Constancy of an Infinite Cyclotomic Product via Ramanujan Sums

#A96 INTEGERS 25 (2025) CONSTANCY OF AN INFINITE CYCLOTOMIC PRODUCT VIA RAMANUJAN SUMS Hartosh Singh Bal The Caravan, Jhandewalan Extn., New Delhi, India [email protected] Received: 4/23/25, Accepted: 10/6/25, Published: 11/5/25 Abstract We show that the infinite product defined by P(z) = − ∞ Y n=1 (Φn(z))−1/n, where Φn(z) is the n-th cyclotomic polynomial, is constant inside the unit disk. The proof translates a result of Ramanujan on Ramanujan sums, equivalent to the prime number theorem, to the setting of infinite products. We also show that similar identities proved by Ramanujan lead to additional results on infinite cyclotomic products. 1. Introduction The study of cyclotomic integers and their associated products plays an important role in number theory, with connections to modular forms, transcendence questions, and classical arithmetic functions. In this paper we revisit one such product, showing that its logarithmic derivative has a surprisingly simple description in terms of Ramanujan sums. Our main result is stated in Theorem 1 below, which provides the starting point for the rest of the paper. Theorem 1. The function P(z) = − ∞ Y n=1 (Φn(z))−1/n, where Φn(z)denotes the n-th cyclotomic polynomial, is identically constant, P(z)≡ 1, inside the unit disk |z|<1. DOI: 10.5281/zenodo.17535241 INTEGERS: 25 (2025) 2 Our approach uses a classical result by Ramanujan on Ramanujan sums [4], which is equivalent to the prime number theorem. It carries forward the idea of using the logarithmic derivative to derive information on an associated infinite product, as in [1]. In subsequent sections, we construct new infinite product identities involving cyclotomic polynomials by invoking other results from Ramanujan’s paper. To our knowledge, results of this type for infinite cyclotomic products have not appeared previously. A related example is [2], where the products are of the form Qk≥0Φℓ(zpk) involving a single cyclotomic factor. 2. Proof of Theorem 1 Before we begin the proof, we recall that the n-th cyclotomic polynomial is defined as Φn(z) = Y 1≤k≤n gcd(k,n)=1 (z−e2πik/n), and Ramanujan sums, denoted by cn(m), are given by cn(m) = X 1≤k≤n gcd(k,n)=1 e2πikm/n. These sums were first studied in detail by Ramanujan [4], who used them to express arithmetic functions in terms of trigonometric sums. For convenience, define ˆ Φn(z) := Y 1≤k≤n (k,n)=11−z ζ k n, so that ˆ Φ1(z) = 1 −zand ˆ Φn(z)=Φn(z) for all n≥2. Proof of Theorem 1. We write the primitive n-th roots of unity e2πik/n as ζnk; hence cn(m) = X 1≤k≤n gcd(k,n)=1 ζm nk. We begin from the definition Φn(z) = Y 1≤k≤n gcd(k,n)=1z−ζk n, INTEGERS: 25 (2025) 3 where ζn=e2πi/n. For each factor we write z−ζk n=ζk nz ζk n−1, so that Φn(z) = Y (k,n)=1 ζk n!Y (k,n)=1z ζk n−1. For n > 2, the number of primitive residues φ(n) is even and the primitive roots occur in inverse pairs ξ, ξ−1, so their product is 1. For n= 2, the only primitive root is −1, giving a product of −1, and for n= 1 the product is 1. Pulling out the minus signs from each term z ζk n−1 = −(1 −zζ−k n) contributes a factor (−1)φ(n), which is +1 for n > 2 and cancels the prefactor for n= 2. Thus, for all n≥2 we obtain Φn(z) = Y (k,n)=11−zζ−k n. Since the map k7→ k−1permutes the primitive residue classes, the set {ζ−k n}is the same as {ζk n}, and therefore Φn(z) = Y (k,n)=11−zζ k n, n ≥2. Finally, for the two base cases we record explicitly: Φ2(z) = z+ 1 = 1 −(−1)z, Φ1(z) = z−1 = −(1 −z). In the case n= 1 the product form gives 1 −z, which differs from Φ1(z) by an overall sign. Consider the logarithm of P(z), which gives − ∞ X n=1 log(ˆ Φn(z)) n= ∞ X n=1 P1≤k≤n gcd(k,n)=1 log(1 −zζnk) n. Expanding the inner sum for |z|<1 yields − ∞ X m=1 cn(m) n zm m. Summing over ngives log P(z) = ∞ X m=1 zm m ∞ X n=1 cn(m) n. INTEGERS: 25 (2025) 4 For |z|<1, the double series converges absolutely: for each fixed mthe inner sum P∞ n=1 cn(m) nconverges, and the factors zm/m