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ON VALUES OF THE HIGHER DERIVATIVES OF THE BARNES

ZETA FUNCTION AT NON-POSITIVE INTEGERS

SHINPEI SAKANE AND MIHO AOKI

Abstract. Let x be a complex number which has a positive real part, and w1, . . . , wNbe positive rational numbers. We write wsζN(s, x | w1, . . . , wN) as a finite linear combi-nation of the Hurwitz zeta function over Q(x), where ζN(s, x | w1, . . . , wN ) is the Barneszeta function and w is a positive rational number explicitly determined by w1, . . . , wN .Furthermore, in the case that x is a positive rational number, we give an explicit formulafor the values at non-positive integers for higher order derivatives of the Barnes zetafunction involving the generalized Stieltjes constants and the values at positive integersof the Riemann zeta function. At the end of the paper, we give some tables of numericalexamples.

The Barnes zeta function [3, 4] is defined by the multiple Dirichlet series:

ζN (s, x | w1, . . . , wN ) :=

∞∑

n1=0

· · ·

∞∑

nN=0

1

(x+ n1w1 + · · ·+ nNwN )s, Re(s) > N,

where N is a positive integer and x,w1, . . . , wN are fixed complex number which havepositive real parts. The series on the right converges absolutely for Re(s) > N , and thefunction of s has a meromorphic continuation to the entire complex plane with simplepoles at s = 1, 2, . . . , N . For N = 1, we have ζ1(s, x | w) = w

−sζ(s, x/w) where

ζ(s, x) :=

∞∑

n=0

(n+ x)−s, Re(s) > 1

is the Hurwitz zeta function.We put

ζN (s, x) := ζN (s, x | 1, . . . , 1︸ ︷︷ ︸

N

),

and let

V (x) := 〈{ζ(s − k, y) | k ∈ Z≥0, y ∈ Q(x), Re(y) > 0}〉Q(x)

be the vector space over the field Q(x) generated by the Hurwitz zeta functions. Koyamaand Kurokawa [8] generalized Kummer’s formula on the gamma function to the multiplegamma function ΓN (x) :=exp(ζ

′N (0, x)) by using some formulae for ζ

′N (s, x). In this paper,

we follow their method.We establish wsζN (s, x | w1, . . . , wN ) ∈ V (x) for any positive rational numbers

w1 =r1q1, . . . , wN =

rNqN

(ri, qi ∈ Z>0, gcd(ri, qi) = 1 for i = 1, . . . , N)

and w := lcm(r1, . . . , rN )/gcd(q1, . . . , qN ).

Lemma 1. Let w1 = r1/q1, . . . , wN = rN/qN (ri, qi ∈ Z>0 for i = 1, 2, . . . , N) be posi-tive rational numbers (gcd(ri, qi) = 1 is not necessary). For any common multiple q of

2020 Mathematics Subject Classification. Primary 11M41; Secondary 11M06, 11M35, 33B15.

1

http://arxiv.org/abs/2011.08667v2

2 SHINPEI SAKANE AND MIHO AOKI

q1, . . . , qN and any common multiple ℓ of qw1, . . . , qwN (∈ Z), we have

ζN (s, x | w1, . . . , wN ) =(q

ℓ

)sℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

ζN

(

s,q

ℓ(x+ k1w1 + · · ·+ kNwN )

)

,

where ℓ1 := ℓ/(qw1), . . . , ℓN := ℓ/(qwN ) (∈ Z).

Proof. For s ∈ C with Re(s) > N , we have

ζN (s, x | w1, . . . , wN )

=

∞∑

n1=0

· · ·

∞∑

nN=0

(x+ n1w1 + · · · + nNwN )−s

=(q

ℓ

)s∞∑

n1=0

· · ·

∞∑

nN=0

(q

ℓ(x+ n1w1 + · · ·+ nNwN )

)−s

=(q

ℓ

)sℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

∞∑

n1=0n1≡k1 (mod ℓ1)

· · ·∞∑

nN=0

nN≡kN (mod ℓN )

{q

ℓ((x+ k1w1 + · · ·+ kNwN )

+(n1 − k1)w1 + · · ·+ (nN − kN )wN )}−s

=(q

ℓ

)sℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

∞∑

n1=0

· · ·

∞∑

nN=0

(q

ℓ(x+ k1w1 + · · · + kNwN )

+ n1 + · · ·+ nN

)−s

=(q

ℓ

)sℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

ζN

(

s,q

ℓ(x+ k1w1 + · · ·+ kNwN )

)

,

and we get the assertion. �

Proposition 2. Let w1 = r1/q1, . . . , wN = rN/qN (ri, qi ∈ Z>0 for i = 1, 2, . . . , N) bepositive rational numbers with gcd(ri, qi) = 1 for i = 1, . . . , N and put

w := lcm(r1, . . . , rN )/gcd(q1, . . . , qN ) (∈ Q),

ℓ1 := w/w1, . . . , ℓN := w/wN . We have ℓ1, . . . , ℓN ∈ Z and

ζN (s, x | w1, . . . , wN ) = w−s

ℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

ζN(s,w−1(x+ k1w1 + · · · + kNwN )

).

Proof. We put q := lcm(q1, . . . , qN ) and ℓ := lcm(qw1, . . . , qwN ). It is enough to showq/ℓ = w−1 = gcd(q1, . . . , qN )/lcm(r1, . . . , rN ) from Lemma 1. To show the equality, wecheck ordp(ℓ gcd(q1, . . . , qN )) = ordp(q lcm(r1, . . . , rN )) for any prime number p, whereordp(a) (∈ Z) for a ∈ Q

× is defined by

a = pordp(a)c

b, b, c ∈ Z, p ∤ b, p ∤ c.

Let p be a prime number and ν (1 ≤ ν ≤ N) be an integer which satisfies ordp(wν) =max {ordp(w1), . . . , ordp(wN )}. Since ℓ = lcm(qw1, . . . , qwN ), we have

ordp(ℓ) = max {ordp(qw1), . . . , ordp(qwN )}

= ordp(q) + ordp(wν).(1)

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 3

(i) Consider the case p|qν . From the assumption gcd(rν , qν) = 1, we have p ∤ rν , andhence

max {ordp(w1), . . . , ordp(wN )} = ordp(wν) = −ordp(qν) < 0.

It concludes that ordp(w1) < 0, . . . , ordp(wN ) < 0, that is, p ∤ r1, . . . , p ∤ rN and p|q1, . . . , p|qN .From these facts and (1), we have

ordp(wν) = max {ordp(w1), . . . , ordp(wN )}

= max {−ordp(q1), . . . ,−ordp(qN )}

= −min {ordp(q1), . . . , ordp(qN )}

= −ordp(gcd(q1, . . . , qN )),

and

ordp(ℓ gcd(q1, . . . , qN )) = ordp(ℓ) + ordp(gcd(q1, . . . , qN ))

= ordp(q) + ordp(wν) + ordp(gcd(q1, . . . , qN ))

= ordp(q)

= ordp(q) + ordp(lcm(r1, . . . , rN ))

= ordp(q lcm(r1, . . . , rN )).

(ii) Consider the case p ∤ qν . If p|ri for some i, then p ∤ qi from gcd(ri, qi) = 1 and

ordp(ri) = ordp(wi) ≤ max {ordp(w1), . . . ordp(wN )}

= ordp(wν) = ordp(rν).

Therefore, we have

ordp(lcm(r1, . . . , rN )) = max {ordp(r1), . . . , ordp(rN )} = ordp(rν) = ordp(wν).

From these facts and (1), we have

ordp(ℓ gcd(q1, . . . , qN )) = ordp(ℓ) + ordp(gcd(q1, . . . , qN ))

= ordp(ℓ) (since p ∤ qν .)

= ordp(q) + ordp(wν)

= ordp(q) + ordp(lcm(r1, . . . , rN ))

= ordp(q lcm(r1, . . . , rN )).

�

For a positive integer N , let CN (t) (∈ Z[t]) be the polynomial defined by

CN (t) :=

{

1 (N = 1),

(t+N − 1)(t+N − 2) · · · (t+ 1) (N ≥ 2),

and put

CN,x(t) := CN (t− x) ∈ (Z[x])[t]

for x ∈ C.

Lemma 3. We have

ζN (s, x) =1

(N − 1)!

N−1∑

k=0

C(k)N,x(0)

k!ζ(s− k, x).

4 SHINPEI SAKANE AND MIHO AOKI

Proof. For s ∈ C with Re(s) > N , we have

ζN (s, x) =∞∑

n1=0

· · ·∞∑

nN=0

(x+ n1 + · · ·+ nN )−s

=

∞∑

n=0

(n+N − 1N − 1

)

(n+ x)−s

(

since |{(n1, . . . , nN ) ∈ ZN | n1 + · · ·+ nN = n, n1, . . . , nN ≥ 0}| =

(n+N − 1N − 1

))

=1

(N − 1)!

∞∑

n=0

CN,x(n+ x)× (n + x)−s

=1

(N − 1)!

∞∑

n=0

N−1∑

k=0

C(k)N,x(0)

k!(n + x)−(s−k)

=1

(N − 1)!

N−1∑

k=0

C(k)N,x(0)

k!ζ(s− k, x),

and we get the assertion. �

From Proposition 2 and Lemma 3, we get the following theorem and corollary.

Theorem 4. Let x be a complex number which has a positive real part, w1 = r1/q1,. . . , wN = rN/qN (ri, qi ∈ Z>0 for i = 1, 2, . . . , N) positive rational numbers with gcd(ri, qi) =1 for i = 1, . . . , N and put

w := lcm(r1, . . . , rN )/gcd(q1, . . . , qN ), ℓ1 := w/w1, . . . , ℓN := w/wN .

We have

ζN (s, x | w1, . . . , wN )

=1

ws(N − 1)!

ℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

N−1∑

k=0

C(k)N,y(k1,...,kN )

(0)

k!ζ(s− k, y(k1, . . . , kN )).

where y(k1, . . . , kN ) := w−1(x+ k1w1 + · · ·+ kNwN ).

