0.4) for t ≥ 2 due to the well known formula

m

X

k=0

n + k n



= n + m + 1 n + 1



we obtain, for all p ≥ 1,

(0.5) β

p+1,t

= (2t − 1)!! p + 1 2t

 .

We note that β

1,0

= 1, so (0.5) holds if p = 0. If t = 0, taking (−1)!! = 1 the above formula holds by a direct computation. For t = 1, (0.5) follows directly from (0.4).

By using formula (0.1) we obtain the corrected form for the function Φ

L

(s) given in the statement of the Theorem in [1].

In the proof of the Theorem of [1] we have to replace page 270, from the fifth line starting with “By Cauchy’s theorem. . . ” up to the end of the page, with the following:

By Cauchy’s theorem we have I

N

(s) = − X

−N ≤2n≤N n6=0

Res



H(z)I

s−1/2

 δ 2 z



z

^s−1/2

; 2πni

 .

If we put

A(z) = I

_s−1/2

 δ 2 z

 z

^s−1/2

, its Taylor series at s = 2πni, n 6= 0, is

A(z) =

∞

X

m=0

1

m! A

^(m)

(2πni)(z − 2πni)

^m

. Then we have

Res(H(z)A(z); 2πni)

= X

p+l=−1 p≥−(d+1)

l≥0

1

l! α

ⁿ_p

A

^(l)

(2πni) =

d

X

p=0

1

p! α

ⁿ_−p−1

A

^(p)

(2πni).

Errata 101

By (0.1), A

^(p)

(z) =

[p/2]

X

l=0

(2l − 1)!!  p 2l

 δ 2



p−l

z

^s−1/2−l

I

s−1/2−(p−l)

 δ 2 z

 . Therefore

I

N

(s) = − X

−N ≤2n≤N n6=0

d

X

p=0 [p/2]

X

l=0

(2l − 1)!!

p!

 p 2l

 δ 2



p−l

× α

ⁿ_−p−1

(2nπi)

^s−1/2−l

I

s−1/2−(p−l)

(δnπi).

By (2) and (3) of [1] the series X

n∈Zn6=0

α

ⁿ_−p−1

(2πni)

^s−1/2−l

I

s−1/2−(p−l)

(δnπi)

converges absolutely and uniformly on compact subsets of σ < 0. Thus, for σ < 0, we have

I(s) = − X

n∈Zn6=0 d

X

p=0 [p/2]

X

l=0

(2l − 1)!!

p!

 p 2l

 δ 2



p−l

× α

ⁿ_−p−1

(2πni)

^s−1/2−l

I

s−1/2−(p−l)

(δnπi).

Then we derive the final formula for Φ

L

(s) in σ > 1:

Φ

L

(s) = I(1 − s) = −

d

X

p=0 [p/2]

X

l=0

X

n∈Zn6=0

(2l − 1)!!

p!

 p 2l

 δ 2



p−l

× α

ⁿ_−p−1

(2πni)

^1/2−s−l

I

1/2−s−(p−l)

(δnπi).

References

[1] E. C a r l e t t i and G. M o n t i B r a g a d i n, On a functional equation satisfied by certain Dirichlet series, Acta Arith. 71 (1995), 265–272.

[2] I. S. G r a d s h t e y n and I. M. R y z h i k, Tables of Integrals, Series and Products, fifth ed., Academic Press, 1993.

Dipartimento di Matematica Universit` a di Genova Via Dodecaneso 35 I-16146 Genova, Italy

E-mail: carletti@dima.unige.it monti@dima.unige.it

Received on 8.10.1996 (3054)