Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz

Apostol, Tom M.

Vu, Thiennu H.

English

Tom M. Apostol

Thiennu H. Vu

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Pacific Journal of Mathematics

Elementary proofs of Berndt’s reciprocity laws

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PACIFIC JOURNAL OF MATHEMATICS
Vol. 98, No. 1, 1982
ELEMENTARY PROOFS OF BERNDT'S
RECIPROCITY LAWS
TOM M. APOSTOL AND THIENNU H. VU
Using analytic functional equations, Berndt derived
three reciprocity laws connecting five arithmetical sums
analogous to Dedekind sums. This paper gives elementary
proofs of all three reciprocity laws and obtains them all
from a common source, a polynomial reciprocity formula of
L. Carlitz.
1* Introduction* The classical Dedekind sums
**>%5.((f)X(τ ))
where k and k are integers, k > 0, ((#)) = x — [x] — 1/2 if x Φ integer,
and ((#)) = 0 for integer x, occur in the transformation formula for
the logarithm of the Dedekind eta function
η(τ) = e«
ίτ/12 Π (1 - e2πίnτ) . (Im (r) > 0)
n-l
Dedekind's formula which describes the behavior of log Ύ](τ) under a
unimodular substitution implies a reciprocity law relating s(h, k) and
s(k, h) when (h, k) = 1. (See [1], Chapter 3.)
Berndt [2] derived transformation formulas for the logarithm of
the theta function
θ(τ) = Π (1 - e2πinτ)(l + e**-
1)xivγ
and related functions, and introduced five new arithmetical sums
which are analogous to (but quite different from) the Dedekind sums,
and showed that the analytic functional equations imply reciprocity
laws for these sums. The sums in question are
( l ) S(h,k)= Σ *
r=l
(2) βl(A,fc) = Σ ( - 1
(4) 8 , < M ) = Σ < - i
17
18 TOM M. APOSTOL AND THIENNU H. VU
and
(5) 8t(h,k)= Σ ( - l ) C A r / * ] .
Berndt's reciprocity laws, which occur, respectively, as Theorems
4.2, 6.2, and 8.2 in [2], can be stated as follows:
THEOREM 1. If h and k have opposite parity and (h, k) = 1,
then
(6 ) S(h, k) + Sφ, h) = 1 .
THEOREM 2. If h is odd, k is even, and (h, k) — 1, then
( 7 ) 2s2(h, k) - Sl(&, h) = i-( A . + f -
2 V ft& k
THEOREM 3. If k is odd and (h, k) = 1, £frβw
( 8 ) 288(fc, fc) - 84(fc, Λ) = 1 - - r
k
Since these theorems concern arithmetical sums, it seems desira-
ble to have proofs independent of the theory of theta functions. An
elementary proof of (6) has been given by Berndt, Evans and others
[3]. This paper gives elementary proofs of all three reciprocity laws
and, moreover, obtains them all from a common source, a polynomial
reciprocity formula of L. Carlitz ([4], Eq. (5.11)) which states that
( 9 ) (u-1) Σ uk-r-1vίhr/kl - 0 - 1) Σ vΛ--V*r/Λ] = uk-1 - vh~ι .
r=l r=l
Here h and k are coprime positive integers and u, v are arbitrary
complex numbers.
In [4] Carlitz gives an elementary proof of (9). We give a
different elementary proof involving lattice points in a triangle and
then use (9) to deduce Theorems 1, 2 and 3. We also show that in
the cases not covered by Berndt's theorems the sums in question
vanish. Thus we have the following companion theorems.
THEOREM la. If both h and k are odd and (h, k) — 1, then
(10) S{h, k) = S(k, fc) = 0 .
THEOREM 2a. If k is odd and (h, k) = 1, then
(11) 82(h, k) = s^k, h) = 0 .
ELEMENTARY PROOFS OF BERNDT'S RECIPROCITY LAWS 19
THEOREM 3a. If k is even and (h, k) — 1, then
(12) s3(h, k) = 84(k, h) = 0 .
2* Proof of Carlitz's reciprocity formula (9)* We have
= (^ - 1 ) Σ w*-1-' - (v - l) Σ vh~λ-r.
r=l r=l
