Download 2.1) cr,,ih,k)= E MtIMt - American Mathematical Society

DEDEKIND SUMS AND LAMBERT SERIES
L. CARLITZ
1. Introduction.
Apostol
[l] has proved a transformation
formula
for the function
oo
GP(x) = E
»-p*m"
(| x\ < 1),
m,n=l
where p is a fixed odd integer
(1.1)
c.(h, k) =
E
M(mod t)
>1. In the formula occur the numbers
3P+1_,(4) £ (t)
\ k /
(P&s£p+1),
\ k/
where (A, k) = l, the summation
is over a complete
(mod k), and Ss(x) is the Bernoulli function. Put
(1.2)
/(A, A:r) = E(
)(*r-
«=o \
then Apostol's
s
residue
system
A)»-c.(ft, *);
/
formula can be put in the form [2, §2]
i2Tci)P
(1.3) Gp(e*"0 = (*t - h)p-*GPie2™')+ „/
,',/(*,
*: 0,
2(/> + 1)!
where r = (AV+*')/(>--A),
that
hh'+kk'
(1.3) implies the following
(1.4)
/(A, ft; t) = r^/(-
+ l =0. It is also shown in [2]
transformation
formula
for/(A,
k; t) :
*, A; - —) + — (5 + tB)"1.
The purpose of this note is to give an elementary
proof of (1.4)
depending
on the representation
of eg(A, k) by means of "Eulerian"
numbers (see (2.3) below).
2. It is convenient
(2.1)
to use, in place of (1.1), the fuller notation
cr,,ih,k)=
E MtIMt)'
,,(mod it)
\ k/
\ k/
where now r, s are arbitrary
non-negative
integers;
that c„(A, k) =cp+i^StSih, k). If we define the Eulerian
thus it is clear
numbers Hmia)
by means of
Presented
to the Society, February
27, 1954; received by the editors January
1954.
580
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19,
DEDEKIND SUMS AND LAMBERT SERIES
1 -
(2.2)
a
"
-
ex — a
„
t,
Xm
= E Hm(a) — ,
m=0
then we have the representation
(2.3)
m\
[2, §6]
BrB,
rs
cr..(h, k) =-E-■'
ffr_i(f*) H^r1)
where 5m denotes a Bernoulli number
the summation
is over all &th roots
formula (2.3) is proved for r^l, 5^1
the right member as 0 for rs = 0, then
valid for all non-negative r, s. In the
"
xry°
P,,_o
rlsl
E k'CrAh, k)—^ = -"
x
in the even suffix notation and
of unity distinct from 1. The
but if we interpret the sum in
it is easily verified that (2.3) is
next place we have, using (2.2),
y
~—,~
ez — 1 e»lk — 1
00
+ *y£ n—^71—*
x*i (1 - r*)(l
^
581
y
=-\gx _ 1 g»/* -
%T
00
ySfe—n
S #r(r*) - E ^.(r1) L—
- f) r=o
r! «-o
v-
f"1
f*
5!
xy y.' ~ e* - f * eW* - f-1
1
and therefore
oo
(2.4)
~r
^.s
yh
>-1
E *'*..(*,*)- -y = *yE -^—-
r,.=o
r!
where the summation
5!
f
..
e1 - f* e"'* - r1
>
is now over all &th roots of unity.
If we put
(2.5)
bT,s(h,*) = E (-!)-'(
* ) *-«Cr+.-M(A, *)
(-0
\
/ /
or, what is the same thing,
(2• 6)
cr,,(h, £) = E ( * ) A-'Jr+._M(A, *),
(=0 W /
then a straight-forward
(2.7)
"
xr y*
yields
"
(£x + hy)r
E *'*..(*, *) — — = E »r..(*.*)-—
r,«-o
For brevity
(2.8)
computation
r!
5.'
T,.=0
we put
fcs= kx+ hy
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»"!
y'
—•
5!
