Sign changes of the partial sums of a random multiplicative function II (Q6621736)

A Rademacher random multiplicative function \(f\) is defined as follows: Over the primes, the values \(f(p)\) are independent and identically distributed random variables taking \(\pm 1\) values with probability \(1/2\) in each instance, and at the other values of \(n\), is defined in terms of the Möbius function \(\mu\) by \(f(n):=\mu^2(n)\prod_{p|n}f(p)\). Also, a Rademacher random completely multiplicative function \(f^*\), is defined by \(f^*(n)=\prod_{p^a\| n}f(p)^a\). In the paper under review, the author studies sign changes for the following weighted sums of \(f\) and \(f^*\), with \(\alpha\geq 0\),\N\[\N\sum_{n\leq x}\frac{f(n)}{n^\alpha},\quad\sum_{n\leq x}\frac{f^{*}(n)}{n^\alpha}.\N\]\NFor \(\alpha>1/2\), the partial sums converge to a non-vanishing Euler product, and hence, these partial sums become positive for all \(x\) sufficiently large, almost surely. The main result of the present paper focuses on the case \(0\leq\alpha\leq 1/2\), for which the author shows that if \(f\) is a Rademacher random multiplicative function, then for each \(0\leq\alpha\leq 1/2\) the partial sum \(\sum_{n\leq x}f(n)/n^\alpha\) changes sign infinitely often as \(x\to\infty\), almost surely. Also, he shows that if \(f^*\) is a Rademacher random completely multiplicative function, then for each \(0\leq\alpha<1/2\) the partial sum \(\sum_{n\leq x}f^*(n)/n^\alpha\) changes sign infinitely often as \(x\to\infty\), almost surely. The case \(\sum_{n\leq x}f^*(n)/\sqrt{n}\) is left as an open question. The proof follows the lines of [the author et al., Bull. Lond. Math. Soc. 55, No. 1, 78--89 (2023; Zbl 1535.11130)], and the present paper is indeed a sequel to this joint work.