The \(L_1\)-norm of a trigonometric sum
The estimate of the \(L_1\)-norm of the exponential sum \[ \int_0^1 \left| \sum_{n\leq N}\alpha_n \exp(2i\pi x f(n))\right| \,dx\tag{*} \] where \(| \alpha_n| = 1\), was investigated by many authors for different functions \(f\). For example, \textit{S.~V.~Bochkarev} [Proc. Steklov Inst. Math. 218, 69--71 (1997); translation from Tr. Mat. Inst. Steklova 218, 74--76 (1997; Zbl 0962.11029)], \textit{A. A. Karatsuba} [Math. Notes 64, No.~3, 401--404 (1998); translation from Mat. Zametki 64, No.~3, 465--468 (1998; Zbl 0924.11065)] and \textit{M. Z. Garaev} [Math. Notes 68, No.~6, 713--720 (2000); translation from Mat. Zametki 68, No.~6, 842--850 (2000; Zbl 1005.11037)]. In this paper, the author considers the sequence of coefficients \((\varepsilon_k)\) in the binary expansion of the integer \(n\) and splits the set of positive integers \(\mathbb{N}\) into two disjoint subsets \(N_0\) and \(N_1\) as follows: \[ \text{ For }i=0,1; \; n\in N_i\Longleftrightarrow\sum_{k=0}^{\infty}\varepsilon_k\equiv i \pmod 2. \] Then, he investigates the \(L_1\)-norm \((*)\) when \(\alpha_n=1 \text{ if }n\in N_0\text{ or } -1 \text{ if } n\in N_1\) and shows that \[ N^{\Delta_2}\leq \int_0^1 \left| \sum_{n\leq N}\alpha_n \exp(2i\pi x n)\right| \,dx\leq N^{\Delta_1} \] where \(\Delta_1=0.407...\), \(\Delta_2=0.390...\) and \(N=2^{24Q-1}-1\) for an arbitrary positif integer \(Q\). He deduces an improvement of the remainder term to \(O(x^{\omega}\log^2 x)\), (\(\omega\leq 0.907...\)) for asymptotic expansion of the sums \(\sum_{n\leq x,n\in N_i}\tau(n)\) and \(\sum_{n\leq x,n\in N_i}r(n)\) where \(\tau(n)\) is the number of natural divisors of \(n\) and \(r(n)\) is the number of solutions of the equation \(a^2+b^2=n\).
scientific article; zbMATH DE number 2121449