Generalized divisor problem
C. Calderón*
M.J. Zárate
∗
The authors study the generalized divisor problem by means of the residue theorem
of Cauchy and the properties of the Riemann Zeta function.
By an elementary argument similar to that used by Dirichlet in the divisor
problem, Landau [7] proved that if α and β are fixed positive numbers and if α 6= β,
then
X
1 = ζ(β/α)x1/α + ζ(α/β)x1/β + O(x1/(α+β)).
mα nβ ≤x
In 1952 H.E. Richert by means of the theory of Exponents Pairs (developed by J.G.
van der Korput and E. Phillips ) improved the above O-term ( see [8] or [4] pag. 221 ).
In 1969 E. Krätzel studied the three-dimensional problem. Besides, M.Vogts (1981)
and A. Ivić (1981) got some interesting results which generalize the work of P.G.
R
Schmidt of 1968. In 1987 A.Ivić obtained Ω-results for 1T ∆2k (a1, . . . , ak , x)dx where
∆k (a1, . . . , ak , x) is the error-term of the summatory function of d(a1, . . . , ak , n).
Moreover, he proved that his results hold for the function
ζ q1 (b1 s)ζ q2 (b2s)ζ q3 (b3 s) . . . ,
1 ≤ b1 < b2 < b3 . . .
qj , bj being some positive integers. In 1983 E.Krätzel studied the many-dimensional
problem
X
ma11 . . . mann ,
b
m11 ...mbnn ≤x
using the properties of the Riemann Zeta-function. M. Vogts (1985) also studied
that problem but using elementary methods. Here, we will analyse some sums of
the above kind by means of non-elementary methods.
∗
Supported by Grant PB91-0449 of the DGICYT and the University of the Basque Country
Received by the editors November 1993
Communicated by Y. Félix
AMS Mathematics Subject Classification : 11N37, 30B50
Bull. Belg. Math. Soc. 2 (1995), 59–63
60
C. Calderón – M.J. Zárate
Theorem. Let aj > −1, bj > 0, (j = 1, . . . , n),
j
Suppose that the fractions 1+a
belong to the gap
bj
(n ≥ 2) be real numbers .
(c −
1
1
1
, c + min{0,
−
}], j = 1, . . . , n
2b1
2bj
2b1
and
1 + a1
1 + aj
= max
.
1≤j≤n
b1
bj
c=
If
n
X
1
≤1
1
j=1 m(bj (c − 2b1 ) − aj )
(1)
where
Z
T
m(σ) = sup{m :
1
|ζ(σ + it)|m T 1+, ∀ > 0}
then we have
X
ma11
. . . mann
=
When the fractions
and we have
1+ak
bk
n
Y

ζ(bj s − aj )
j=1
1+aj
bj
are different, the point s =



Res
1+a
s=
Res 
s=
b
m11 ...mbnn ≤x
X

X
k
bk
n
Y
ζ(bj s − aj )
j=1
1+aj
bj
xs
c− 1 +
+ O(x 2b1 ).
s
is a simple pole of ζ(bj s−aj )
n Y
n
1+ak
xs X
1 + ak
=
ζ(bj
− aj )x bk .
s
bk
k=1 j=1
j6=k
When
1 + ar
1 + ar+1
1 + ar+q−1
=
= ... =
br
br+1
br+q−1
(1 ≤ r < r + q − 1 ≤ n), the sum
r+q−1
X
Y
ζ(bj
k=r j=1
j6=k
1+ak
1 + ak
− aj )x bk
bk
1+ar
is replaced by x br Pr,q (log x) where Pr,q (log x) is a polynomial in log x of degree
q − 1.
Q
Proof. The function nj=1 ζ(bj s − aj ) can be expresed as a Dirichlet series absolutely convergent for σ > c

∞
X


N =1

X
b
m11 ...mbnn =N

ma11 . . . mann  N −s .
Generalized divisor problem
61
By Lemma 3.12 [10] if x is the half of an odd integer we have
X
ma11 . . . mann =
b
m11 ...mbnn ≤x
Z
1
2πi
c++iT
c+−iT
n
Y
ζ(bj s − aj )
j=1
We will consider the rectangle RT with vertices c −
Z
1
2πi
=
c+−iT

