Ramanujan sums are nearly orthogonal to powers (Q401980)

The author studies the \textit{orthogonality} properties of integer powers (say, \(j^r\)) and Ramanujan sums, namely NEWLINE\[NEWLINE c_k(j):=\sum_{{1\leq n\leq k}\atop {(n,k)=1}}e_k(jn), NEWLINE\]NEWLINE where, as usual, \(e_k(m):=e^{2\pi im/k}\), for all integers \(m\) and all natural numbers \(k\); here the summation restricts to reduced residue classes modulo \(k\), i.e., \((n,k)=1\) means \(n\) and \(k\) have no common prime factors.NEWLINENEWLINEThe author defines NEWLINE\[NEWLINE a_r(k):={1\over {k^{r+1}}}\sum_{j\leq k}j^r c_k(j) NEWLINE\]NEWLINENEWLINENEWLINE\noindent and proves that, in various possible meanings, \(a_r(k)\) tends to be both small and positive (see the paper's results); thus, the Ramanujan sums tend to be orthogonal to integer powers, as alluded by the title. As an example, if the power is low (\(r=2,3,4,5,6\)), then these coefficients have an explicit positive lower bound, uniformly for \(k\geq 1\) integer (see Corollary 2). The paper also contains many other results, concerning partial sums of Ramanujan sums (a version for \(c_k(j)\) instead of characters \(\chi(j)\), like in the style of \textit{R. E. A. C. Paley} [J. Lond. Math. Soc. 7, 28--32 (1932; JFM 58.0192.01)]) and related optimality results for linear combinations of \(e_k(jn)\) and \(c_k(j)\) itself, in the style of \textit{G. Bachman}'s results [Proc. Am. Math. Soc. 125, No. 4, 1001--1003 (1997; Zbl 0873.11049)]. The last result of the paper goes much in the direction of technical combinatorial calculations, especially involving Stirling numbers.NEWLINENEWLINEThe author proves his results in an elementary fashion, applying (his and) known formulæ, for Ramanujan sums, that involve Bernoulli numbers (think about a kind of very elaborate partial summation, with sums of powers) and zeta values at odd integers, like is transparent in Corollary 2 lower bounds.