Constancy of an Infinite Cyclotomic Product via Ramanujan Sums

#A96 INTEGERS 25 (2025)
CONSTANCY OF AN INFINITE CYCLOTOMIC PRODUCT VIA
RAMANUJAN SUMS
Ha osh Singh Bal
The Ca a an, Jhandewalan Ex n., New Delhi, India
[email p o ec ed]
Recei ed: 4/23/25, Accep ed: 10/6/25, Published: 11/5/25
Abs ac
We show ha he in ini e p oduc de ined by
P(z) = −
∞
Y
n=1
(Φn(z))−1/n,
whe e Φn(z) is he n- h cyclo omic polynomial, is cons an inside he uni disk. The
p oo ansla es a esul o Ramanujan on Ramanujan sums, equi alen o he p ime
numbe heo em, o he se ing o in ini e p oduc s. We also show ha simila
iden i ies p o ed by Ramanujan lead o addi ional esul s on in ini e cyclo omic
p oduc s.
1. In oduc ion
The s udy o cyclo omic in ege s and hei associa ed p oduc s plays an impo an
ole in numbe heo y, wi h connec ions o modula o ms, anscendence ques ions,
and classical a i hme ic unc ions. In his pape we e isi one such p oduc , show-
ing ha i s loga i hmic de i a i e has a su p isingly simple desc ip ion in e ms o
Ramanujan sums. Ou main esul is s a ed in Theo em 1 below, which p o ides
he s a ing poin o he es o he pape .
Theo em 1. The unc ion
P(z) = −
∞
Y
n=1
(Φn(z))−1/n,
whe e Φn(z)deno es he n- h cyclo omic polynomial, is iden ically cons an , P(z)≡
1, inside he uni disk |z|<1.
DOI: 10.5281/zenodo.17535241
INTEGERS: 25 (2025) 2
Ou app oach uses a classical esul by Ramanujan on Ramanujan sums [4], which
is equi alen o he p ime numbe heo em. I ca ies o wa d he idea o using he
loga i hmic de i a i e o de i e in o ma ion on an associa ed in ini e p oduc , as in
[1].
In subsequen sec ions, we cons uc new in ini e p oduc iden i ies in ol ing
cyclo omic polynomials by in oking o he esul s om Ramanujan’s pape . To
ou knowledge, esul s o his ype o in ini e cyclo omic p oduc s ha e no ap-
pea ed p e iously. A ela ed example is [2], whe e he p oduc s a e o he o m
Qk≥0Φℓ(zpk) in ol ing a single cyclo omic ac o .
2. P oo o Theo em 1
Be o e we begin he p oo , we ecall ha he n- h cyclo omic polynomial is de ined
as
Φn(z) = Y
1≤k≤n
gcd(k,n)=1
(z−e2πik/n),
and Ramanujan sums, deno ed by cn(m), a e gi en by
cn(m) = X
1≤k≤n
gcd(k,n)=1
e2πikm/n.
These sums we e i s s udied in de ail by Ramanujan [4], who used hem o exp ess
a i hme ic unc ions in e ms o igonome ic sums.
Fo con enience, de ine
ˆ
Φn(z) := Y
1≤k≤n
(k,n)=11−z ζ k
n,
so ha ˆ
Φ1(z) = 1 −zand ˆ
Φn(z)=Φn(z) o all n≥2.
P oo o Theo em 1. We w i e he p imi i e n- h oo s o uni y e2πik/n as ζnk; hence
cn(m) = X
1≤k≤n
gcd(k,n)=1
ζm
nk.
We begin om he de ini ion
Φn(z) = Y
1≤k≤n
gcd(k,n)=1z−ζk
n,
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whe e ζn=e2πi/n. Fo each ac o we w i e
z−ζk
n=ζk
nz
ζk
n−1,
so ha
Φn(z) = Y
(k,n)=1
ζk
n!Y
(k,n)=1z
ζk
n−1.
