On the rearranged block-orthonormal systems.

Nadibaidze, G.

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Let $H (x) = \sum_{n \leq x} \frac{φ (n)}{n} - \frac{6}{π^{2}} x$ . Motivated by a conjecture of Erdös, Lau developed a new method and proved that $# {n \leq T : H (n) H (n + 1) < 0} ≫ T .$ We consider arithmetical functions $f (n) = \sum_{d ∣ n} \frac{b_{d}}{d}$ whose summation can be expressed as $\sum_{n \leq x} f (n) = α x + P (log (x)) + E (x)$ , where $P (x)$ is a polynomial, $E (x) = - \sum_{n \leq y (x)} \frac{b_{n}}{n} ψ (\frac{x}{n}) + o (1)$ and $ψ (x) = x - ⌊ x ⌋ - 1 / 2$ . We generalize Lau’s method and prove results about the number of sign changes for these error terms.