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**The family of ternary cyclotomic polynomials with one free prime.**
*(English)*
Zbl 1292.11052

Ternary cyclotomic polynomials are cyclotomic polynomials \(\Phi_{pqr}\), where \(p\), \(q\) and \(r\) are distinct odd primes, \(3\leq p<q<r\) say. Following the standard convention, we write \(A(pqr)\) for the greatest absolute value of the coefficients of \(\Phi_{pqr}\), its height. This function is probably the most studied question about the coefficients of these polynomials, but a number of interesting and challenging problems remain open. One also writes \(M(p)=\max_{q,r}A(pqr)\) and \(M(p,q)=\max_{r}A(pqr)\). The paper under review considers certain functional properties of \(M(p,q)\). It helps to recall the following result due to N. Kaplan [J. Number Theory 127, No. 1, 118–126 (2007; Zbl 1171.11015)]. For fixed \(p\) and \(q\), the value of \(A(pqr)\) depends only on the residue class of \(r\) to the modulus \(pq\). It is only natural to consider whether, for a fixed \(p\), \(M(p,q)\) is determined by \(q\bmod p\). This is false, as is shown by the example \(M(7,13)=3\) and \(M(7,41)=M(7)=4\). Nonetheless, the authors conjecture that the appropriate analogue of Kaplan’s result for the function \(M(p,q)\) does exist.

Let us present their conjecture in the form of two questions: Is it true that for (i) \(q>p^3\) the value of \(M(p,q)\) depends only on (ii) the residue class of \(q\) to the modulus \(p^3\)? What are the correct quantities that should be used in place of \(p^3\) in (i) and in (ii)?

The conjecture is supported by the following result. For a fixed \(p\) and an arbitrary integer \(a\), there are integers \(q_a\) and \(d_a\) such that the value of \(M(p,q)\) remains unchanged for all \(q\geq q_a\) with \(q\equiv a\pmod{d_ap}\). Observing an application of this result to the equation \(M(p,q)=M(p)\), the authors conclude that the set of primes \(q\) satisfying this equation always contains an arithmetic progression and thus has a positive lower asymptotic density in the set of primes. For certain small primes \(p\) they were able to obtain a more complete analysis of this problem, lending further support for their conjecture. We give their results for \(3, 5\) and \(7\), which are complete and particularly simple, if somewhat confusing. For \(p=3\) or \(5\), we have \(M(p,q)\equiv M(p)\). That is, in these cases \(M(p,q)\) is determined by \(q\bmod p\), but happens to be constant. Similarly, \(M(7,q)=M(7)\) for all \(q\neq13\), and we again observe periodicity modulo \(p\) for all \(q\geq17\).

In addition, the paper contains a number of other results on \(M(p,q)\) – too numerous to mention here – and a collection of open questions.

Let us present their conjecture in the form of two questions: Is it true that for (i) \(q>p^3\) the value of \(M(p,q)\) depends only on (ii) the residue class of \(q\) to the modulus \(p^3\)? What are the correct quantities that should be used in place of \(p^3\) in (i) and in (ii)?

The conjecture is supported by the following result. For a fixed \(p\) and an arbitrary integer \(a\), there are integers \(q_a\) and \(d_a\) such that the value of \(M(p,q)\) remains unchanged for all \(q\geq q_a\) with \(q\equiv a\pmod{d_ap}\). Observing an application of this result to the equation \(M(p,q)=M(p)\), the authors conclude that the set of primes \(q\) satisfying this equation always contains an arithmetic progression and thus has a positive lower asymptotic density in the set of primes. For certain small primes \(p\) they were able to obtain a more complete analysis of this problem, lending further support for their conjecture. We give their results for \(3, 5\) and \(7\), which are complete and particularly simple, if somewhat confusing. For \(p=3\) or \(5\), we have \(M(p,q)\equiv M(p)\). That is, in these cases \(M(p,q)\) is determined by \(q\bmod p\), but happens to be constant. Similarly, \(M(7,q)=M(7)\) for all \(q\neq13\), and we again observe periodicity modulo \(p\) for all \(q\geq17\).

In addition, the paper contains a number of other results on \(M(p,q)\) – too numerous to mention here – and a collection of open questions.

Reviewer: Gennady Bachman (Las Vegas)