decay exponentially in m. Hence log P(z) is well-defined on the unit disk. Ramanujan proved [4] that ∞ X n=1 cn(m) n= 0 for all m≥1,(1) a statement equivalent to the prime number theorem [3]. Therefore, for |z|<1, log P(z)=0, and, since P(0) = 1, we conclude that P(z)≡1 on the unit disk. Conversely, assume that P(z)≡1 on |z|<1. Since the logarithmic series converges uniformly on compact subsets of the unit disk, we may differentiate term by term to obtain zP′(z) P(z)= ∞ X m=1 ∞ X n=1 cn(m) n!zm. Comparing coefficients shows that P∞ n=1 cn(m) n= 0 for all m≥1. Remark 1 (Partial products and boundary behavior).Although the analytic function P(z) collapses to the constant 1, the truncations PN(z) = − N Y n=1 Φn(z)−1/n form a natural approximation scheme. On |z|<1 they converge uniformly to 1, while on the unit circle they acquire genuine zeros and poles at roots of unity of order at most N, remaining close to 1 elsewhere. As N→ ∞, these spikes become increasingly dense, so the sequence {PN}exhibits a “near–0/near–1” behavior around the circle, resembling an approximate delta function concentrated on |z|= 1. Thus the representation, though analytically trivial in the limit, encodes nontrivial boundary concentration phenomena that may be of independent interest. Theorem 2. For any real parameter s > 1, Y i≥11−zi−1/is =Y i≥1 ˆ Φi(z)−ζ(s)/is. Proof. A classical result of Ramanujan on cyclotomic sums asserts that σ(n) ns=ζ(s+ 1) X i≥1 cn(i) is+1 ,(s > 0), INTEGERS: 25 (2025) 5 where σ(n) is the divisor-sum function. Multiplying both sides by zn/n and summing over n≥1, we obtain two expressions. The left-hand side produces X n≥1 σ(n) ns zn n= log Y i≥1 (1 −zi)−1/is+1 !, the logarithm of an Euler-type product convergent for |z|<1. On the other hand, the right-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 the two logarithmic series agree, the corresponding products agree as well. Replacing sby s−1 gives the stated identity. Remark 2. Rearranging the argument above also yields, for |z|<1, ζ(s+ 1) = Pi≥11 is+1 ln(1 −zi) Pi≥21 is+1 ln(Φi(z)) + ln(1 −z), an alternative expression that makes explicit the link between ζ(s+ 1) and cyclotomic factors. 3. Infinite Cyclotomic Products and the Theta Function We conclude with a classical identity of Ramanujan concerning the number of representations of a positive integer nas a sum of two squares. Let r2(n) denote this number of representations. Ramanujan proved π ∞ X i=0 (−1)ic2i+1(n) 2i+ 1 =r2(n). A classical divisor-class formula gives r2(n) = 4d1(n)−d3(n), where d1(n) and d3(n) count the divisors of ncongruent to 1 and 3 modulo 4, respectively. Applying the cyclotomic product method developed in the previous section, we arrive at the infinite product identity ∞ 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. INTEGERS: 25 (2025) 6 Taking the logarithmic derivative of this product gives us back π ∞ X n=1 zn ∞ X i=0 (−1)ic2i+1(n) 2i+ 1 = ∞ X n=1 r2(n)zn. The right-hand side equals θ(z)2, the square of the Jacobi theta function, which is a modular form of weight 1 on Γ0(4) [5]. Thus the logarithmic derivative of this cyclotomic product transforms as a modular function. Taking logarithms of the infinite product identity 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, rearranging terms leads to the striking logarithmic identity 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 . References [1] H. S. Bal and G. Bhatnagar, Glaisher’s divisors and infinite products, J. Integer Seq. 27 (2024), 1, Paper 24.1.6. [2] W. Duke and H. N. Nguyen, Infinite products of cyclotomic polynomials, Bull. Aust. Math. Soc. 91 (3) (2015), 400–411. [3] G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, AMS Chelsea, Providence, RI, 1999. [4] S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Trans. Cambridge Philos. Soc. 22 (1918), 259–276. [5] D. Zagier, Elliptic modular forms and their applications, in The 1-2-3 of Modular Forms (K. Ranestad, ed.), Universitext, Springer, Berlin, 2008, 1–103.