A Survey on Coefficients of Cyclotomic Polynomials

Let $n$ be a positive integer. The $n$th cyclotomic polynomial $\Phi_{n}(X)$ is defined as the monic polynomial whose roots are the $n$th primitive roots of unity, that is,

The word “cyclotomic” comes from the ancient Greek words “cyclo” (circle) and “tomos” (cutting), and refers to how the $n$th roots of unit divide the circle into equal parts. Note that, incidentally, the Greek letter $\Phi$ looks a bit like a cut circle. The degree of $\Phi_{n}(X)$ is equal to $\varphi(n)$, where $\varphi$ is the Euler totient function. Despite its definition in terms of complex numbers, it can be proved that $\Phi_{n}(X)$ has integer coefficients. Furthermore, $\Phi_{n}(X)$ is irreducible over $\mathbb{Q}$ and, consequently, it is the minimal polynomial of any primitive $n$th root of unity. The irreducibility of $\Phi_{n}(X)$ for $n$ prime was first proved by Gauss (1801) [64], and the irreducibility of $\Phi_{n}(X)$ in general was first proved by Kronecker (1854) [85]. Weintraub (2013) [127] presented proofs of the irreducibility of $\Phi_{n}(X)$ due to Gauss, Kronecker, Schönemann, and Eisenstein, for $n$ prime, and Dedekind, Landau, and Schur,¹¹1Perhaps curiously, Schur’s proof of the irreducibility of $\Phi_{n}(X)$ was set to rhymes [42, pp. 38–41]. for every $n$.

From (1) it follows easily that

which in turn, by the Möbius inversion formula, yields that

where $\mu$ is the Möbius function. In particular, we have that

for every prime number $p$.

The next lemma collects some important elementary identities, which can be proved either using (3) or checking that both sides have the same zeros [91, 122].

Starting from $\Phi_{1}(X)=X-1$ and using Lemma 1.1’s (i) and (ii), one can inductively compute the cyclotomic polynomials. The first ten cyclotomic polynomials are:

A natural observation is that the coefficients of the cyclotomic polynomials are very small, and one could be even tempted to conjecture that they are always in $\{-1,0,+1\}$. The first counterexample to this conjecture occurs for $n=105$, since we have

It is no coincidence that $105=3\cdot 5\cdot 7$ is the smallest odd positive integer having three different prime factors. Indeed, every cyclotomic polynomial $\Phi_{n}(X)$ such that $n$ has less than three odd prime factors has all its coefficients in $\{-1,0,+1\}$ (see Section 2).

For every positive integer $n$, let us write

so that $a_{n}(j)$ is the coefficient of $X^{j}$ in $\Phi_{n}(X)$. (Note that $a_{n}(j)=0$ for $j\notin[0,\varphi(n)]$.) The peculiarity of the smallness of the coefficients of the cyclotomic polynomials was very well explained by D. H. Lehmer (1966) [89], who wrote: “The smallness of $|a_{n}(j)|$ would appear to be one of the fundamental conspiracies of the primitive $n$th roots of unity. When one considers that $a_{n}(j)$ is a sum of $\binom{\varphi(n)}{j}$ unit vectors (for example $73629072$ in the case of $n=105$, $j=7$) one realizes the extent of the cancellation that takes place.”

In light of Lemma 1.1’s (iii) and (iv), for the purpose of studying the coefficients of $\Phi_{n}(X)$ it suffices to consider only odd squarefree integers $n$. A squarefree positive integer $n$, or a cyclotomic polynomial $\Phi_{n}(X)$, is binary, ternary, …if the number of prime factors of $n$ is $2$, $3$, …, respectively. The order of $\Phi_{n}(X)$ is the number of prime factors of $n$. From Lemma 1.1’s (v) we have that for every integer $n>1$ the cyclotomic polynomial $\Phi_{n}(X)$ is palindromic, that is,

We conclude this section by defining the main quantities that have been considered in the study of the coefficients of the cyclotomic polynomials. First, we have $A(n)$, $A^{+}(n)$, and $A^{-}(n)$, which are defined as follows