Corollary 5. For positive rational numbers w1, . . . , wN , we have wsζN (s, x | w1, . . . , wN ) ∈

V (x) := 〈{ζ(s − k, y) | k ∈ Z≥0, y ∈ Q(x), Re(y) > 0}〉Q(x).

Example 6. (1)

ζ1(s, x) = ζ(s, x),

ζ2(s, x) = (1− x)ζ(s, x) + ζ(s− 1, x),

ζ3(s, x) =1

2((x2 − 3x+ 2)ζ(s, x) + (3− 2x)ζ(s − 1, x) + ζ(s− 2, x)).

(2)

ζ1(s, x | 1) = ζ(s, x),

ζ2(s, x | 1, 2)

= 2−s((

1−x

2

)

ζ(

s,x

2

)

+ ζ(

s− 1,x

2

)

+

(1

2−x

2

)

ζ

(

s,x

2+

1

2

)

+ζ

(

s− 1,x

2+

1

2

))

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 5

(3)

ζ1(s, x | 1) = ζ(s, x),

ζ2

(

s, x

∣∣∣∣1,

1

2

)

= (1− x) ζ (s, x) + ζ (s− 1, x) +

(1

2− x

)

ζ

(

s, x+1

2

)

+ ζ

(

s− 1, x+1

2

)

,

ζ3

(

s, x

∣∣∣∣1,

1

2,1

3

)

=1

2

((x2 − 3x+ 2

)ζ (s, x) + (3− 2x) ζ (s− 1, x) + ζ (s− 2, x)

+

(

x2 −7x

3+

10

9

)

ζ

(

s, x+1

3

)

+

(7

3− 2x

)

ζ

(

s− 1, x+1

3

)

+ ζ

(

s− 2, x+1

3

)

+

(

x2 −5x

3+

4

9

)

ζ

(

s, x+2

3

)

+

(5

3− 2x

)

ζ

(

s− 1, x+2

3

)

+ ζ

(

s− 2, x+2

3

)

+

(

x2 − 2x+3

4

)

ζ

(

s, x+1

2

)

+ (2− 2x) ζ

(

s− 1, x+1

2

)

+ ζ

(

s− 2, x+1

2

)

+

(

x2 −4x

3+

7

36

)

ζ

(

s, x+5

6

)

+

(4

3− 2x

)

ζ

(

s− 1, x+5

6

)

+ ζ

(

s− 2, x+5

6

)

+

(

x2 −2x

3−

5

36

)

ζ

(

s, x+7

6

)

+

(2

3− 2x

)

ζ

(

s− 1, x+7

6

)

+ ζ

(

s− 2, x+7

6

))

.

The surface on the left in Figure 1 shows a graph of ζ2(s, x | 1, 1/2) with −2 < s < 1and 0 < x < 1, and the curve on the right shows the section of the surface cut by theplane x = 1/3. From the figure, we can know that ζ2(s, 1/3 | 1, 1/2) has a zero arounds = 0.25. In fact, the computation shows that the zero exists at s ; 0.2558028917231215.

Figure 1. ζ2(s, x | 1, 1/2) and ζ2(s, 1/3 | 1, 1/2)

The surface on the left in Figure 2 shows a graph of ζ3(s, x | 1, 1/2, 1/3) with −2 < s < 1and 0 < x < 1, and the curve on the right shows the section of the surface cut by theplane x = 1/3. From the figure, we can know that ζ3(s, 1/3 | 1, 1/2, 1/3) has a zero arounds = 0. In fact, we have ζ3(0, 1/3 | 1/2, 1/3) = 0. We can know this fact by a result ofthe Barnes on the relation between the values of Barnes zeta function at non-positiveintegers and the Bernoulli-Barnes polynomials Bk(X | w1, . . . , wN ) ∈ Q(w1, . . . , wN )[X]([4, p.383]) defined by

tNe(w1+···+wN−X)t

(ew1t − 1) · · · (ewN t − 1)=

∞∑

k=0

Bk(X | w1, . . . , wN )tk

k!.

6 SHINPEI SAKANE AND MIHO AOKI

Theorem 7. (Barnes [4, p.405]) For ℓ ∈ Z≥0, we have

ζN (−ℓ, x | w1, . . . , wN ) =(−1)ℓℓ!

(N + ℓ)!BN+ℓ(x | w1, . . . , wN ).

By Theorem 7 and

B3(X | w1, w2, w3) =3

4+w1 + w24w3

+w2 + w34w1

+w3 + w14w2

−

{3

2w1+

3

2w2+

3

2w3+

w12w2w3

+w2

2w3w1+

w32w1w2

}

X

+

{3

2w1w2+

3

2w2w3+

3

2w1w3

}

X2 −1

w1w2w3X3,

we have

ζ3(0, 1/3 | 1, 1/2, 1/3) =1

3!B3(1/3 | 1, 1/2, 1/3) = 0.

Figure 2. ζ3(s, x | 1, 1/2, 1/3) and ζ3(s, 1/3 | 1, 1/2, 1/3)

Next, we calculate the values at non-positive integers s of higher order derivatives ofζN (s, x | w1, . . . , wN ) for positive rational numbers w1, . . . , wN and x. From Theorem 4,we have

ζ(n)N (s, x | w1, . . . , wN ) =

1

(N − 1)!

n∑

j=0

(nj

)

(− logw)n−jw−s

×

ℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

N−1∑

k=0

C(k)N,y(k1,...,kN )

(0)

k!ζ(j)(s− k, y(k1, . . . , kN )),(2)

for n ∈ Z≥0. Therefore, the calculation of the values at non-positive integers s of thehigher order derivatives of ζN (s, x | w1, . . . , wN ) for positive rational numbers w1, . . . , wNand x is reduced to that of the values of ζ(n)(s, y) at non-positive integers s for y ∈ Q>0.

Let u and v be integers with 1 ≤ u ≤ v. Consider the functional equation [1, Theo-rem 12.8]:

ζ(

1− s,u

v

)

=2Γ(s)

(2πv)sH(

s,u

v

)

,(3)

where

H(

s,u

v

)

:=v∑

k=1

{

cos

(πs

2−

2πku

v

)

ζ

(

s,k

v

)}

.

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 7

We have

(4) ζ(s, x) =1

s− 1+

∞∑

n=0

(−1)n

n!γn(x)(s − 1)

n,

where γn(x) is the generalized Stieltjes constant. For the Riemann zeta function ζ(s) =ζ(s, 1),

γn = γn(1) = limm→∞

(m∑

k=1

logn k

k−

logn+1m

n+ 1

)

,

and

γ := γ0 = limm→∞

(m∑

k=1

1

k− logm

)

; 0.57721

is the Euler’s constant. Berndt [5] proved the following theorem.

Theorem 8 ([5, Theorem 1]). For x ∈ R with 0 < x ≤ 1 and n ∈ Z with n ≥ 0, we have

γn(x) = limm→∞

(m∑

k=0

logn(k + x)

k + x−

logn+1(m+ x)

n+ 1

)

.

Lemma 9. H(s, u

v

)is an entire function.

Proof. It is enough to show that H(s, u

v

)is holomorphic at s = 1. Since

cos

(πs

2−

2πku

v

)(n)

= −(π

2

)n

sin

(π(s+ n− 1)

2−

2πku

v

)

,

we have

(5) cos

(πs

2−

2πku

v

)(n)

s = 1

= −(π

2

)n

sin

(πn

2−

2πku

v

)

.

Therefore, we obtain the Taylor expansion at s = 1:

(6) cos

(πs

2−

2πku

v

)

= −

∞∑

n=0

1

n!

(π

2

)n

sin

(πn

2−

2πku

v

)

(s− 1)n.

From (6) and (4), the coefficient of (s−1)−1 in the expansion ofH(s, u

v

)is

v∑

k=1

sin

(2πku

v

)

= 0,

and hence H(s, u

v

)is holomorphic at s = 1 and

H(

s,u

v

)

= −

∞∑

n=1

v∑

k=1

1

n!

(π

2

)n

sin

(nπ

2−

2πku

v

)

(s− 1)n−1

+v∑

k=1

{

cos

(πs

2−

2πku

v

) ∞∑

n=0

(−1)n

n!γn

(k

v

)

(s− 1)n

}

.(7)

�

By Leibniz’s rule for (3), we have

(8) (−1)nζ(n)(

1− s,u

v

)

= 2∑

a+b+c=na,b,c≥0

{(n

a, b, c

)

Γ(a)(s)((2πv)−s

)(b)H(c)

(

s,u

v

)}

,

8 SHINPEI SAKANE AND MIHO AOKI

where

(n

a, b, c

)

:= n!/(a!b!c!). Consider the values of Γ(a)(s), ((2πv)−s)(b) and H(c)(s, u

v

)

in (8) at positive integers s. Let m be a positive integer. First, we have

(9) ((2πv)−s)(b)s=m = (− log (2πv))b(2πv)−m.

Next, we consider the values of H(c)(m, u

v

)for positive integers m. For m ≥ 2, since

ζ(s, k

v

)is holomorphic at s = m, we have

H(c)(

m,u

v

)

=v∑

k=1

c∑

i=0

(ci

)

× cos

(πs

2−

2πku

v

)(c−i)

s = m

× ζ(i)(

m,k

v

)

= −v∑

k=1

c∑

i=0

{(ci

)

×(π

2

)c−isin

(π(m+ c− i− 1)

2−

2πku

v

)

×ζ(i)(

m,k

v

)}

,

ζ(i)(

m,k

v

)

=

∞∑

ℓ=0

(

− log

(

ℓ+k

v

))i(

ℓ+k

v

)−m.

