Magic partition functions: Sign smoothing convolutions with Dirichlet invertible arithmetic functions

Abstract

Sign changes in sums of arithmetic functions and their inverses are a subtle topic with room to grow new results. Suppose that $S_f(x):={\sum }_{n\le x}f(n)$ is the summatory function of some arithmetic function $f$ such that $f(1)\ne 1$. There are known lower bounds on the limiting growth of $V(S_f,Y)$ -- the number of sign changes of $S_f(y)$ on the interval $y\in (0,Y]$ as $Y\to \infty$. We observe a partition theoretic sign smoothing by discrete convolution of the local oscillatory properties of the Dirichlet inverse of $f$, $S_{f^{-1}}(x)$. These so-called invertible ``magic partition function`` encodings lead to a sequence of convolution sums which have predictable sign properties provided the sequence of $f(n)$ ($f^{-1}(n)$, respectively) has reasonable asymptotic upper bounds with respect to $n$.

Keywords: Arithmetic functions; Dirichlet inverse; Dirichlet convolution; Dirichlet series; sign changes of arithmetic function; smoothing transformations; Cauchy product; discrete convolution