In general, the height of a polynomial $P\in\mathbb{C}[X]$ is defined as the maximum of the absolute values of the coefficients of $P$, and $P$ is flat if its height is not exceeding $1$. Thus, $A(n)$ is the height of $\Phi_{n}(X)$. We also let $\mathcal{A}(n):=\{a_{n}(j):0\leq j\leq\varphi(n)\}$ be the set of coefficients of $\Phi_{n}(X)$. Moreover, we let $\theta(n):=\#\{j\geq 0:a_{n}(j)\neq 0\}$ be the number of nonzero coefficients of $\Phi_{n}(X)$. The maximum gap of a nonzero polynomial $P(X)=\sum_{i=1}^{k}c_{k}X^{e_{k}}\in\mathbb{C}[X]$, where $c_{1},\dots,c_{k}\in\mathbb{C}^{*}$ and $e_{1}<\cdots<e_{k}$, is defined as $G(P):=\max\{e_{j+1}-e_{j}:j<k\}$. We let $G(n):=G(\Phi_{n})$ denote the maximum gap of $\Phi_{n}(X)$. Note that by (4) we have that $A(p)=A^{+}(p)=A^{-}(p)=1$, $\mathcal{A}(p)=\{1\}$, $\theta(p)=p$, and $G(p)=1$, for every prime number $p$. Thus, the first interesting case in the study of these quantities is the one of binary cyclotomic polynomials.

Upper bounds for the height of ternary cyclotomic polynomials have been studied by many authors. Bang (1895) [17] proved that $A(pqr)\leq p-1$. Beiter (1968) [21] made the following conjecture, which is known as Beiter’s conjecture.

Beiter (1968) [21] proved her conjecture in the case in which $q\equiv\pm 1\pmod{p}$ or $r\equiv\pm 1\pmod{p}$. As a consequence, Beiter’s conjecture is true for $p=3$. Also, Bloom (1968) [25] showed that Beiter’s conjecture is true for $p=5$. Beiter (1971) [22] improved Bang’s bound to $A(pqr)\leq p-\lfloor(p+1)/4\rfloor$. Möller (1971) [103] proved that for every odd prime number $p$ there exists a ternary cyclotomic polynomial $\Phi_{pqr}(X)$, with $p<q<r$, having a coefficient equal to $(p+1)/2$. This shows that Beiter’s conjecture, if true, is the best possible. Bachman (2003) [10] proved an upper bound for $A^{+}(pqr)$ and a lower bound for $A^{-}(pqr)$ in terms of $p$ and the inverses of $q$ and $r$ modulo $p$. As corollaries, he deduced that: Beiter’s conjecture is true if $q$ or $r$ is equal to $\pm 1,\pm 2$ modulo $p$; we have $A(pqr)\leq p-\lceil p/4\rceil$; and $A^{+}(pqr)-A^{-}(pqr)\leq p$, in particular either $A^{+}(pqr)\leq(p-1)/2$ or $A^{-}(pqr)\geq-(p-1)/2$. Note that the first two corollaries improve the previous results of Beiter [21, 22]. Regarding the third, Bachman (2004) [11] also proved that for every odd prime number $p$ there exist infinitely many ternary cyclotomic polynomials $\Phi_{pqr}(X)$, with $p<q<r$, such that $\mathcal{A}(pqr)=[-(p-1)/2,(p+1)/2]\cap\mathbb{Z}$, and similarly for the interval $[-(p+1)/2,(p-1)/2]$. Leher (2007) [88, p. 70] found a counterexample to Beiter’s conjecture, that is, $A(17\cdot 29\cdot 41)=10>(17+1)/2$. Let $M(p):=\max_{p<q<r}A(p)$. For every odd prime $p$, Gallot and Moree (2009) [61] defined an effectively computable set of natural numbers $\mathcal{B}(p)$ such that if $\mathcal{B}(p)$ is nonempty then

and so Beiter’s conjecture is false for $p$. Then, for $p\geq 11$ they showed that $\mathcal{B}(p)$ is nonempty and $\max(\mathcal{B}(p))=(p-3)/2$. Moreover, for every $\varepsilon>0$, they proved that

for all sufficiently large $p$. In light of these results, they formulated the following:

Zhao and Zhang (2010) [138] gave a sufficient condition for the Corrected Beiter conjecture and proved it when $p=7$. (Note that for $p=7$ the Corrected Beiter Conjecture is equivalent to the original Beiter Conjecture.) Moree and Roşu (2012) [108] showed that for each odd integer $\ell\geq 1$ there exist infinitely many odd primes $p<q<r$ such that

This provides a family of cyclotomic polynomials that contradict the Beiter conjecture and have the largest coefficient range possible. Bzdȩga (2010) [27] improved Bachman’s bounds [10] by giving the following theorem.