Fo n > 2, he numbe o p imi i e esidues φ(n) is e en and he p imi i e oo s
occu in in e se pai s ξ, ξ−1, so hei p oduc is 1. Fo n= 2, he only p imi i e
oo is −1, gi ing a p oduc o −1, and o n= 1 he p oduc is 1. Pulling ou he
minus signs om each e m z
ζk
n−1 = −(1 −zζ−k
n) con ibu es a ac o (−1)φ(n),
which is +1 o n > 2 and cancels he p e ac o o n= 2. Thus, o all n≥2 we
ob ain
Φn(z) = Y
(k,n)=11−zζ−k
n.
Since he map k7→ k−1pe mu es he p imi i e esidue classes, he se {ζ−k
n}is he
same as {ζk
n}, and he e o e
Φn(z) = Y
(k,n)=11−zζ k
n, n ≥2.
Finally, o he wo base cases we eco d explici ly:
Φ2(z) = z+ 1 = 1 −(−1)z, Φ1(z) = z−1 = −(1 −z).
In he case n= 1 he p oduc o m gi es 1 −z, which di e s om Φ1(z) by an
o e all sign.
Conside he loga i hm o P(z), which gi es
−
∞
X
n=1
log(ˆ
Φn(z))
n=
∞
X
n=1 P1≤k≤n
gcd(k,n)=1
log(1 −zζnk)
n.
Expanding he inne sum o |z|<1 yields
−
∞
X
m=1
cn(m)
n
zm
m.
Summing o e ngi es
log P(z) =
∞
X
m=1
zm
m
∞
X
n=1
cn(m)
n.
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Fo |z|<1, he double se ies con e ges absolu ely: o each ixed m he inne sum
P∞
n=1
cn(m)
ncon e ges, and he ac o s zm/m decay exponen ially in m. Hence
log P(z) is well-de ined on he uni disk.
Ramanujan p o ed [4] ha
∞
X
n=1
cn(m)
n= 0 o all m≥1,(1)
a s a emen equi alen o he p ime numbe heo em [3]. The e o e, o |z|<1,
log P(z)=0,
and, since P(0) = 1, we conclude ha P(z)≡1 on he uni disk.
Con e sely, assume ha P(z)≡1 on |z|<1. Since he loga i hmic se ies
con e ges uni o mly on compac subse s o he uni disk, we may di e en ia e e m
by e m o ob ain
zP′(z)
P(z)=
∞
X
m=1 ∞
X
n=1
cn(m)
n!zm.
Compa ing coe icien s shows ha P∞
n=1
cn(m)
n= 0 o all m≥1.
Rema k 1 (Pa ial p oduc s and bounda y beha io ).Al hough he analy ic unc-
ion P(z) collapses o he cons an 1, he unca ions
PN(z) = −
N
Y
n=1
Φn(z)−1/n
o m a na u al app oxima ion scheme. On |z|<1 hey con e ge uni o mly o 1,
while on he uni ci cle hey acqui e genuine ze os and poles a oo s o uni y o o de
a mos N, emaining close o 1 elsewhe e. As N→ ∞, hese spikes become inc eas-
ingly dense, so he sequence {PN}exhibi s a “nea –0/nea –1” beha io a ound he
ci cle, esembling an app oxima e del a unc ion concen a ed on |z|= 1. Thus he
ep esen a ion, hough analy ically i ial in he limi , encodes non i ial bounda y
concen a ion phenomena ha may be o independen in e es .
Theo em 2. Fo any eal pa ame e s > 1,
Y
i≥11−zi−1/is
=Y
i≥1
ˆ
Φi(z)−ζ(s)/is.
P oo . A classical esul o Ramanujan on cyclo omic sums asse s ha
σ(n)
ns=ζ(s+ 1) X
i≥1
cn(i)
is+1 ,(s > 0),
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whe e σ(n) is he di iso -sum unc ion.