On the other hand, for m = 1, since{

cos

(πs

2−

2πku

v

) ∞∑

n=0

(−1)n

n!γn

(k

v

)

(s − 1)n

}(c)

=

c∑

i=0

(ci

)

cos

(πs

2−

2πku

v

)(c−i)( ∞∑

n=0

(−1)n

n!γn

(k

v

)

(s− 1)n

)(i)

,

and by (5), we have

hc,uv:=

v∑

k=1

{

cos

(πs

2−

2πku

v

) ∞∑

n=0

(−1)n

n!γn

(k

v

)

(s− 1)n

}(c)

s = 1

=

v∑

k=1

c∑

i=0

(−1)i+1(ci

)(π

2

)c−isin

(π(c− i)

2−

2πku

v

)

γi

(k

v

)

.(10)

From (7),(10) and

v∑

k=1

sin2πku

v= 0,

v∑

k=1

cos2πku

v=

{

0 (u 6= v)

v (u = v),

we get

H(c)(

1,u

v

)

= εc,uv+ hc,u

v,

εc,uv:= −

v∑

k=1

1

c+ 1

(π

2

)c+1sin

((c+ 1)π

2−

2πku

v

)

=

(−1)c2+1

c+ 1

(π

2

)c+1v (c ∈ 2Z, u = v),

0 (otherwise).

We established the following lemma.

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 9

Lemma 10. For c ∈ Z≥0, m ∈ Z>0 and y = u/v ∈ Q with 1 ≤ u ≤ v, we have

H(c)(m, y) =

−v∑

k=1

c∑

i=0

{(ci

)

×(π

2

)c−i

× sin

(π(m+ c− i− 1)

2− 2πky

)

× ζ(i)(

m,k

v

)}

(m ≥ 2),

εc,y +

v∑

k=1

c∑

i=0

(−1)i+1(ci

)(π

2

)c−i

× sin

(π(c− i)

2− 2πky

)

γi

(k

v

)

(m = 1),

where

ζ(i)(

m,k

v

)

=∞∑

ℓ=0

(

− log

(

ℓ+k

v

))i(

ℓ+k

v

)−m(m ≥ 2),

εc,y =

(−1)c2+1

c+ 1

(π

2

)c+1v (c ∈ 2Z, y = 1),

0 (otherwise).

Finally we consider the value Γ(a)(m) for positive integers m. Coffey [6] pointed outthat the value is given by using the complete Bell polynomials Bn. For a positive integern, we define the complete Bell polynomial Bn = Bn(X1, . . . ,Xn) by

exp

∞∑

j=1

Xjtj

j!

=

∞∑

n=0

Bn(X1, . . . ,Xn)tn

n!,

and let B0 := 1 (see [7, Chap. III, §3.3]). It is known that these polynomials are integraland have the following explicit formulae.

Lemma 11. ([7, Chap. III, §3.3, Theorem A]) Let n ≥ 1. We have Bn(X1, . . . ,Xn) ∈Z[X1, . . . ,Xn] and

Bn(X1, . . . ,Xn) =∑

π(n)

n!

k1!ks! · · · kn!

(X11!

)k1(X12!

)k2

· · ·

(Xnn!

)kn

where

π(n) := {(k1, k2, . . . , kn) ∈ Zn | k1, k2, . . . , kn ≥ 0, k1 + 2k2 + · · ·+ nkn = n}.

We use Bn to give an explicit formula for the higher order derivatives of holomorphicfunctions.

Lemma 12. ([6, p3, Lemma 1]) Let f(s) and g(s) be holomorphic functions on an opensubset D of C satisfying f ′(s) = f(s)g(s). We have

f (n)(s)

(

:=

(d

ds

)n

f(s)

)

= f(s) Bn(g(s), g′(s), . . . , g(n−1)(s)).

By applying Lemma 11 for the gamma function Γ(s) and the psi function ψ(s) :=Γ′(s)/Γ(s), we have

Γ(n)(s) = Γ(s) Bn(ψ(s), ψ′(s), . . . , ψ(n−1)(s))

10 SHINPEI SAKANE AND MIHO AOKI

= Γ(s)∑

π(n)

n!

k1!k2! · · · kn!

(ψ(s)

1!

)k1(ψ′(s)

2!

)k2

· · ·

(

ψ(n−1)(s)

n!

)kn

.

Since

ψ(1) =Γ′(1)

Γ(1)=

∫ ∞

0e−t log t dt = −γ,

ψ(s + 1) =Γ′(s+ 1)

Γ(s+ 1)=

Γ(s) + sΓ′(s)

sΓ(s)=

1

s+ ψ(s),

we have

ψ(m+ 1) = −γ +

m∑

k=1

1

k(m ∈ Z≥1).

On the other hand, the value ψ(n)(m) for n ≥ 1 is obtained as follows. From theWeierstrass factorization theorem of the gamma function:

1

Γ(s)= seγs

∞∏

n=1

(

1 +s

n

)

e−sn ,

we have

ψ(s) = −γ +

∞∑

n=1

(1

n−

1

s+ n− 1

)

,

and hence

ψ(n)(s) =

∞∑

k=1

(−1)n+1n!

(k + s− 1)n+1(n ∈ Z≥1).

Therefore, we obtain

ψ(n)(m) = (−1)n+1n!

(

ζ(n+ 1)−m−1∑

k=1

1

kn+1

)

(n,m ∈ Z≥1).

We established the following lemma.

Lemma 13. For n ∈ Z≥0 and m ∈ Z>0, we have

Γ(n)(m) = (m− 1)!∑

π(n)

n!

k1!k2! · · · kn!

(ψ(m)

1!

)k1(ψ′(m)

2!

)k2

· · ·

(

ψ(n−1)(m)

n!

)kn

,

where

π(n) = {(k1, k2, . . . , kn) ∈ Zn | k1, k2, . . . , kn ≥ 0, k1 + 2k2 + · · ·+ nkn = n},

ψ(ℓ)(m) =

−γ +∞∑

n=1

(1

n−

1

m+ n− 1

)

(ℓ = 0),

(−1)ℓ+1ℓ!

(

ζ(ℓ+ 1)−m−1∑

k=1

1

kℓ+1

)

(ℓ ≥ 1).

Corollary 14. For n ∈ Z≥0, we have

Γ(n)(1) =∑

π(n)

(−1)nn!

k1!k2! · · · kn!

(γ

1

)k1(ζ(2)

2

)k2(ζ(3)

3

)k3

· · ·

(ζ(n)

n

)kn

.

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 11

Let y be a positive rational number, and put y = Y + u/v where Y ∈ Z≥0 and u, v ∈Z, 1 ≤ u ≤ v. We have

ζ(s, y) = ζ(

s, Y +u

v

)

= ζ(

s,u

v

)

−

Y−1∑

m=0

(

m+u

v

)−s.

By differentiating, we have

ζ(j)(s, y) = ζ(j)(

s,u

v

)

−Y−1∑

m=0

{

− log(

m+u

v

)}j (

m+u

v

)−s.

From (8), we have

ζ(j)(1− s, y) =(−1)j2∑

a+b+c=ja,b,c>0

{(j

a, b, c

)

Γ(a)(s)((2πv)−s

)(b)H(c)

(

s,u

v

)}

−

Y−1∑

m=0

{

− log(

m+u

v

)}j (

m+u

v

)s−1.(11)

Let w1 = r1/q1, . . . , wN = rN/qN (ri, qi ∈ Z>0 for i = 1, 2, . . . , N) be positive rationalnumbers with gcd(ri, qi) = 1 for i = 1, 2, . . . , N and put

w := lcm(r1, . . . , rN )/gcd(q1, . . . , qN ), ℓ1 := w/w1, . . . , ℓN := w/wN .

Furthermore, for a positive rational number x and non-negative integers k1, . . . , kN , weput

y(k1, . . . , kN ) := w−1(x+ k1w1 + · · ·+ kNwN )

= Y (k1, . . . , kN ) +u(k1, . . . , kN )

v(k1, . . . , kN ),

where Y (k1, . . . , kN ) ∈ Z≥0 and u(k1, . . . , kN ), v(k1, . . . , kN ) ∈ Z with 1 ≤ u(k1, . . . , kN ) ≤v(k1, . . . , kN ). From (2) and (11), we have

ζ(n)N (−s, x | w1, . . . , wN ) =

1

(N − 1)!

n∑

j=0

(nj

)

(− logw)n−jws

×

ℓ1−2∑

k1=0

· · ·

ℓN−1∑

kN=0

N−1∑

k=0

C(k)N,y(k1,...,kN )

(0)

k!

(−1)

j2∑

a+b+c=ja,b,c>0

{(j

a, b, c

)

Γ(a)(s+ k + 1)

×(

(2πv(k1, . . . , kN ))−(s+k+1)

)(b)H(c)

(

s+ k + 1,u(k1, . . . , kN )

v(k1, . . . , kN )

)}

−

Y (k1,...,kN )−1∑

m=0

{

− log

(

m+u(k1, . . . , kN )

v(k1, . . . , kN )

)}j (

m+u(k1, . . . , kN )

v(k1, . . . , kN )

)s+k

.(12)

From (9), (12), and Lemmas 10, 13, we established the following theorem.

Theorem 15. Let w1 = r1/q1, . . . , wN = rN/qN (ri, qi ∈ Z>0 for i = 1, 2, . . . , N) bepositive rational numbers with gcd(ri, qi) = 1 for i = 1, . . . , N and put

w := lcm(r1, . . . , rN )/gcd(q1, . . . , qN ), ℓ1 := w/w1, . . . , ℓN := w/wN .

For any positive rational number x and ℓ, n ∈ Z≥0, we have

ζ(n)N (−ℓ, x | w1, . . . , wN )

12 SHINPEI SAKANE AND MIHO AOKI

=1

(N − 1)!

n∑

j=0

(nj

)

(− logw)n−jwℓℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

N−1∑

k=0

C(k)N,y(k1,...,kN )

(0)

k!