As an application of Theorem 3.2, Bzdȩga proved a density result showing that the Corrected Beiter conjecture holds for at least $25/27+O(1/p)$ of all the ternary cyclotomic polynomials with the smallest prime factor dividing their order equal to $p$. He also proved that for these polynomials the average value of $A(pqr)$ does not exceed $(p+1)/2$. Moreover, for every prime $p\geq 13$, he provided some new classes of ternary cyclotomic polynomials $\Phi_{pqr}(X)$ for which the set of coefficients is very small. Luca, Moree, Osburn, Saad Eddin, and Sedunova (2019) [93], using Theorem 3.2 and some analytic estimates for constrained ternary integers that they developed, showed that the relative density of ternary integers for which the correct Sister Beiter conjecture holds true is at least $25/27$.

Gallot, Moree, and Wilms (2011) [63] initiated the study of

They remarked that $M(p,q)$ can be effectively computed for any given odd primes $p<q$. For $p=3,5,7,11,13,19$, they proved that the set $\mathcal{Q}_{p}$ of primes $q$ with $M(p,q)=M(p)$ has a subset of positive density, which they determined, and they also conjectured the value of the natural density of $\mathcal{Q}_{p}$. Moreover, they computed or bound $M(p,q)$ for $p$ and $q$ satisfying certain conditions, and they posed several problems regarding $M(p,q)$ [63, Section 11]. Cobeli, Gallot, Moree, and Zaharescu (2013) [41], using techniques from the study of the distribution of modular inverses, in particular bounds on Kloosterman sums, improved the lower bound in (6) to

for every prime $p$, and

for infinitely many primes $p$, where $C>0$ is a constant. Moreover, they proved that

Duda (2014) [47] put $M_{q^{\prime}}(p):=\max\{M(p,q):q\equiv q^{\prime}\pmod{p}\}$ and proved one of the main conjectures on $M(p,q)$ of Gallot, Moree, and Wilms [63, Conjecture 8], that is, for all distinct primes $p$ and $q^{\prime}$ there exists $q_{0}\equiv q^{\prime}\pmod{p}$ such that for every prime $q\geq q_{0}$ with $q\equiv q^{\prime}\pmod{p}$ we have $M(p,q)=M_{q^{\prime}}(p)$. Also, he gave an effective method to compute $M_{q}(p)$, from which it follows an algorithm to effectively compute $M(p)$, since $M(p)=\max\{M_{q}(p):q<p\}$.

Kosyak, Moree, Sofos, and Zhang (2021) [84] conjectured that every positive integer is of the form $A(n)$, for some ternary integer $n$. They proved this conjecture under a stronger form of Andrica’s conjecture on prime gaps, that is, assuming that $p_{n+1}-p_{n}<\sqrt{p_{n}}+1$ holds for every $n\geq 31$, where $p_{n}$ denotes the $n$th prime number. Furthermore, they showed that almost all positive integers are of the form $A(n)$ where $n=pqr$ with $p<q<r$ primes is a ternary integer and $\#\mathcal{A}(n)=p+1$ (which is the maximum possible value for this cardinality). A nice survey regarding these connections between cyclotomic polynomials and prime gaps was given by Moree (2021) [107].