Mul iplying bo h sides by zn/n and summing o e n≥1, we ob ain wo exp es-
sions. The le -hand side p oduces
X
n≥1
σ(n)
ns
zn
n= log
Y
i≥1
(1 −zi)−1/is+1 !,
he loga i hm o an Eule - ype p oduc con e gen o |z|<1. On he o he hand,
he igh -hand side yields
ζ(s+ 1) X
n≥1X
i≥1
ci(n)
is+1
zn
n= log
Y
i≥1
ˆ
Φi(z)−ζ(s+1)/is+1 !.
Since he wo loga i hmic se ies ag ee, he co esponding p oduc s ag ee as well.
Replacing sby s−1 gi es he s a ed iden i y.
Rema k 2. Rea anging he a gumen abo e also yields, o |z|<1,
ζ(s+ 1) = Pi≥11
is+1 ln(1 −zi)
Pi≥21
is+1 ln(Φi(z)) + ln(1 −z),
an al e na i e exp ession ha makes explici he link be ween ζ(s+ 1) and cyclo-
omic ac o s.
3. In ini e Cyclo omic P oduc s and he The a Func ion
We conclude wi h a classical iden i y o Ramanujan conce ning he numbe o ep-
esen a ions o a posi i e in ege nas a sum o wo squa es. Le 2(n) deno e his
numbe o ep esen a ions. Ramanujan p o ed
π
∞
X
i=0
(−1)ic2i+1(n)
2i+ 1 = 2(n).
A classical di iso -class o mula gi es
2(n) = 4d1(n)−d3(n),
whe e d1(n) and d3(n) coun he di iso s o ncong uen o 1 and 3 modulo 4,
espec i ely.
Applying he cyclo omic p oduc me hod de eloped in he p e ious sec ion, we
a i e a he in ini e p oduc iden i y
∞
Y
i=0
(1 −z4i+3)4/(4i+3)
(1 −z4i+1)4/(4i+1) =
∞
Y
i=0
Φ4i+3(z)π/(4i+3)
ˆ
Φ4i+1(z)π/(4i+1) ,|z|<1.

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Taking he loga i hmic de i a i e o his p oduc gi es us back
π
∞
X
n=1
zn
∞
X
i=0
(−1)ic2i+1(n)
2i+ 1 =
∞
X
n=1
2(n)zn.
The igh -hand side equals θ(z)2, he squa e o he Jacobi he a unc ion, which is
a modula o m o weigh 1 on Γ0(4) [5]. Thus he loga i hmic de i a i e o his
cyclo omic p oduc ans o ms as a modula unc ion. Taking loga i hms o he
in ini e p oduc iden i y yields
X
i≥0"4
4i+ 3 log(1 −z4i+3)−4
4i+ 1 log(1 −z4i+1)#
=X
i≥0"π
4i+ 3 log Φ4i+3(z)−π
4i+ 1 log ˆ
Φ4i+1(z)#.
Finally, ea anging e ms leads o he s iking loga i hmic iden i y
π
4=X
i≥0
(−1)i+1 log(1 −z2i+1)
2i+ 1
X
i≥0
(−1)i+1 log ˆ
Φ2i+1(z)
2i+ 1
.
Re e ences
[1] H. S. Bal and G. Bha naga , Glaishe ’s di iso s and in ini e p oduc s, J. In ege Seq. 27
(2024), 1, Pape 24.1.6.
[2] W. Duke and H. N. Nguyen, In ini e p oduc s o cyclo omic polynomials, Bull. Aus . Ma h.
Soc. 91 (3) (2015), 400–411.
[3] G. H. Ha dy, Ramanujan: Twel e Lec u es on Subjec s Sugges ed by His Li e and Wo k, AMS
Chelsea, P o idence, RI, 1999.
[4] S. Ramanujan, On ce ain igonome ical sums and hei applica ions in he heo y o num-
be s, T ans. Camb idge Philos. Soc. 22 (1918), 259–276.
[5] D. Zagie , Ellip ic modula o ms and hei applica ions, in The 1-2-3 o Modula Fo ms (K.
Ranes ad, ed.), Uni e si ex , Sp inge , Be lin, 2008, 1–103.