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Consider the function $$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial upper bounds of the function $f_n(t)$ in terms of $n$ and possibly $\phi(n)?$ An estimation of the bound will help in the computation of bounds of coefficients of cyclotomic polynomials. I tried a lot but could not find some good bound. I will highly appreciate any help in this regard.

** PS: The above function appears in the paper "Andrica, Dorin & Bagdasar, Ovidiu. (2018). The Cauchy integral formula with applications to polynomials, partitions and sequences. **

YCor
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AgnostMystic
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    Which sorts of bounds are you interested in? It is $O(c^{\phi(n)})$ for any $c > \frac{1}{2}$ because the coprime residues are asymptotically equidistributed https://mathoverflow.net/questions/22953/distribution-of-coprime-integers and it can't be less than $\left(\frac{1}{2}\right)^{\phi(n)}$ because that's the value of the logarithmic integral. – Aleksei Kulikov Mar 01 '24 at 09:46
  • Without specific properties or constraints on $n$, it's challenging to provide a general nontrivial upper bound. Depending on the specifics of your problem, you might need to combine some of strategies like periodicity and bounding sin or explore additional techniques tailored to your particular context. If you have more specific information about $n$ or the context in which $f_n(t)$ arises, that could guide the choice of strategy for obtaining bounds. – zeraoulia rafik Mar 01 '24 at 19:01

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