Cyclotomic polynomials with 7$^{th}$ coefficient greater than 1 in absolute value

Question

It is known that the seventh coefficient of $\Phi_{105}(x)$ is $-2$ and that's the first occurrence of a coefficient with absolute value greater than $1$ for a cyclotomic polynomial. When I did a quick check for the seventh coefficient of $\Phi_n(x)$ where $n=105k$ with $\gcd(105,k)=1$ and $\mu(k)\neq 0$ they all came out to be $2$ in absolute value whenever they are nonzero. Is it true in general or there is a counterexample to this?

Answer 1

$k=11$ is the smallest counterexample - the 7'th coefficient is 0. Here are the details:

We have the following identity: $$\Phi_n(x) = \prod_{d \mid n} (1-x^d)^{\mu(n/d)},$$ valid for $n>1$.

If we are interested only in the first $m+1$ coefficients ($x^0$ to $x^{m}$), it suffices to look at the following product, going only over divisors $\le m$: $$\Phi_n(x) = \prod_{d \mid n, d \le m} (1-x^d)^{\mu(n/d)} \mod {x^{m+1}}.$$ Hence, $$[x^7] \Phi_{105 k}(x) = [x^7]\prod_{d \mid 105k, d \le 7} (1-x^d)^{\mu(105k/d)}.$$ Since you assume $\gcd(k,105)=1$ and $\mu(k)\neq 0$, we actually have 4 cases, according to the parity of $k$ and according to $\mu(k)$.

When $2 \nmid k$, the set $\{d : d\mid 105k, d \le 7\}$ is $\{1,3,5,7\}$ and we find $$[x^7] \Phi_{105 k}(x) = [x^7] (1-x)^{\mu(105k)}(1-x^3)^{\mu(35k)}(1-x^5)^{\mu(21k)}(1-x^7)^{\mu(15k)}$$ $$ = [x^7] ((1-x)^{-1}(1-x^3) (1-x^5)(1-x^7))^{\mu(k)}.$$ When $\mu(k)=1$, we get $-2$. When $\mu(k)=-1$, we get $0$.

When $2 \mid k$, the set $\{d : d\mid 105k, d \le 7\}$ is $\{1,2,3,5,6,7\}$ and we find $$[x^7] \Phi_{105 k}(x) = [x^7] (1-x)^{\mu(105k)}(1-x^2)^{\mu(105k/2)}(1-x^3)^{\mu(35k)}(1-x^5)^{\mu(21k)}(1-x^6)^{\mu(35k/2)}(1-x^7)^{\mu(15k)}$$ $$ = [x^7] ((1-x)(1-x^2)^{-1}(1-x^3)^{-1}(1-x^5)^{-1}(1-x^6)(1-x^7)^{-1})^{\mu(k/2)}.$$ Again, only two cases to check. When $\mu(k/2)=1$ we get 2, and when $\mu(k/2)=-1$ we get 0.

Answer 2

Actually for all $11 \leq k \leq 100$ which satisfy the conditions in the question (i.e. $\gcd(k,105) = 1$ and $\mu(k) \neq 0$), the coefficient of $x^7$ in $\Phi_{105k}(x)$ equals $0$. Hence so far there are only counterexamples!

This can be checked with GAP as follows:

gap> ks  := Filtered([1..100],k -> Gcd(k,105) = 1
>                              and Factors(k) = Set(Factors(k)));
[ 1, 2, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 
  53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97 ]
gap> c7s := List(ks,k->[k,CoefficientsOfUnivariatePolynomial(
>                           CyclotomicPolynomial(Rationals,105*k))[8]]);
[ [ 1, -2 ], [ 2, 2 ], [ 11, 0 ], [ 13, 0 ], [ 17, 0 ], [ 19, 0 ], 
  [ 22, 0 ], [ 23, 0 ], [ 26, 0 ], [ 29, 0 ], [ 31, 0 ], [ 34, 0 ],
  [ 37, 0 ], [ 38, 0 ], [ 41, 0 ], [ 43, 0 ], [ 46, 0 ], [ 47, 0 ],
  [ 53, 0 ], [ 58, 0 ], [ 59, 0 ], [ 61, 0 ], [ 62, 0 ], [ 67, 0 ],
  [ 71, 0 ], [ 73, 0 ], [ 74, 0 ], [ 79, 0 ], [ 82, 0 ], [ 83, 0 ],
  [ 86, 0 ], [ 89, 0 ], [ 94, 0 ], [ 97, 0 ] ]

(Note that the $8$ is not a typo, since the first list entry is the coefficient of $x^0$.)