×

(−1)

j2∑

a+b+c=ja,b,c≥0

{(j

a, b, c

)

Γ(a)(ℓ+ k + 1)(− log (2πv(k1, . . . , kN )))b

× (2πv(k1, . . . , kN ))−(ℓ+k+1)H(c)

(

ℓ+ k + 1,u(k1, . . . , kN )

v(k1, . . . , kN )

)}

−

Y (k1,...,kN )−1∑

m=0

{

− log

(

m+u(k1, . . . , kN )

v(k1, . . . , kN )

)}j (

m+u(k1, . . . , kN )

v(k1, . . . , kN )

)ℓ+k

,

where

y(k1, . . . , kN ) := w−1(x+ k1w1 + · · ·+ kNwN )

= Y (k1, . . . , kN ) + u(k1, . . . , kN )/v(k1, . . . , kN ),

Y (k1, . . . , kN ) ∈ Z≥0, u(k1, . . . , kN ), v(k1, . . . , kN ) ∈ Z>0,

1 ≤ u(k1, . . . , kN ) ≤ v(k1, . . . , kN ),

and H(c) (ℓ+ k + 1, u(k1, . . . , kN )/v(k1, . . . , kN )), Γ(a)(ℓ+ k+ 1) are given by Lemmas 10

, 13, respectively.

Especially, considering ℓ = 0, we get the following result.

Corollary 16. Let w1 = r1/q1, . . . , wN = rN/qN (ri, qi ∈ Z>0 for i = 1, 2, . . . , N) bepositive rational numbers with gcd(ri, qi) = 1 for i = 1, . . . , N and put

w := lcm(r1, . . . , rN )/gcd(q1, . . . , qN ), ℓ1 := w/w1, . . . , ℓN := w/wN .

For any positive rational number x and n ∈ Z≥0, we have

ζ(n)N (0, x | w1, . . . , wN )

=1

(N − 1)!

n∑

j=0

(nj

)

(− logw)n−jℓ1−1∑

k1=0

· · ·

ℓN−1∑

kN=0

N−1∑

k=0

C(k)N,y(k1,...,kN )

(0)

k!

×

(−1)j2

∑

a+b+c=ja,b,c≥0

{(j

a, b, c

)

Γ(a)(k + 1)(− log (2πv(k1, . . . , kN )))b

×(2πv(k1, . . . , kN ))−(k+1)H(c)

(

k + 1,u(k1, . . . , kN )

v(k1, . . . , kN )

)}

−

Y (k1,...,kN )−1∑

m=0

{

− log

(

m+u(k1, . . . , kN )

v(k1, . . . , kN )

)}j (

m+u(k1, . . . , kN )

v(k1, . . . , kN )

)k

,

where

y(k1, . . . , kN ) := w−1(x+ k1w1 + · · ·+ kNwN )

= Y (k1, . . . , kN ) + u(k1, . . . , kN )/v(k1, . . . , kN ),

Y (k1, . . . , kN ) ∈ Z≥0, u(k1, . . . , kN ), v(k1, . . . , kN ) ∈ Z>0,

1 ≤ u(k1, . . . , kN ) ≤ v(k1, . . . , kN ),

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 13

and H(c) (k + 1, u(k1, . . . , kN )/v(k1, . . . , kN )), Γ(a)(k + 1) are given by Lemmas 10, 13,

respectively.

The figures below show graphs of ζ2(0, x | w1, w2) with x = 1/10, 1/20, 1/30 and 0 <w1 < 1, 0 < w2 < 1 drawn using Corollary 16.

Figure 3. ζ2(0, 1/10 | w1, w2)

Figure 4. ζ2(0, 1/20 | w1, w2)

Figure 5. ζ2(0, 1/30 | w1, w2)

14 SHINPEI SAKANE AND MIHO AOKI

The figures below show graphs of ζ ′2(0, x | w1, w2) with x = 1/10, 1/20, 1/30 and 0 <w1 < 1, 0 < w2 < 1 drawn using Corollary 16.

Figure 6. ζ ′2(0, 1/10 | w1, w2)

Figure 7. ζ ′2(0, 1/20 | w1, w2)

Figure 8. ζ ′2(0, 1/30 | w1, w2)

By applying Corollary 16 for w1 = · · · = wN = 1, we get the values of ζ(n)N (0, x) =

ζ(n)N (0, x | 1, . . . , 1).

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 15

Corollary 17. For any positive rational number x, we put x = X + u/v with X ∈Z≥0, u, v ∈ Z, 1 ≤ u ≤ v. For n ∈ Z≥0, we have

ζ(n)N (0, x)

=1

(N − 1)!

N−1∑

k=0

C(k)N,x(0)

k!

(−1)n2

∑

a+b+c=na,b,c≥0

{(n

a, b, c

)

Γ(a)(k + 1)(− log (2πv))b

×(2πv)−(k+1)H(c)(

k + 1,u

v

)}

−

X−1∑

m=0

{

− log(

m+u

v

)}n (

m+u

v

)k

]

,

where H(c)(k + 1, u/v) and Γ(a)(k + 1) are given by Lemmas 10 and 13, respectively.

Since ζ(s, x) = ζ1(s, x | 1) = ζ1(s, x) and ζ(s) = ζ(s, 1), we obtain the values at s = 0of higher order derivatives of the Hurwitz zeta function and the Riemann zeta function.

Corollary 18. For any positive rational number, we put x = X+u/v with X ∈ Z≥0, u, v ∈Z, 1 ≤ u ≤ v. For n ∈ Z≥0, we have

ζ(n)(0, x) =(−1)n

πv

∑

a+b+c=na,b,c≥0

{(n

a, b, c

)

Γ(a)(1)(− log (2πv))bH(c)(

1,u

v

)}

−

X−1∑

m=0

{

− log(

m+u

v

)}n

,

where

Γ(a)(1) =∑

π(a)

(−1)aa!

k1!k2! · · · ka!

(γ

1

)k1(ζ(2)

2

)k2(ζ(3)

3

)k3

· · ·

(ζ(a)

a

)ka

,

and

H(c)(

1,u

v

)

= εc,uv+

v∑

k=1

c∑

i=0

(−1)i+1(ci

)(π

2

)c−isin

(π(c− i)

2−

2πku

v

)

γi

(k

v

)

,

εc,uv=

(−1)c2+1

c+ 1

(π

2

)c+1v (c ∈ 2Z, u = v),

0 (otherwise).

Example 19. Let u, v ∈ Z with 1 ≤ u ≤ v.

ζ(

0,u

v

)

=

−1

2(u = v)

1

πv

v∑

k=1

γ0

(k

v

)

sin

(2πku

v

)

(u 6= v)

ζ ′(

0,u

v

)

=

−1

2(γ + log (2πv)) +

1

2v

v∑

k=1

γ0

(k

v

)

(u = v)

1

πv

{

(γ + log (2πv))

v∑

k=1

γ0

(k

v

)

sin

(2πku

v

)

+v∑

k=1

π

2γ0

(k

v

)

cos

(2πku

v

)

+ γ1

(k

v

)

sin

(2πku

v

)}

(u 6= v)

16 SHINPEI SAKANE AND MIHO AOKI

The equations ζ(0, x) = 12−x and ζ′(0, x)

(= ∂

∂sζ(s, x)s=0

)= log

(Γ(x)√2π

)

([9]) are known,

but similar simple equations for the higher order derivatives ζ(n)(0, x) = ∂n

∂snζ(s, x)s=0 are

unknown.

Corollary 20. For any n ∈ Z≥0, we have

ζ(n)(0) =(−1)n

π

∑

a+b+c=na,b,c≥0

{(n

a, b, c

)

Γ(a)(1)(− log (2π))bH(c)(1, 1)

}

,

where

Γ(a)(1) =∑

π(a)

(−1)aa!

k1!k2! · · · ka!

(γ

1

)k1(ζ(2)

2

)k2(ζ(3)

3

)k3

· · ·

(ζ(a)

a

)ka

,

and

H(c)(1, 1) = εc,1 +

c∑

i=0

(−1)i+1(ci

)(π

2

)c−iγi sin

π(c− i)

2,

εc,1 =

(−1)c2+1

c+ 1

(π

2

)c+1(c ∈ 2Z),

0 (c 6∈ 2Z).

Example 21.

ζ(0) = −1

2

ζ ′(0) = −1

2log (2π)

ζ ′′(0) =π3

24−

1

2log2 (2π)−

1

2ζ(2) +

1

2γ2 − γ1

Apostol [2, Theorem 3] has given another formula for ζ(n)(0) involving the coefficientsof the Laurent expansion of Γ(s)ζ(s) at s = 1.

Acknowledgements The authors would like to thank Prof. Shinya Koyama and Prof.Nobushige Kurokawa for useful advice.

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 17

Table 1: γn(x) for n = 0, . . . , 10

n γn(1)

0 0.577215664901532

1 −0.728158454836767 (-1)

2 −0.969036319287231 (-2)

3 0.205383442030334 (-2)

4 0.232537006546730 (-2)

5 0.793323817301062 (-3)

6 −0.238769345430199 (-3)

7 −0.527289567057751 (-3)

8 −0.352123353803039 (-3)

9 −0.343947744180880 (-4)

10 0.205332814909064 (-3)

n γn(12

)

0 1.963510026021423

1 −1.353459680804941

2 0.968864475220290

3 −0.667424273711380

4 0.459547445076771

5 −0.320812026677865

6 0.222019509755189

7 −0.153229125022779

8 0.106924377837182

9 −0.738304956274036 (-1)

10 0.509976576405138 (-1)

n γn(13

)

0 3.132033780020806

1 −3.259557515917910

2 3.619163909618964

3 −3.982394799471162

4 4.368702081667627

5 −4.800731842019795

6 5.275354220225243

7 −5.794291007961231

8 6.366130571415645

9 −6.994208805691668

10 7.683328767950750

n γn(23

)

0 1.318234415786588

1 −0.598906284285989

2 0.257412160840898

3 −0.973035546647373 (-1)

4 0.397278869525733 (-1)

5 −0.176373125264944 (-1)

6 0.612722828123768 (-2)

7 −0.260388126751116 (-2)

8 0.146782221630731 (-2)

9 −0.131712367415337 (-3)

10 0.264294195941530 (-3)

n γn(14

)

0 4.227453533376265

1 −5.518076350199403

2 7.679704425808516

3 −1.066143123379588 (1)

4 1.477301119821657 (1)

5 −2.047938290778997 (1)

6 2.839256318031732 (1)

7 −3.935918438256303 (1)