This identity reduces to (9) if, and only if, we have
(13) (u - 1) Σ ^ - ^ ( l - v^m) = (v - 1) Σ ^ - ^ ( l - u^
Now if ftr/λ; ̂  1 we have
and there is a corresponding formula for 1 — uίlcr/hΊ if kr/h ^ 1.
Hence (13) is equivalent to the identity
fc-l [Ar/fc]-l Λ - l [&β/Λ]-l
(14) Σ Σ ^fc"1"rt;w = Σ Σ v^^u™ .
r = l 97=0 s=l m=0
hrlk^l fcβ/Λ^l
Because of symmetry in A and ά, we can assume that h < k so
the condition fts/ft ̂  1 is automatically satisfied. Let L denote the
left member of (14). In the sum over r introduce a new index of
summation, m = k — 1 — r. Then
Γ hr Ί _ Γ h(k — 1 — m) ~| , 1 Γ h(l + m) Ί
b π - L Λ̂  J - h ~1 ~ L— i—J
and we get
k-2 h-l-[h(l+m,)lk]
"̂ -> ">-~» m n
— 2-x 2a U V .
Now replace the index n hy s — h — 1 — n. This gives
k-2 h-1
L = Σ> Σ
m=O s = ["
This double sum is extended over the lattice points (m, s) in the α?2/-
plane which lie inside or on the boundary of the right triangle
bounded by the lines
as = 0 , y = h — 1 , and y — h(l + x)/k .
Interchanging the order of summation we find
20 TOM M. APOSTOL AND THIENNU H. VU
ft-l [fcβ/λ]-l
L = Σ Σ w^v"-1-*,
which proves (14), and hence (9).
3. Proof of Theorems 1 and la. Taking u = i; = — 1 in (9)
and dividing by 2( —l)fc+1 we obtain
fe-l fc-1 1 ( 1 Uτli
y1 / ]\r+l+lhr/k] i / ]\h-k+l y / J\r+l+[fcr/Λ] __ •*• V -U
r=l r=l 2
If /i and k have opposite parity this implies Berndt's Theorem 1,
and if h and k have the same parity (both odd since (h, k) — 1), we
obtain
S(h, k) - S(k, h) = 0.
But if h and k are both odd we have
S(h, k) = Σ(-l) (*- r ) + 1 + C Λ (*- r ) /* 3
r=l
= (-1)* Σί-l)"'"1"1"1"*"1"1*''*3 = ( - l ) * " " - ^ , k) = -S(h, k) ,
r=l
so S(h, k) = 0 and hence also S(k, h) = 0.
4* Proof of Theorems 3 and 3a* We differentiate each member
of (9) with respect to v and then put v — 1 to obtain
(15) Σ (uk-r - w * - " - 1 ) 4 2 1 - Σ u l k r m = l - h .
r = l L fe J f=l
When % = — 1 this becomes
(16) 2(-l) f c Σ ( - l ) f ψ\ - sβc, Λ) = 1 - Λ .
But [fcr/fc] - fer/fc - 1/2 - ((hr/k)) so
Σ (-1)1 γ = y - Σ (-i)r» ~ -T Σ (-Dr - «,(Λ, fc)
since
Σ(-D'r = Γ-|-l(-l)*-1 and Σ(-D r = - ( ~ 1 } *
r=l L 2 J r=l 2
Using (17) in (16) we obtain
ELEMENTARY PROOFS OF BERNDT'S RECIPROCITY LAWS 21
When k is odd this gives Berndt's Theorem 3, and when k is even
it gives
(18) -2s8(Λ, k) - s4(k, h) - 0 .
But when k is even it is easy to see that sjjύ, h) = 0 because
s4(fe, h) = Σ (
r=l r=l
h-1
= - E ( - i ) [ i r / " ] = -««(&, A ) .
r=l
Therefore s4(fc, h) — 0 and (18) shows that s8(/&, fc) = 0 when k is even.
5* Proof of Theorems 2 and 2a* Start with Equation (15)
and rewrite it as follows:
k-l Γ Z , r Ί h-1
(u - l) Σ w*-H — - Σ ^ [ / ί r A ] = l - λ .
r=l L fc J r=l
Replace r by k — r in the first sum and note that [h(k — r)/k] =
h — 1 — [fcr/fc] to obtain
h-1 7c-1 Γ TΪ/Ϊ ' Ί k - 1
- l) Σ ^ r - 1 - (w - l) Σ ^ r "Ί — - Σ
r=l r=l L & J r=l
or
fc-i Γ Z,Λ. Ί h-i
(19) (w - l) Σ * H -^- + Σ ^ [ / c r A ] = ^"'(Λ - i)
r=l L k J • ^ = 1
Differentiate with respect to w and multiply by w to obtain
(u - 1) g ru-f ^ Ί + Σ «H"-^ 1 + Σ Γ-̂ -V*"1"
r=i L A J r=l L fc J r=lL fe J
= (& - l ) ^ - 1 ^ - 1) .
Now multiply by (w — 1) and use (19) to obtain
fc-l Γ j L ^ " I A - l A - l Γ JU/v. " I
(« - l)2 Σ ntH — - Σ «[ir/Λ] + ( » - i ) Σ τ k*r/*]
r=l L β J r=l r=lLft J
When « = — 1 this gives us
4 Σ ί - i r ' r Γ - ^ l - «*(fc, A) - 2 Σ(-l)[*'/»iΓ-^L
(20) r=l L & J r=l L ft
22 TOM M. APOSTOL AND THIENNU H. VU
If k is even, s4(fc, ft) = 0 by Theorem 3a, and (20) becomes
