J. S. Littlewood conjectured that there exist complex numbers \(a_1,a_2,\dots,a_N\) with \(|a_n|=1\) \((n=1,2,\dots,N)\) such that for all \(x\in\mathbb{R}\) we have \[ A_1\sqrt{N} \leq\left|\sum_{n=1}^N a_ne^{2\pi inx}\right|\leq A_2\sqrt{N}, \] where \(A_1,A_2\) are absolute positive constants. This conjecture was proved by T. W. Körner [“On a polynomial of Byrnes”, Bull. Lond. Math. Soc. 12, 219-224 (1980; Zbl 0435.30004)].
In this paper a new method for proving the conjecture of Littlewood is given. This method is effective and it yields numerical values for \(A_1,A_2\). Further, the author gives a positive answer to the following question of W. OrIicz: Does there exist a trigonometric series \(\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\) which is everywhere divergent and \[ \sum_{n=1}^\infty (|a_n|^{2+\varepsilon} +|b_n|^{2+\varepsilon})<+\infty\quad (\varepsilon>0)? \]