Some remarks on the coefficients of cyclotomic polynomials (Q6610525)

It is well acknowledged that if \(n\in \mathbb N\) and \(\zeta_n=\cos\frac{2\pi}{n}+i\sin\frac{2\pi}{n}\) denotes the first root of order \(n\) of the unity, then the \(n^{th}\) cyclotomic polynomial is given as \N\[\N\Phi_n(z)=\displaystyle\prod_{1\leq k\leq n-1, \gcd(k, n)=1}(z-\zeta_n^k)=\sum_{j=0}^{\phi(n)}c_j^{(n)}z^j,\N\]\Nwhere \(\phi\) is the Euler totient function and also the degree of the polynomial. In this article, vital results concerning the coefficients of cyclotomic polynomials are reviewed. Also an important lemma on formal binomial coefficients is established that can be used to perform fast computations of coefficients of cyclotomic polynomials. In addition, new formulas for coefficients of cyclotomic polynomials are presented in Theorems 7, 8 in the article. Connections between Ramanujan sums and Möbius functions are readdressed. Formal power series have been industriously engaged in proving a beneficial identity involving symmetric and power sums. It is worth mentioning here that every integer can be found among the coefficients of some cyclotomic polynomial, more precisely we have Suzuki's Theorem [\textit{J. Suzuki}, Proc. Japan Acad., Ser. A 63, 279--280 (1987; Zbl 0641.10008)] which is mentioned below.\N\NTheorem A. For every integer \(s,\) there exists \(n, i\in \mathbb N\) such that \(c_i^{(n)}=s.\) \N\NThe aforementioned result based on [\textit{J. Suzuki}, Proc. Japan Acad., Ser. A 63, 279--280 (1987; Zbl 0641.10008)] via a consequence of the Prime Number Theorem is established. A sound contemplation of facts associated to the Prime Number Theorem is imperative to obtain some crucial and stronger results concerning the coefficients of cyclotomic polynomials. In this context, Andrica's conjecture needs to be discussed as it claims that for \(n\geq 1, \sqrt{p_{n+1}}-\sqrt{p_n}<1,\) where \(p_n, p_{n+1}\) denote the \(n^{th}, (n+1)^{th}\) primes respectively. In fact the conjecture implies that every natural number occurs as the largest coefficient of some cyclotomic polynomial (see Theorem 2). Stronger versions of this conjecture were studied by \textit{M. Visser} [Int. Math. Forum 14, No. 4, 181--188 (2019; \url{doi:0.12988/imf.2019.9729})] along with some numerical simulations and affinity to other conjectures.\N\NFor the entire collection see [Zbl 1539.11005].