8 5.456344856174961 (1)

9 −7.564166352341146 (1)

10 1.048609204739940 (2)

n γn(34

)

0 1.085860879786472

1 −0.391298902404549

2 0.119376626018584

3 −0.276661223223528 (-1)

4 0.937466019667217 (-2)

5 −0.357873936586486 (-2)

6 0.931326315228971 (-5)

7 −0.398672736245695 (-3)

8 0.290439256227467 (-3)

9 0.321479008596310 (-3)

10 0.219468578979273 (-3)

18 SHINPEI SAKANE AND MIHO AOKI

n γn(15

)

0 5.289039896592188

1 −8.030205511035976

2 1.294072189077527 (1)

3 −2.084865313923871 (1)

4 3.354832937201646 (1)

5 −5.399227025508818 (1)

6 8.689978858054436 (1)

7 −1.398587197999496 (2)

8 2.250936208372715 (2)

9 −3.622750759614857 (2)

10 5.830585578550786 (2)

n γn(25

)

0 2.561384544585115

1 −2.252808348497526

2 2.101723619623312

3 −1.926843649385201

4 1.760280019163508

5 −1.614874737399961

6 1.480202985145566

7 −1.355194441202952

8 1.242397051138749

9 −1.138425674804981

10 1.042680217903939

n γn(35

)

0 1.540619213893190

1 −0.830775485631480

2 0.445558885156065

3 −0.221000716029528

4 0.111965267766928

5 −0.591535886676098 (-1)

6 0.293449824908595 (-1)

7 −0.148191171354038 (-1)

8 0.814560226265020 (-2)

9 −0.371711703342400 (-2)

10 0.197805498252393 (-2)

n γn(45

)

0 0.965008566706138

1 −0.298163895437231

2 0.691268758704496 (-1)

3 −0.937007424313524 (-2)

4 0.398239404923361 (-2)

5 −0.137070601820030 (-2)

6 −0.627806476775229 (-3)

7 −0.365802273421131 (-3)

8 0.117098770639237 (-3)

9 0.310284896148254 (-3)

10 0.267047492693664 (-3)

n γn(16

)

0 6.332127505374915

1 −1.074258252954789 (1)

2 1.924993463979145 (1)

3 −3.451705746778220 (1)

4 6.184089238317568 (1)

5 −1.108012883745778 (2)

6 1.985320949131804 (2)

7 −3.557208608936910 (2)

8 6.373656573492860 (2)

9 −1.142006945298026 (3)

10 2.046201110829653 (3)

n γn(56

)

0 0.890729412672261

1 −0.246169003811390

2 0.448970524427169 (-1)

3 −0.266403147496967 (-2)

4 0.260762771145478 (-2)

5 −0.699649349006206 (-3)

6 −0.721017707414172 (-3)

7 −0.423192917324565 (-3)

8 0.251354361344485 (-4)

9 0.276380944841368 (-3)

10 0.287490064659336 (-3)

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 19

n γn(17

)

0 7.363980242224343

1 −1.362107052365825 (1)

2 2.649252111626674 (1)

3 −5.158107797983580 (1)

4 1.003677378100677 (2)

5 −1.953030740865246 (2)

6 3.800452158591942 (2)

7 −7.395331240804790 (2)

8 1.439064247993295 (3)

9 −2.800290799753117 (3)

10 5.449113683502010 (3)

n γn(27

)

0 3.685517980285815

1 −4.352556865459653

2 5.487899916675097

3 −6.886097431842962

4 8.619799249401384

5 −1.079897953461351 (1)

6 1.353029437152432 (1)

7 −1.694894294793400 (1)

8 2.123324386362879 (1)

9 −2.660072869760077 (1)

10 3.332374327198948 (1)

n γn(37

)

0 2.365818757294982

1 −1.939717510550862

2 1.679584427180398

3 −1.422359865429945

4 1.200489230755341

5 −1.019296928020903

6 0.863881400940876

7 −0.730973216427048

8 0.620042325457718

9 −0.525286352378200

10 0.444699423346678

n γn(47

)

0 1.648770734921077

1 −0.954605391777175

2 0.558221384139058

3 −0.306252038583967

4 0.169874901634284

5 −0.971636996174147 (-1)

6 0.536144972600240 (-1)

7 −0.296951535140083 (-1)

8 0.172492135056757 (-1)

9 −0.923832840667151 (-2)

10 0.518136728724386 (-2)

n γn(57

)

0 1.180181440339498

1 −0.471136327491078

2 0.168563618211536

3 −0.497603188404758 (-1)

4 0.177220775284988 (-1)

5 −0.712980817549124 (-2)

6 0.135331047274477 (-2)

7 −0.746428063630727 (-3)

8 0.529666001567594 (-3)

9 0.261692624503150 (-3)

10 0.189801688907283 (-3)

n γn(67

)

0 0.840395877730673

1 −0.213259276331916

2 0.311699364453950 (-1)

3 0.289212194168603 (-3)

4 0.220043971643164 (-2)

5 −0.386964870066445 (-3)

6 −0.717282021995928 (-3)

7 −0.467019935129325 (-3)

8 −0.381865633875891 (-4)

9 0.243894289901007 (-3)

10 0.294710945078304 (-3)

20 SHINPEI SAKANE AND MIHO AOKI

n γn(18

)

0 8.388492663295854

1 −1.664171976360931 (1)

2 3.457864853977310 (1)

3 −7.193566477005933 (1)

4 1.495825302477861 (2)

5 −3.110439548091731 (2)

6 6.468007881336385 (2)

7 −1.344983870540879 (3)

8 2.796814410283284 (3)

9 −5.815813199921666 (3)

10 1.209364300472224 (4)

n γn(38

)

0 2.753999049145139

1 −2.577714402801239

2 2.566555681303016

3 −2.520207569548424

4 2.466146772558618

5 −2.420596446033305

6 2.374955632046392

7 −2.328242949240230

8 2.284195795705497

9 −2.240532126288400

10 2.197070670781933

n γn(58

)

0 1.452708764576566

1 −0.735380945923567

2 0.364318586143685

3 −0.164359032929736

4 0.767998354852073 (-1)

5 −0.379082612651699 (-1)

6 0.168702134274392 (-1)

7 −0.787273056952799 (-2)

8 0.421910898760802 (-2)

9 −0.152404257152755 (-2)

10 0.849663574077101 (-3)

n γn(78

)

0 0.804017071547695

1 −0.190659230534731

2 0.225351940036588 (-1)

3 0.174303989746742 (-2)

4 0.208255440133306 (-2)

5 −0.198869685584874 (-3)

6 −0.691680347581824 (-3)

7 −0.496083191761831 (-3)

8 −0.847309021478239 (-4)

9 0.215809157859313 (-3)

10 0.295874810380760 (-3)

n γn(19

)

0 9.407943277131832

1 −1.978671549677987 (1)

2 4.343592540110386 (1)

3 −9.547167002656365 (1)

4 2.097700742462369 (2)

5 −4.609072613142302 (2)

6 1.012719882034104 (3)

7 −2.225172598262015 (3)

8 4.889202868529442 (3)

9 −1.074267787866837 (4)

10 2.360407533835460 (4)

n γn(29

)

0 4.761235072755231

1 −6.746392522535578

2 1.017074247887400 (1)

3 −1.531608907429460 (1)

4 2.302993658432073 (1)

5 −3.463774665220799 (1)

6 5.210023484697115 (1)

7 −7.836159790939911 (1)

8 1.178617609313827 (2)

9 −1.772739899542268 (2)

10 2.666331072777717 (2)

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 21

n γn(49

)

0 2.266764187562898

1 −1.787926283609225

2 1.484947387960652

3 −1.202552897301359

4 0.970859833957256

5 −0.789492863557758

6 0.640313541609353

7 −0.518325616508155

8 0.421032124554191

9 −0.341300658412662

10 0.276439332765320

n γn(59

)

0 1.712816640339515

1 −1.031210137055049

2 0.631692055795577

3 −0.365498640277425

4 0.212990180044145

5 −0.127354660509799

6 0.741726328911679 (-1)

7 −0.432110690842741 (-1)

8 0.260562043339681 (-1)

9 −0.149228119600288 (-1)

10 0.874287265745430 (-2)

n γn(79

)

0 1.017230741372017

1 −0.337126606346973

2 0.891423036618103 (-1)

3 −0.160420645600365 (-1)

4 0.573107939178407 (-2)

5 −0.209899668885806 (-2)

6 −0.451311449453632 (-3)

7 −0.349057780582776 (-3)

8 0.184560727597335 (-3)

9 0.323014751941544 (-3)

10 0.247623084874959 (-3)

n γn(89

)

0 0.776488400269347

1 −0.174225220071331

2 0.166907047829024 (-1)

3 0.250962713382615 (-2)

4 0.206164677597227 (-2)

5 −0.684684161648000 (-4)

6 −0.662003643296816 (-3)

7 −0.515246671465318 (-3)

8 −0.120173772863621 (-3)

9 0.192046427416416 (-3)

10 0.294232904787688 (-3)

22 SHINPEI SAKANE AND MIHO AOKI

Table 2: Γ(n)(s) for n = 0, . . . , 10

Γ(0)(s) = Γ(s),

Γ(1)(s) = Γ(s)ψ(s),

Γ(2)(s) = Γ(s)(ψ(s)2 + ψ(1)(s)),

Γ(3)(s) = Γ(s)(ψ(s)3 + 3ψ(s)ψ(1)(s) + ψ(2)(s)),

Γ(4)(s) = Γ(s)(ψ(s)4 + 6ψ(s)2ψ(1)(s) + 4ψ(s)ψ(2)(s) + 3ψ(1)(s)2 + ψ(3)(s)),

Γ(5)(s)

= Γ(s)(ψ(s)5 + 10ψ(s)3ψ(1)(s) + 10ψ(s)2ψ(2)(s) + 15ψ(s)ψ(1)(s)2 + 5ψ(s)ψ(3)(s)

+10ψ(1)(s)ψ(2)(s) + ψ(4)(s)),

Γ(6)(s)

