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Let $f: \mathbb{N} \rightarrow (0,\infty)$ be a multiplicative function, and let $\omega>0$. We consider the frequency with which patterns of the form $$ f(n)>\omega \max\{f(n-1),f(n+1)\}, \quad f(n)<\frac{1}{\omega}\min\{f(n-1),f(n+1)\} $$ occur in the sequence $(f(n))_n$, and give necessary conditions on $f$ so that...
Let $f: \mathbb{N} \rightarrow (0,\infty)$ be a multiplicative function, and let $\omega>0$. We consider the frequency with which patterns of the form $$ f(n)>\omega \max\{f(n-1),f(n+1)\}, \quad f(n)<\frac{1}{\omega}\min\{f(n-1),f(n+1)\} $$ occur in the sequence $(f(n))_n$, and give a sharp classification of all positi...
Let f:N→D$f:\mathbb {N}\rightarrow \mathbb {D}$ be a multiplicative function. Under the merely necessary assumption that f$f$ is nonpretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts h1,h2$h_1,h_2$ , the two‐point correlation 1x∑n⩽xf(n+h1)f¯(n+h2)$$\begin{equ...
Let be a multiplicative function. Under the merely necessary assumption that is nonpretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts , the two‐point correlation tends to 0 along a set of of full upper logarithmic density. We also show that the same re...
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