Geodesic
An estimate of the free term of a~non-negative trigonometric polynomial with integer coefficients
Izvestiya. Mathematics , Tome 60 (1996) no. 6, pp. 1123-1182

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We denote by $M_Z^{\downarrow}(n)$ (resp., $K_Z^{\downarrow}(n)$) the smallest value of $a_0$ that can occur in a non-negative trigonometric polynomial $$ \sum_{k=0}^n a_k\cos(kx) $$ with non-negative integer coefficients $a_1\geqslant a_2\geqslant\dots\geqslant a_n$ such that $a_n\geqslant 1$ (resp., $\sum_{k=1}^n a_k=n$). We prove that for all natural numbers $n\geqslant 3$ $$ \dfrac{\ln^2 n}{\ln\ln n}\ll K_Z^\downarrow(n)\ll M_Z^\downarrow(n)\ll(\ln n)^3. $$ 
@article{IM2_1996_60_6_a1,
     author = {A. S. Belov and S. V. Konyagin},
     title = {An estimate of the free term of a~non-negative trigonometric polynomial with integer coefficients},
     journal = {Izvestiya. Mathematics },
     pages = {1123--1182},
     publisher = {mathdoc},
     volume = {60},
     number = {6},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1996_60_6_a1/}
}
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A. S. Belov; S. V. Konyagin. An estimate of the free term of a~non-negative trigonometric polynomial with integer coefficients. Izvestiya. Mathematics , Tome 60 (1996) no. 6, pp. 1123-1182. http://geodesic.mathdoc.fr/item/IM2_1996_60_6_a1/