= Γ(s)(ψ(s)6 + 15ψ(s)4ψ(1)(s) + 20ψ(s)3ψ(2)(s) + 45ψ(s)2ψ(1)(s)2

+15ψ(s)2ψ(3)(s) + 60ψ(s)ψ(1)(s)ψ(2)(s) + 6ψ(s)ψ(4)(s) + 15ψ(1)(s)3

+15ψ(1)(s)ψ(3)(s) + 10ψ(2)(s)2 + ψ(5)(s)),

Γ(7)(s)

= Γ(s)(ψ(s)7 + 21ψ(s)5ψ(1)(s) + 35ψ(s)4ψ(2)(s) + 105ψ(s)3ψ(1)(s)2

+35ψ(s)3ψ(3)(s) + 210ψ(s)2ψ(1)(s)ψ(2)(s) + 21ψ(s)2ψ(4)(s) + 105ψ(s)ψ(1)(s)3

+105ψ(s)ψ(1)(s)ψ(3)(s) + 70ψ(s)ψ(2)(s)2 + 7ψ(s)ψ(5)(s) + 105ψ(1)(s)2ψ(2)(s)

+21ψ(1)(s)ψ(4)(s) + 35ψ(2)(s)ψ(3)(s) + ψ(6)(s)),

Γ(8)(s)

= Γ(s)(ψ(s)8 + 28ψ(s)6ψ(1)(s) + 56ψ(s)5ψ(2)(s) + 210ψ(s)4ψ(1)(s)2

+70ψ(s)4ψ(3)(s) + 560ψ(s)3ψ(1)(s)ψ(2)(s) + 56ψ(s)3ψ(4)(s) + 420ψ(s)2ψ(1)(s)3

+420ψ(s)2ψ(1)(s)ψ(3)(s) + 280ψ(s)2ψ(2)(s)2 + 28ψ(s)2ψ(5)(s)

+840ψ(s)ψ(1)(s)2ψ(2)(s) + 168ψ(s)ψ(1)(s)ψ(4)(s) + 280ψ(s)ψ(2)(s)ψ(3)(s)

+8ψ(s)ψ(6)(s) + 105ψ(1)(s)4 + 210ψ(1)(s)2ψ(3)(s) + 280ψ(1)(s)ψ(2)(s)2

+28ψ(1)(s)ψ(5)(s) + 56ψ(2)(s)ψ(4)(s) + 35ψ(3)(s)2 + ψ(7)(s)),

Γ(9)(s)

= Γ(s)(ψ(s)9 + 36ψ(s)7ψ(1)(s) + 84ψ(s)6ψ(2)(s) + 378ψ(s)5ψ(1)(s)2

+126ψ(s)5ψ(3)(s) + 1260ψ(s)4ψ(1)(s)ψ(2)(s) + 126ψ(s)4ψ(4)(s)

+1260ψ(s)3ψ(1)(s)3 + 1260ψ(s)3ψ(1)(s)ψ(3)(s) + 840ψ(s)3ψ(2)(s)2

+84ψ(s)3ψ(5)(s) + 3780ψ(s)2ψ(1)(s)2ψ(2)(s) + 756ψ(s)2ψ(1)(s)ψ(4)(s)

+1260ψ(s)2ψ(2)(s)ψ(3)(s) + 36ψ(s)2ψ(6)(s) + 945ψ(s)ψ(1)(s)4

+1890ψ(s)ψ(1)(s)2ψ(3)(s) + 2520ψ(s)ψ(1)(s)ψ(2)(s)2 + 252ψ(s)ψ(1)(s)ψ(5)(s)

+504ψ(s)ψ(2)(s)ψ(4)(s) + 315ψ(s)ψ(3)(s)2 + 9ψ(s)ψ(7)(s) + 1260ψ(1)(s)3ψ(2)(s)

+378ψ(1)(s)2ψ(4)(s) + 1260ψ(1)(s)ψ(2)(s)ψ(3)(s) + 36ψ(1)(s)ψ(6)(s) + 280ψ(2)(s)3

+84ψ(2)(s)ψ(5)(s) + 126ψ(3)(s)ψ(4)(s) + ψ(8)(s)),

Γ(10)(s)

= Γ(s)(ψ(s)10 + 45ψ(s)8ψ(1)(s) + 120ψ(s)7ψ(2)(s) + 630ψ(s)6ψ(1)(s)2

+210ψ(s)6ψ(3)(s) + 2520ψ(s)5ψ(1)(s)ψ(2)(s) + 252ψ(s)5ψ(4)(s)

+3150ψ(s)4ψ(1)(s)3 + 3150ψ(s)4ψ(1)(s)ψ(3)(s) + 2100ψ(s)4ψ(2)(s)2

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 23

+210ψ(s)4ψ(5)(s) + 12600ψ(s)3ψ(1)(s)2ψ(2)(s) + 2520ψ(s)3ψ(1)(s)ψ(4)(s)

+4200ψ(s)3ψ(2)(s)ψ(3)(s) + 120ψ(s)3ψ(6)(s) + 4725ψ(s)2ψ(1)(s)4

+9450ψ(s)2ψ(1)(s)2ψ(3)(s) + 12600ψ(s)2ψ(1)(s)ψ(2)(s)2

+1260ψ(s)2ψ(1)(s)ψ(5)(s) + 2520ψ(s)2ψ(2)(s)ψ(4)(s) + 1575ψ(s)2ψ(3)(s)2

+45ψ(s)2ψ(7)(s) + 12600ψ(s)ψ(1)(s)3ψ(2)(s) + 3780ψ(s)ψ(1)(s)2ψ(4)(s)

+12600ψ(s)ψ(1)(s)ψ(2)(s)ψ(3)(s) + 360ψ(s)ψ(1)(s)ψ(6)(s) + 2800ψ(s)ψ(2)(s)3

+840ψ(s)ψ(2)(s)ψ(5)(s) + 1260ψ(s)ψ(3)(s)ψ(4)(s) + 10ψ(s)ψ(8)(s) + 945ψ(1)(s)5

+3150ψ(1)(s)3ψ(3)(s) + 6300ψ(1)(s)2ψ(2)(s)2 + 630ψ(1)(s)2ψ(5)(s)

+2520ψ(1)(s)ψ(2)(s)ψ(4)(s) + 1575ψ(1)(s)ψ(3)(s)2 + 45ψ(1)(s)ψ(7)(s)

+2100ψ(2)(s)2ψ(3)(s) + 120ψ(2)(s)ψ(6)(s) + 210ψ(3)(s)ψ(5)(s) + 126ψ(4)(s)2

+ψ(9)(s)).

Table 3: Γ(n)(1) for n = 0, . . . , 10

Γ(0)(1) = 1,

Γ(1)(1) = −γ,

Γ(2)(1) = γ2 + ζ(2),

Γ(3)(1) = −γ3 − 3γζ(2)− 2ζ(3),

Γ(4)(1) = γ4 + 6γ2ζ(2) + 8γζ(3) + 3ζ(2)2 + 6ζ(4),

Γ(5)(1)

= −γ5 − 10γ3ζ(2)− 20γ2ζ(3)− 15γζ(2)2 − 30γζ(4) − 20ζ(2)ζ(3) − 24ζ(5),

Γ(6)(1)

= γ6 + 15γ4ζ(2) + 40γ3ζ(3) + 45γ2ζ(2)2 + 90γ2ζ(4) + 120γζ(2)ζ(3)

+144γζ(5) + 15ζ(2)3 + 90ζ(2)ζ(4) + 40ζ(3)2 + 120ζ(6),

Γ(7)(1)

= −γ7 − 21γ5ζ(2)− 70γ4ζ(3)− 105γ3ζ(2)2 − 210γ3ζ(4)

−420γ2ζ(2)ζ(3)− 504γ2ζ(5)− 105γζ(2)3 − 630γζ(2)ζ(4) − 280γζ(3)2

−840γζ(6) − 210ζ(2)2ζ(3)− 504ζ(2)ζ(5) − 420ζ(3)ζ(4) − 720ζ(7),

Γ(8)(1)

= γ8 + 28γ6ζ(2) + 112γ5ζ(3) + 210γ4ζ(2)2 + 420γ4ζ(4) + 1120γ3ζ(2)ζ(3)

+1344γ3ζ(5) + 420γ2ζ(2)3 + 2520γ2ζ(2)ζ(4) + 1120γ2ζ(3)2 + 3360γ2ζ(6)

+1680γζ(2)2ζ(3) + 4032γζ(2)ζ(5) + 3360γζ(3)ζ(4) + 5760γζ(7) + 105ζ(2)4

+1260ζ(2)2ζ(4) + 1120ζ(2)ζ(3)2 + 3360ζ(2)ζ(6) + 2688ζ(3)ζ(5)

+1260ζ(4)2 + 5040ζ(8),

Γ(9)(1)

= −γ9 − 36γ7ζ(2)− 168γ6ζ(3)− 378γ5ζ(2)2 − 756γ5ζ(4)

−2520γ4ζ(2)ζ(3) − 3024γ4ζ(5)− 1260γ3ζ(2)3 − 7560γ3ζ(2)ζ(4)− 3360γ3ζ(3)2

−10080γ3ζ(6)− 7560γ2ζ(2)2ζ(3)− 18144γ2ζ(2)ζ(5) − 15120γ2ζ(3)ζ(4)

−25920γ2ζ(7)− 945γζ(2)4 − 11340γζ(2)2ζ(4)− 10080γζ(2)ζ(3)2

−30240γζ(2)ζ(6) − 24192γζ(3)ζ(5) − 11340γζ(4)2 − 45360γζ(8)

24 SHINPEI SAKANE AND MIHO AOKI

−2520ζ(2)3ζ(3)− 9072ζ(2)2ζ(5)− 15120ζ(2)ζ(3)ζ(4) − 25920ζ(2)ζ(7)

−2240ζ(3)3 − 20160ζ(3)ζ(6) − 18144ζ(4)ζ(5) − 40320ζ(9),

Γ(10)(1)