(21) 4 Σ ί - i r ' r Γ - T 1 ! - 2 Σ (-l) [ f c r /*f-^1 = (fc - l)(2fc - 1) .
r=l L & J r=l L ft J
Theorem 2 now follows at once from (21) and the following lemma:
LEMMA. // k is even and ft is odd, (ft, k) = 1, then we have
(22) - 2 Σ (-1)[*/*3Γ * r 1 = l _ JL - 2 / ^ , fc) ,
r=l L ft J ft
(23) 4 Σ ( - l ) r " ^ Γ — 1 = 4Λ*2(fc, k) + 2hk-2h-k
r=l L & J
To prove (22) we evaluate the sum
in two ways. On the one hand we have (since s4(k, h) — 0 if k is
even)
ft
2 Σ ( i ) f 4
r=l L ft
- -2 Σ ( - 1 ) ^ 4 ^ 1 + 2Λ»i(fc, Λ)
r=l L fc J
On the other hand we have, since A; is even,
t(h, Jc) = r=έO(modfc) \ \ ft
Write kr = qh + ^, where g = [fer/ft] and 0 < p < ft. Since k is even
and ft is odd we have qh + p = q + p== 0(mod2) so (—1)' = ( —I) 9 .
Hence
ft- 1 , 1
ft
Equating the two expressions for t(h, k) we obtain (22V
To prove (23) we write
ELEMENTARY PROOFS OF BERNDT'S RECIPROCITY LAWS 23
η 1 J. ^ I J 1 i 1 Jg J
M _|_ 2Jc V. ( — 1)Ί + V ( — l ) r r 2
r=l L fc J r=l L fc J fc r=l
Λ-l k-l
r = l r = l
Now if fc is even we have
k-l fcΓfc J\ k~1 fo
Σ(-DV = - , Σ(-i)rr = - - ,
and Σ(-Dr = " I ,
r=l
SO
(24) 4fcs (/fc fc) = 4 Σ ( l)rτ\ + 2fc V, (-—-In + 2Λ Λfc
r=l L fc J r=l L fc J
L e t S denote t h e second sum on the r ight . Replacing r by fc — r
we find (since fc is even),
Q — X^ / 1 \k-r(h 1 Γ ^
r=l \ L fc
so 2S = 1 - h and 2fcS = k - hk. Therefore (24) reduces to (23).
This completes the proof of the lemma and also of Theorem 2.
Finally, to prove Theorem 2a we replace the index r by fc — r
in (3) to obtain
k-l
Q (I, IΛ — ^ / 1 \fc~i
O2\*t', Π/y x j ^ X.J
Therefore s2(/ι, fc) = 0 if fc is odd and (fc, fc) = 1. A similar argument
shows that 8t(h, fc) = 0 if h is odd and {h, fc) = 1. This implies
Theorem 2a.
REFERENCES
1. Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory.
Graduate texts in Mathematics, 41. Springer-Verlag , New York, 1976.
2. Bruce C. Berndt, Analytic Eisenstein series, theta-functions, and series relations
in the spirit of Ramanujan, J. fur die reine und angewandte Math. 303/304 (1978),
332-365. MR 80b, #10035.
3. Bruce C. Berndt and Ronald J. Evans, Problem E2758, American Math. Monthly,
v. 86(1979), p. 128, and v. 87(1980), p. 404.
4. Leonard Carlitz, Some polynomials associated with Dedekind sums, Acta Math.
Acad. Sci. Hungaricae, 26(1975), 311-319. MR 52, #5540.
Received January 21, 1981.
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CA 91125



Elementary proofs of Berndt's reciprocity laws

Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J. fur die reine und angewandte Math.

Bruce C. Berndt and Ronald J. Evans, Problem E2758, American Math. Monthly, v. 86(1979), p. 128, and v. 87(1980), p. 404.

Bruce C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J. fur die reine und angewandte Math. 303/304 (1978), 332-365. MR 80b, #10035.

Leonard Carlitz, Some polynomials associated with Dedekind sums, Acta Math. Acad. Sci. Hungaricae, 26(1975), 311-319. MR 52, #5540. Received January 21, 1981. CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125

Modular Functions and

Problem E2758,

Some polynomials associated with Dedekind sums,

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory. Graduate texts in Mathematics, 41. Springer-Verlag , New York, 1976.

http://authors.library.caltech.edu/543/1/APOpjm82.pdf

Elementary proofs of Berndt's reciprocity laws

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