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 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $n\geqslant 3$ $$ \dfrac{\ln^2 n}{\ln\ln n}\ll K_Z^\downarrow(n)\ll M_Z^\downarrow(n)\ll(\ln n)^3. $$

[1] Belov A. S., “Novye primery neotritsatelnykh trigonometricheskikh polinomov s tselymi koeffitsientami”, Konstruktivnaya teoriya funktsii i ee prilozheniya, Makhachkala, 1994, 17–19

[2] Belov A. S., “Novye primery neotritsatelnykh trigonometricheskikh polinomov s tselymi koeffitsientami”, Fund. i prikl. matem., 1:3 (1995), 581–612 | MR | Zbl

[3] Odlyzko A. M., “Minima of cosine sums and maxima of polynomials on the unit circle”, J. Lond. Math. Soc., 26:3 (1982), 412–420 | DOI | MR | Zbl

[4] Kolountzakis M. N., “On nonnegative cosine polynomials with nonnegative integral coefficients”, Proc. Am. Math. Soc., 120:1 (1994), 157–163 | DOI | MR | Zbl

[5] Erdös P., Szekeres G., “On the product $\prod _{k=1}^n(1-z^{a_k})$”, Acad. Serbe Sci. Publ. Inst. Math., 13 (1959), 29–34 | MR | Zbl

[6] Belov A. S., “O neravenstve Sidona–Zigmunda v teorii lakunarnykh trigonometricheskikh ryadov”, Matem. zametki, 30:4 (1981), 501–515 | MR | Zbl

[7] Chandrasekkharan K., Vvedenie v analiticheskuyu teoriyu chisel, Mir, M., 1974 | MR

[8] Rosser J. B., Schoenfeld L., “Approximate formulas for some functions of prime numbers”, Illinois J. Math., 6:1 (1962), 64–94 | MR | Zbl

[9] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[10] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965 | MR

[11] Kusis P., Vvedenie v teoriyu prostranstv $H^p$, Mir, M., 1984 | MR