Recall that a cyclotomic polynomial $\Phi_{n}(X)$ is flat if $A(n)=1$. Several families of flat ternary cyclotomic polynomials have been constructed, but a complete classification is still not known. Beiter (1978) [23] characterized the primes $r>q>3$ such that $\Phi_{3qr}(X)$ is flat. In particular, there are infinitely many such primes. Bachman (2006) [12] proved that if $p\geq 5$, $q\equiv-1\pmod{p}$, and $r\equiv 1\pmod{pq}$ then $\Phi_{pqr}(X)$ is flat. Note that, for every prime $p\geq 5$, the existence of infinitely many primes $q$ and $r$ satisfying the aforementioned congruences is guaranteed by Dirichlet’s theorem on primes in arithmetic progressions. Flanagan (2007) [58] improved Bachman’s result by relaxing the congruences to $q\equiv\pm 1\pmod{p}$ and $r\equiv\pm 1\pmod{pq}$. Kaplan (2007) [77] used Lemma 3.1 to show that last congruence suffices, that is, the following holds:

Luca, Moree, Osburn, Saad Eddin, and Sedunova (2019) [93] proved some asymptotic formulas for ternary integers that, together with Theorem 3.3, yield that for every sufficiently large $N>1$ there are at least $CN/\log N$ ternary integers $n\leq N$ such that $\Phi_{n}(X)$ is flat, where $C:=1.195\dots$ is an explicit constant.

Ji (2010) [72] considered odd primes $p<q<r$ such that $2r\equiv\pm 1\pmod{pq}$ and showed that in such a case $\Phi_{pqr}(X)$ is flat if and only if $p=3$ and $q\equiv 1\pmod{3}$. For $a\in\{3,4,5\}$, Zhang (2017) [132] gave similar characterizations for the odd primes such that $ar\equiv\pm 1\pmod{pq}$ and $\Phi_{pqr}(X)$ is flat (see also [137] for a weaker result for the case $a=4$). For $a\in\{6,7\}$, Zhang (2020, 2021) [135, 136] characterized the odd primes such that $q\equiv\pm 1\pmod{p}$, $ar\equiv\pm 1\pmod{pq}$, and $\Phi_{pqr}(X)$ is flat. Zhang (2017) [133] also showed that if $p\equiv 1\pmod{w}$, $q\equiv 1\pmod{pw}$, and $r\equiv w\pmod{pq}$, for some integer $w\geq 2$, then $A^{+}(pqr)=1$. (See also the unpublished work of Elder (2012) [50].) Furthermore, for $q\not\equiv 1\pmod{p}$ and $r\equiv-2\pmod{pq}$, Zhang (2014) [130] constructed an explicit $j$ such that $a_{pqr}(j)=-2$, so that $\Phi_{pqr}(X)$ is not flat. Regarding nonflat ternary cyclotomic polynomials with small heights, Zhang (2017) [131] showed that for every prime $p\equiv 1\pmod{3}$ there exist infinitely many $q$ and $r$ such that $A(pqr)=3$.

Gallot and Moree (2009) [60] proved that neighboring coefficients of ternary cyclotomic polynomials differ by at most one. They called this property jump one property.

Gallot and Moree used the jump one property to give a different proof of Bachman’s result [11] on ternary polynomials with optimally large set of coefficients. Their proof of the jump one property makes use of Kaplan’s lemma. Previously, Leher (2007) [88, Theorem 57] proved the bound $|a_{n}(j)-a_{n}(j-1)|\leq 4$ using methods from the theory of numerical semigroups. A different proof of the jump one property was given by Bzdȩga (2010) [27]. Furthermore, for every ternary integer $n$, Bzdȩga (2014) [31] gave a characterization of the positive integers $j$ such that $|a_{n}(j)-a_{n}(j-1)|=1$. A coefficient $a_{n}(j)$ is jumping up, respectively jumping down, if $a_{n}(j)=a_{n}(j-1)+1$, respectively $a_{n}(j)=a_{n}(j-1)-1$. Since cyclotomic polynomials are palindromic, the number of jumping up coefficients is equal to the number of jumping down coefficients. Let $J_{n}$ denote such number. Bzdȩga [31] proved that $J_{n}>n^{1/3}$ for all ternary integers $n$. As a corollary, $\theta_{n}>n^{1/3}$. Also, he showed that Schinzel Hypothesis H implies that for every $\varepsilon>0$ we have $J_{n}<10n^{1/3+\varepsilon}$ for infinitely many ternary integers $n$. Camburu, Ciolan, Luca, Moree, and Shparlinski (2016) [37] gave an unconditional proof that $J_{n}<n^{7/8+\varepsilon}$ for infinitely many ternary integers $n$.