582
L. CARLITZ
[August
and
(2.9)
Fih, ft; z, y) = E or..(A, k) — — ■
r,«-o
rl
si
It follows from (2.4) and (2.7) that
(2.10)
Fih,k;kz,y)
3. If T)runs through
(3.1)
= xyE
^
f
,
ex -
f*
„„
evlhi — 1
'
the Ath roots of unity, then
_^_r^-r_^.
x* — 1
, x — ij
,
*ij — 1
Thus
*f»
=
e- _ f*
A
=
1
e*r* _ !
j
^/^f-1
_
f
- 1 "" ,
c1"1'?- f
and (2.10) becomes
xy ,-,
(3.2)
f
7?(A,k; kz, y) = — E-
1
But since
f
1
/
ex/*»?— f e"'*f — 1
the right member
*? y,
vex,h
1
\elMJ? — f
\
1
c1,/*f — 1/ exlh+«'kri — 1
of (3.2) is equal to
<?exM_1_,^Zx^_
_J_L__
A 7r ex'hri - f ew»+w*, _i
h Xi e»'*f - 1 ««/*+»/*,,- 1
Then using (2.8) and (3.1) we get
kxy _
F(A, ft; kz, y) =-
1
E-
i3.3)
1
kxy
+-
(C - l)(e' - 1)
Returning to (2.14), we replace A, ft, x, y by ft, A, —ftx/A, 2 respectively; we get
"
(-ftx)r
2*
ftxz -_
1
1
(3.4) T,cr.(k,h)--=--EZU
r! see http://www.ams.org/journal-terms-of-use
s!
A , «r*x'V* - 1 «''*ij - 1
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1954]
DEDEKIND SUMS AND LAMBERT SERIES
583
But by (2.7) the left-hand member of (3.4) is equal to
*
(kz -
kx)r
E bT,(k, h)--—r,._o
Comparison
z*
-
r\
"
(hy)T
r,s=o
r\
= E bF,(k, k)U^-
s\
z'
- •
si
with (3.3) leads at once to
(3.5) zF(h, k; kz, y) = yF(k, h; hy, z) + (kz - hy) —-—-
e' — 1 e" — 1
4. We now compare coefficients
In view of (2.9) we get at once
(4.1)
of zTy" in both members
rk'~1br-.1,,(k, k) = sh'-^-iAk,
for all r, 5^0.
Incidentally
of (3.5).
h) + rkBT-iBs - shBTBa-i
this proves Theorem
1 of [2].
If we put
(4.2)
fm(h, k; r) = E (M ) (*t - *)—1-cM-...(A, k),
—0 \ 5 /
then it follows readily from (2.5) that
(4.3)
(kr - h)fm(h,k; r) - e(*
V*r)—6Mi.(A, k).
8_0 \
5 /
In (4.3) replace h, k, r by —k, h, —1/r, respectively; then
kr-
h
/
1\
—-—M-
k<h'--)
(4.4)
™ /w\/
A\m-'
-£(.)(-t)
»-■•<-*•*>•
But it is clear from (2.1) that
cr.,(-
k, h) = (-
\Ycr..(k, h)
and therefore (2.5) implies
(4.5)
*,..(- 4, A) = (- \)%,,(k, h).
Hence (4.4) becomes
T^(kr(4.6)
h)fj-
k, h;-\
T
= (-l)mE(
)r»A—*_..(*,
s=0 \
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5 /
A).
584
L. CARLITZ
It follows that
(At - A)|/„(A, ft;r) - (- l)^rm-2fj= El
ft, h; - i-U
)rm_'^",_'&»'-».'(A' *)-*-h'-^i.^+iik,
,=o\ s /
\
= E ( m ) rm- \ kBm-.B,_0 \ S /
= ft E (
1.
)
kBrn-.+lBs-.X
OT — 5+1
) £—A - AE (
8_0 \ 5 /
h)\
m —s + 1
,= 1 V
)
) t—5)M+15^i
— 1/
= (k--.yB+TB)~
and therefore
(4.7)
/.(A, ft; r) = (- l)»r»-2/m (-
ft, ft; - —) + — (5 + tB)'".
In particular for m = p-\-l, p odd, (4.7) reduces to (1.4).
We have therefore proved (1.4) and indeed the more general result
(4.7) using only the representation
(2.3) and familiar properties of the
Bernoulli functions.
References
1. T. M. Apostol,
Generalized
Dedekind
sums and transformation
formulae
of cer-
tain Lambert series, Duke Math. J. vol. 17 (1950) pp. 147-157.
2. L. Carlitz,
Some theorems on generalized
Dedekind
Mathematics vol. 3 (1953) pp. 513-522.
Duke University
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sums, Pacific
Journal
of