= γ10 + 45γ8ζ(2) + 240γ7ζ(3) + 630γ6ζ(2)2 + 1260γ6ζ(4) + 5040γ5ζ(2)ζ(3)

+6048γ5ζ(5) + 3150γ4ζ(2)3 + 18900γ4ζ(2)ζ(4) + 8400γ4ζ(3)2 + 25200γ4ζ(6)

+25200γ3ζ(2)2ζ(3) + 60480γ3ζ(2)ζ(5) + 50400γ3ζ(3)ζ(4) + 86400γ3ζ(7)

+4725γ2ζ(2)4 + 56700γ2ζ(2)2ζ(4) + 50400γ2ζ(2)ζ(3)2 + 151200γ2ζ(2)ζ(6)

+120960γ2ζ(3)ζ(5) + 56700γ2ζ(4)2 + 226800γ2ζ(8) + 25200γζ(2)3ζ(3)

+90720γζ(2)2ζ(5) + 151200γζ(2)ζ(3)ζ(4) + 259200γζ(2)ζ(7) + 22400γζ(3)3

+201600γζ(3)ζ(6) + 181440γζ(4)ζ(5) + 403200γζ(9) + 945ζ(2)5 + 18900ζ(2)3ζ(4)

+25200ζ(2)2ζ(3)2 + 75600ζ(2)2ζ(6) + 120960ζ(2)ζ(3)ζ(5) + 56700ζ(2)ζ(4)2

+226800ζ(2)ζ(8) + 50400ζ(3)2ζ(4) + 172800ζ(3)ζ(7) + 151200ζ(4)ζ(6)

+72576ζ(5)2 + 362880ζ(10).

Table 4: ζ(n)2 (0, x) for n = 0, . . . , 10

n ζ(n)2 (0, 1)

0 −0.08333333333333331 −0.1654211437004512 −0.2502044241096003 −0.3825315249197724 −0.7474864328370225 −1.873851239295046 −5.626822125842737 −19.68668837240068 −78.74957271962419 −354.37603606952010 −1771.87418572639

n ζ(n)2

(0, 12)

0 0.04166666666666671 −0.1194573558130922 −0.7666507697643553 −3.033535457733914 −12.52293364980485 −61.71592085894156 −365.5118598072677 −2539.611865194768 −20238.69701185149 −181794.33687053710 −1816171.85048653

n ζ(n)2

(0, 13)

0 0.1388888888888891 0.1380476109095092 −0.3933807828526113 −3.039923534479754 −15.29236421857875 −80.27804198318066 −483.8650422616547 −3377.755485337198 −26956.62943472509 −242272.04251902510 −2420969.31389048

n ζ(n)2

(0, 23

)

0 −0.02777777777777781 −0.2192251999980872 −0.7353138417409363 −2.306872810118854 −8.727544235151375 −41.86213250498116 −245.6198126714867 −1699.687308196628 −13518.74828632489 −121314.37452677810 −1211371.87588641

n ζ(n)2

(0, 14)

0 0.1979166666666671 0.3703808625903152 0.1467766884141823 −2.204933290465114 −15.05290494775875 −86.75855689901766 −538.5263423269817 −3789.846313930938 −30305.11070450799 −272495.46400061810 −2723345.65914960

n ζ(n)2

(0, 34)

0 −0.05208333333333331 −0.2311269784884932 −0.6480930364918843 −1.853760282192284 −6.746561523929735 −31.87036853175666 −185.6247054896427 −1279.688553410678 −10158.74867595349 −91074.375396390110 −908971.875589861

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 25

n ζ(n)2

(0, 15

)

0 0.2366666666666671 0.5672829965674902 0.7124243360187893 −1.003299822387314 −13.23521633710145 −87.07961773496326 −564.2452587137527 −4023.713992883398 −32289.73271551879 −290585.91979478210 −2904695.26233644

n ζ(n)2

(0, 25)

0 0.09666666666666671 0.009410863281159032 −0.6194798137368033 −3.195788281405004 −14.44637373874865 −73.23175067722426 −437.0309297415047 −3043.145525474668 −24270.25421576549 −218081.91835358010 −2179051.45839912

n ζ(n)2

(0, 35)

0 −0.003333333333333331 −0.1931678141051932 −0.7764147232915913 −2.637043867522504 −10.28669626824225 −49.83930142391086 −293.6057373587857 −2035.680082919278 −16206.74481526029 −145506.37197930710 −1453291.87496779

n ζ(n)2

(0, 45

)

0 −0.06333333333333331 −0.2289269349263162 −0.5816740273235363 −1.568599853776294 −5.549868289153535 −25.87164816191356 −149.6257079613397 −1027.688320740338 −8142.748693356209 −72930.375703020110 −727531.875282966

n ζ(n)2

(0, 16)

0 0.2638888888888891 0.7352816772618442 1.265232172632393 0.3776142324833084 −10.43739154005175 −83.41079379871506 −572.5370460385307 −4160.398750530128 −33572.52367008359 −302564.04060346510 −3025430.84034503

n ζ(n)2

(0, 56)

0 −0.06944444444444441 −0.2240461982214492 −0.5328331402096313 −1.374857047365344 −4.750432915906875 −21.87204451441016 −125.6261174409527 −859.6880619545688 −6798.748746093129 −60834.375861199810 −606571.875067509

n ζ(n)2

(0, 17

)

0 0.2840136054421771 0.8808472227068292 1.794089665047543 1.848921326338204 −6.977023099352795 −76.83401250610566 −568.4767755767697 −4234.037888275758 −34433.17052381829 −310994.33375851210 −3111393.42894330

n ζ(n)2

(0, 27)

0 0.1717687074829931 0.2588728574600502 −0.1311355272596263 −2.687046692447234 −15.42724105736695 −84.50731192863276 −516.0439301789737 −3614.843880985798 −28872.68493202459 −259546.77396683010 −2593762.35357214

n ζ(n)2

(0, 37)

0 0.07993197278911571 −0.03424587060453442 −0.6805058100109223 −3.185498873222094 −13.95100258294155 −70.01193949834266 −416.6812361273387 −2899.374607513728 −23118.48526412489 −207714.14857563310 −2075371.68506199

n ζ(n)2

(0, 47

)

0 0.008503401360544231 −0.1767420564868732 −0.7837664344028643 −2.765267513124634 −10.94230077922345 −53.24827099165336 −314.1639486348007 −2179.671911104118 −17358.74005913489 −155874.36878085010 −1556971.87318392

n ζ(n)2

(0, 57)

0 −0.04251700680272111 −0.2285746924455282 −0.6895393715356753 −2.052069771913214 −7.598341888696585 −36.15398686756906 −211.3376746051677 −1459.688436399628 −11598.74864659099 −104034.37511230410 −1038571.87577330

n ζ(n)2

(0, 67)

0 −0.07312925170068031 −0.2190173903390062 −0.4960415557578963 −1.235119327991334 −4.178936187157635 −19.01511287527356 −108.4834820359887 −739.6878575315228 −5838.748810266839 −52194.375950716210 −520171.874914798

26 SHINPEI SAKANE AND MIHO AOKI

n ζ(n)2

(0, 18

)

0 0.2994791666666671 1.008877115893712 2.296606015596603 3.363661495219184 −3.045830204346835 −67.99649722934716 −554.7762595150017 −4261.562456585038 −35009.14920020909 −317147.39880585010 −3175460.16831317

n ζ(n)2

(0, 38)

0 0.1119791666666671 0.05290104244475692 −0.5500877558493623 −3.171629422561634 −14.82499829829115 −75.97023822799906 −454.7325475811857 −3168.814257527618 −25277.89473546359 −227153.53357604910 −2269771.05140090

n ζ(n)2

(0, 58)

0 −0.01302083333333331 −0.2048540826810842 −0.7645992699127493 −2.517677845518204 −9.705990202592905 −46.85074358609656 −275.6129914513507 −1909.684104759748 −15198.74688998199 −136434.37340164610 −1362571.87560740

n ζ(n)2

(0, 78

)

0 −0.07552083333333331 −0.2144475149830982 −0.4675154496895533 −1.129711347357394 −3.750146727669025 −16.87240687872496 −95.62647572963347 −649.6876998994558 −5118.748872711029 −45714.376004278910 −455371.874803070

n ζ(n)2

(0, 19)

0 0.3117283950617281 1.122953871155422 2.773425907714463 4.895259079856524 1.230908343113255 −57.33147039967416 −533.1021269529967 −4252.444027568718 −35375.50612835119 −321720.74469889110 −3224749.19860890

n ζ(n)2

(0, 29)

0 0.2191358024691361 0.4729014487881092 0.4298415530137773 −1.634815936801044 −14.29432074273385 −87.51431431777306 −554.0470524137177 −3922.272199970208 −31412.56001073439 −282554.98040866710 −2824112.62238708

n ζ(n)2

(0, 49

)

0 0.07098765432098771 −0.05596871776001062 −0.7071303785020883 −3.165449147400544 −13.65342691998755 −68.19292137851556 −405.3382789319337 −2819.457661427038 −22478.56376293329 −201954.22201035310 −2017771.75287427

n ζ(n)2

(0, 59)

0 0.01543209876543211 −0.1661045465607202 −0.7846748272143683 −2.832040751437894 −11.30183545565345 −55.13804637020616 −325.5818621226737 −2259.664804423598 −17998.73556260969 −161634.36569503010 −1614571.87126871

n ζ(n)2

(0, 79)

0 −0.05864197530864201 −0.2306957312840942 −0.6122917668868413 −1.696261070848054 −6.082191176488855 −28.53789635797096 −165.6253401344067 −1139.688457896698 −9038.748679606939 −80994.375577261410 −808171.875423527

n ζ(n)2

(0, 89

)

0 −0.07716049382716051 −0.2104388305240532 −0.4448195786161643 −1.047419580054404 −3.416576145336575 −15.20586203368156 −85.62656613892407 −579.6875767355088 −4558.748929303809 −40674.376037615210 −404971.874718718

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 27

Table 5: ζ(n)3 (0, x) for n = 0, . . . , 10

n ζ(n)3 (0, 1)