X

Res
1+a
k
s=
n
Y
c++iT
n
Y
ζ(bj s − aj )
j=1

ζ(bj s − aj )
j=1
bk
1
2b1
xs
xc+
ds + O(
).
s
T
± iT, c + ± iT . Therefore,
xs
ds =
s
xs
1
+
(I1 + I2 + I3 )
s
2πi
where I1, I2, I3 are the integrals along the other three sides of the rectangle
I1 + I3 Z
n
Y
c+
c− 2b1 j=1
1
|ζ(bj σ − aj + ibj T )|xσ T −1dσ.
Let
µ(σ) = inf{ξ, ζ(σ + it) = O(|t|ξ ), as t → ∞}
then
I1 + I3 Pn
T −1+
j=1
Z
Pn
c+
T
c− 2b1
1
µ(bj (c+)−aj )+ c+
x
j=1
µ(bj σ−aj )+
Pn
+T
−1+
j=1
T −1xσ dσ µ(bj (c− 2b1 )−aj )+ c− 2b1
1
x
1
From hypothesis (1) we can deduce
−1 +
n
X
µ(bj (c −
j=1
1
) − aj ) ≤ 0
2b1
therefore
I1 + I3 x
c− 2b1
1
xc+
.
T
T +
For the integral I2 we have
I2 x
x
c− 2b1
1
+x
c− 2b1
Z
T
1
c− 2b1
1
+x
n
Y
1
1
1
T
c− 2b1
|ζ(bj (c −
j=1
Z
1
T
n
Y
1
dt
) − aj + bj t)| 2b1
t
|ζ(bj (c −
j=1
1
) − aj + bj t)|dt.
2b1
Calling
mj = m(bj (c −
1
) − aj )
2b1
.
62
C. Calderón – M.J. Zárate
by the Hölder inequality and hypothesis (1) we get
I2 x
c− 2b1
1
T .
Then
X
ma11 . . . mann =

X

Res
1+a
s=
b
m11 ...mbnn ≤x
k
bk
n
Y

ζ(bj s − aj )
j=1
xs
+
s
xc+
c− 1
+O(
) + O(x 2b1 T ).
T
Taking T = xd for some sufficiently large d > 0 we obtain (2) and theorem is proved.
Case A. Let b1 = . . . = bn = 1, aj > −1, 0 ≤ a1 − aj < 1/2 for j = 1, . . . , n. If
n
X
1
≤1
j=1 m((1/2) + a1 − aj )
then
X
m1 ...mn ≤x
ma11 . . . mann =
X

Res 
s=1+aj
n
Y

ζ(s − aj )
j=1
xs
+ O(xa1 +(1/2)+)
s
for every > 0.
For example, let a1 > −1 and aj = a1 − j−1
, (j = 2, . . . , n). be real numbers.
2n
For 4 ≤ n ≤ 10, from Theorem 8.4. [2], the hypothesis (1) is verified and the above
formula holds.
Case B. Let a1 = . . . = an = a > −1 y 0 < b1 ≤ . . . ≤ bn . If
1+a
1
1 1
1 1
( + a) <
≤ ( + a) +
, j = 1, . . . , n
b1 2
bj
b1 2
2bj
and, besides
n
X
1
≤1
1
j=1 m((bj /b1 )( 2 + a) − a)
then
X
a
(m1 . . . mn ) =
X

Res 
s= 1+a
b
b
m11 ...mbnn ≤x
j
1
+O(x b1
In particular if a = 1 and 0 < b1 ≤ bj <
(a+(1/2)+
4b1
3
3
is satisfied, the order of the O-term is x 2b1
n
Y

ζ(bj s − a)
j=1
xs
s
).
and if the condition about the moments
+
for every > 0.
Generalized divisor problem
63
References
[1] A. Ivić, On the number of finite non isomorphic abelian groups in short intervals. Math. Nachr. 101(1981), 257-271.
[2] A. Ivić, The Riemann zeta function. John Wiley and sons. New York 1985.
[3] A. Ivić, The general divisor problem. Journal of number theory, 27 (1987),
73-91.
[4] E. Krätzel, Lattice Points. Kluwer Academic Publishers. Dordrech Boston London. 1988.
[5] E. Krätzel, Teilerproblem in drei dimensionen. Math. Nachr. 42 (1969), 275-288
[6] E. Krätzel, Divisor problems and powerful numbers. Math. Nachr. 114 (1983)
97-104.
[7] E. Landau, Au sujet d0 une certaine expression asymptotique. L’intermédiaire
des math. Vol. 20 (1913), p.155.
[8] H.E. Richert, ”Uber die anzahl abelscher gruppen gegebener ordnung I. Math.
Zeit. 56 (1952), 21-32.
[9] P.G.Schmidt. Zur Anzahl abelscher gruppen gegebener ordnung I. J.Reine
Angew. Math. 229 (1968), 34-42.
[10] E.C.Titchmarsh, The theory of the Riemann zeta function. Clarendon Press ,
Oxford (1951).
[11] M. Vogts, Teilerproblem in drei dimensionen. Math. Nachr. 101(1981), 243-256.
[12] M. Vogts, Many dimensional generalized divisor problems. Math. Nachr. 124
(1985) 103-121.
C.Calderón and M.J. Zárate
Universidad del Paı́s Vasco
Departamento de Matemáticas
E-48080 Bilbao- España
E-mail MTPCAGAC@lg.ehu.es