On the Turán-Kubilius inequality (Q2874924)

Given two functions \(f,g : \mathbb{N} \rightarrow \mathbb{C}\), one always has the trivial bound NEWLINE\[NEWLINE \left|\sum_{n \leq x} f(n) g(n) \right| \leq \sum_{n \leq x} |f(n)| |g(n)| . NEWLINE\]NEWLINE This is sharp in certain cases, but one of the main tasks in analytic number theory is to obtain stronger estimates where \(f, g\) are interesting functions that one expects to be somehow ``uncorrelated''. A classical example dating back at least to Vinogradov is where \(g(n) = e^{2\pi i \theta n}\) for some real \(\theta\), and \(f(n)\) is some other interesting function, for example the Möbius function \(\mu(n)\). If one can obtain non-trivial estimates, these can then be fed into techniques like the circle method or the Erdős-Turán discrepancy inequality and applied to arithmetic problems.NEWLINENEWLINEThe present paper, which describes itself as a ``short survey'', states various theorems that fit into the above framework. So far as the reviewer can see, essentially all of these theorems are taken verbatim from previous papers of the author and his collaborators. These kinds of results have attracted attention recently because of their connection with work of Bourgain, Sarnak and Ziegler [\textit{J. Bourgain} et al., Dev. Math. 28, 67--83 (2013; Zbl 1336.37030)], which considered the case where \(f(n) = \mu(n)\) and \(g(n)\) is a certain kind of function connected with dynamical systems. The Turán-Kubilius inequality, which is mentioned in the title of the paper, is a result from probabilistic number theory that upper bounds the variance of additive functions, and is the key ingredient in the author's approach to these problems.NEWLINENEWLINEFor example, Theorem 1 in the paper asserts that if \(f\) is a multiplicative function bounded in absolute value by 1, and if \(t(n)\) is some other function for which \((1/x) \sum_{m \leq x} e^{2\pi i (t(p_i m) - t(p_j m))}\) is ``small'' for ``many'' pairs of distinct primes \(p_i , p_j\), then \((1/x) \sum_{n \leq x} f(n) e^{2\pi i t(n)}\) must also be small. This is proved by inserting a suitable additive function, counting divisibility by different primes \(p_i\), into the sum. On the one hand this must only change things roughly by multiplying by the mean value of that additive function (since the Turán-Kubilius inequality says additive functions have small variance), but on the other hand it turns the sum into a double sum (over \(n\) and \(p_i\)) that can be rearranged and allows the Cauchy-Schwarz inequality to be usefully applied. In particular, one can apply the Theorem when \(t(n) = \alpha n\) for irrational \(\alpha\), which reproves a result of Daboussi (see [\textit{H. Daboussi} and \textit{H. Delange}, J. Lond. Math. Soc., II. Ser. 26, 245--264 (1982; Zbl 0499.10052)]).NEWLINENEWLINEReviewer's remarks: The reviewer thinks that the idea underlying Theorem 1 is neat and clever, and well worth the attention of those who work in this area. However, the present paper is perhaps not the best place to learn much about it. The proof of Theorem 1 is given in full, but although it is described as ``a shortened variant of the original proof'' it seems to the reviewer it is really exactly the same as in earlier works of the author. None of the other stated results are proved or described much at all, apart from a few lines of discussion about Theorem 3. There are also some typos and small errors that seem to have survived the refereeing process: partway down page 180, the reference to (3.7) should really be to (3.8); at the top of page 181, the sum in the definition of \(B(x)\) should be a supremum; at the start of section 4, the reference given seems to be to the wrong paper (it should perhaps be reference [16] instead of [14]?); in the statement of Theorem 2, presumably one needs to assume that \(f\) is multiplicative, and it should be \(g(n) = e(rn/D)h(n)\) rather than \(e(r/D) h(n)\); and in the statement of Theorem 8, presumably \(\lambda(F_1)\) should really be the product \(\lambda(\mathcal{F}_1) \cdots \lambda(\mathcal{F}_k)\)?NEWLINENEWLINEFor the entire collection see [Zbl 1279.00053].