0 −0.04166666666666671 −0.09793480037942212 −0.1579839701485133 −0.2328719825496034 −0.4229529191691335 −1.017602319059446 −2.979127345486787 −10.22756351789478 −40.39810282744579 −180.26156599601810 −896.179304719780

n ζ(n)3

(0, 12

)

0 0.04166666666666671 −0.06471748563119782 −0.5579975400151123 −2.317018904528644 −9.572146761325465 −46.79982918179766 −275.6754874011027 −1909.995724345988 −15199.72218667939 −136437.40876839910 −1362582.09289135

n ζ(n)3

(0, 13)

0 0.1188271604938271 0.1479249709455402 −0.2595353220908323 −2.471198585425194 −12.79769906151935 −67.34127671476026 −404.9672499187907 −2821.419658304078 −22490.77966509349 −202014.17763992510 −2018074.86791619

n ζ(n)3

(0, 23)

0 −0.007716049382716041 −0.1365877882366352 −0.5076500633432043 −1.611834648557854 −5.986637555491335 −28.29913328545156 −164.8457873573317 −1136.790754602138 −9026.647789560289 −80938.383918370310 −807886.806436587

n ζ(n)3

(0, 14

)

0 0.1692708333333331 0.3537039174079722 0.2236616436937003 −1.773820924330064 −13.03653624288145 −76.06085152292356 −472.5980499530757 −3322.650872340348 −26545.82311124859 −238567.13089497610 −2383598.86912902

n ζ(n)3

(0, 34)

0 −0.02343750000000001 −0.1426591033643262 −0.4325264239639283 −1.232505932891634 −4.360140045871365 −20.23387002472926 −116.8821544208877 −802.6515835473948 −6360.085302121409 −56968.853763820910 −568339.149985826

n ζ(n)3

(0, 15)

0 0.2036666666666671 0.5339968179358932 0.7494070728402083 −0.6750885381855884 −11.58607157637435 −78.12513881818956 −508.4986532519877 −3626.802917473958 −29088.78232285679 −261664.90574795910 −2614933.06598909

n ζ(n)3

(0, 25

)

0 0.08433333333333331 0.03884393979318572 −0.4492422853243123 −2.547563012743574 −11.68925868384455 −59.10342578985916 −351.3576086110007 −2440.697533392588 −19440.75453051289 −174574.82811119410 −1743782.88776024

n ζ(n)3

(0, 35)

0 0.009000000000000001 −0.1192677153660062 −0.5494880197992133 −1.913187088740994 −7.381974642057715 −35.33511159345916 −206.8066629683457 −1429.295749304918 −11361.49462286349 −101928.40628802910 −1017668.93032397

n ζ(n)3

(0, 45)

0 −0.03033333333333331 −0.1398770376260322 −0.3803352896599343 −1.011688764824454 −3.455992505294895 −15.79109027108716 −90.50210294973007 −618.9670210656998 −4894.547620594079 −43796.735498625810 −436706.555908710

n ζ(n)3

(0, 16

)

0 0.2283950617283951 0.6908151667374412 1.273401506980863 0.6288016834980674 −9.066543015585105 −75.78505580726836 −524.5782010144547 −3817.343985057558 −30799.79804367669 −277485.23924366110 −2774031.70649782

n ζ(n)3

(0, 56)

0 −0.03395061728395061 −0.1358848488466022 −0.3438855858897733 −0.8691845665606174 −2.884865751262935 −12.99547261878866 −73.91515923444987 −503.5105101990358 −3973.522559056339 −35519.323284836410 −353991.493150366

28 SHINPEI SAKANE AND MIHO AOKI

n ζ(n)3

(0, 17)

0 0.2469630709426631 0.8284024321638492 1.780661114033453 2.040991804258934 −5.821369247919195 −70.23897711948356 −526.5703888386627 −3932.880520707068 −31993.74828969149 −288906.82716055510 −2889865.05405166

n ζ(n)3

(0, 27

)

0 0.1466229348882411 0.2538007086069792 −0.02833007994903283 −2.191760152503444 −13.18543113381495 −72.74802341154386 −443.9450317008327 −3105.159672190958 −24776.29929212059 −222597.21295459110 −2223866.56812057

n ζ(n)3

(0, 37)

0 0.07106413994169101 0.002962680112195752 −0.4972791968771883 −2.511527079512034 −11.11407461656875 −55.53613578666946 −329.0837444466397 −2284.021424467238 −18187.99271186159 −163308.24749045910 −1631165.63243631

n ζ(n)3

(0, 47)

0 0.01737123420796891 −0.1076123560920102 −0.5599709265026023 −2.037305522203624 −7.999966455465045 −38.50396693560336 −225.7623453029837 −1561.507074492068 −12416.99713003069 −111417.84237304610 −1112512.69628522

n ζ(n)3

(0, 57

)

0 −0.01737123420796891 −0.1418822898595862 −0.4670119741089993 −1.394066227722134 −5.039628336715645 −23.58979737607166 −136.8265139024637 −941.5688081583568 −7468.612215630189 −66932.938190863110 −667916.717112838

n ζ(n)3

(0, 67)

0 −0.03607871720116621 −0.1321333287228422 −0.3173672651659643 −0.7701264787871094 −2.492748477844435 −11.08017819885096 −62.55707234955617 −424.4701109077848 −3343.076123653079 −29853.743110506610 −297377.876882709

n ζ(n)3

(0, 18)

0 0.2613932291666671 0.9504774076715062 2.266451854701823 3.509165207853544 −2.060551840674635 −62.21911847266856 −517.6770574511967 −3993.703155896858 −32834.70450569689 −297438.36326932010 −2977684.86265322

n ζ(n)3

(0, 38

)

0 0.09667968750000001 0.07521673393614602 −0.3925099856667493 −2.550029013430084 −12.15379318834135 −62.22627863983376 −371.2250375762997 −2581.033316754788 −20563.76378666919 −184675.90283562910 −1844752.77750484

n ζ(n)3

(0, 58)

0 0.002278645833333331 −0.1272747118504902 −0.5363156424961013 −1.801474525651274 −6.850222493879035 −32.63700302754946 −190.6982947003957 −1316.982699322478 −10464.92783597939 −93868.148354339310 −937110.634172959

n ζ(n)3

(0, 78)

0 −0.03743489583333331 −0.1288790159505282 −0.2973233316511663 −0.6974577904705374 −2.207409724815195 −9.688351537491296 −54.30633612879397 −367.0647995199428 −2885.241317699809 −25739.557966201010 −256267.664678424

n ζ(n)3

(0, 19

)

0 0.2729195244627341 1.059950922792442 2.729964417354483 5.003230271548774 2.078349691214185 −52.21649591471226 −499.9005655022987 −4011.594345234158 −33416.04396292569 −303941.63109230210 −3046248.50236030

n ζ(n)3

(0, 29)

0 0.1879858253315041 0.4470346771581152 0.4850645446675993 −1.258994137957524 −12.47452853462425 −77.74804933987386 −493.5530354101357 −3492.568058813708 −27951.01531521009 −251296.49122751410 −2511015.05106299

THE HIGHER DERIVATIVES OF THE BARNES ZETA FUNCTION 29

n ζ(n)3

(0, 49)

0 0.06407178783721991 −0.01461892867942282 −0.5173777701043003 −2.479330348396474 −10.78062213333615 −53.56638455162136 −316.9257828923837 −2198.712708819378 −17506.18627349629 −157176.98157652110 −1569880.43247900

n ζ(n)3

(0, 59

)

0 0.02234796524919981 −0.09987636670725792 −0.5633442754941853 −2.103940064823354 −8.347056525891305 −40.30266773259376 −236.5439452600397 −1636.733016258058 −13017.60777533829 −116817.75000965010 −1166483.67644409

n ζ(n)3

(0, 79)

0 −0.02749199817101051 −0.1416251027234562 −0.4039836899667393 −1.108897519243644 −3.850900438838085 −17.72876713438356 −102.0044214731277 −699.0490899126558 −5533.453247178219 −49539.010290984310 −494089.931065099

n ζ(n)3

(0, 89)

0 −0.03835162322816641 −0.1261080208237532 −0.2816849060525523 −0.6419503171229674 −1.990696020723765 −8.632284259903486 −48.04781165242037 −323.5273302902968 −2538.036480538239 −22619.636178563610 −225093.055185326

References

1. T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.2. T. M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Math. Comp. 44, no. 169

(1985), 223–232.3. T. M. Apostol, Zeta and Related Functions, NIST Handbook of Mathematical Functions, U. S. Dept.

Commerce, Washington, DC, 2010, pp. 601–616.3. E. W. Barnes, The theory of the double gamma function, Phil. Trans. Roy. Soc. (A) 196 (1901),

265–387.4. E. W. Barnes, On the theory of the multiple gamma function, Trans. Camb. Philos. Soc. 19 (1904),

374–425.5. B. C. Berndt, On the Hurwitz zeta function, Rochy Mountain J. Math. 2 (1972), 151-157.6. M. W. Coffey, A set of identities for a class of alternating binomial sums arising in computing appli-

cations, Utilitas Mathematica 76 (2008), 79–90.7. L. Comtet, Advanced combinatorics, The art of finite and infinite expansions, Revised and enlarged

edition. D. Reidel Publishing Co., Dordrecht, 1974.8. S. Koyama and N. Kurokawa, Kummer’s formula for multiple gamma functions, J. Ramanujan Math.

Soc. 18 (2003), 87–107.9. M. Lerch, Daľsi studie v oboru Malmsténovských ŕad, Rozpravy České Akad. véd 3 (1894), 1–61.

Department of Information Systems Design and Data Science, Interdisciplinary Faculty

of Science and Engineering, Shimane University, Matsue, Shimane, 690-8504, Japan

Email address: s173040@matsu.shimane-u.ac.jp

Department of Mathematics, Interdisciplinary Faculty of Science and Engineering, Shi-

mane University, Matsue, Shimane, 690-8504, Japan

Email address: aoki@riko.shimane-u.ac.jp

References