Sign changes of the partial sums of a random multiplicative function III: Average

Let $V(x)$ be the number of sign changes of the partial sums up to $x$, say $M_f(x)$, of a Rademacher random multiplicative function $f$. We prove that the averaged value of $V(x)$ is at least $\gg (\log x)(\log\log x)^{-1/2-ε}$. Our new method applies for the counting of sign changes of the partial sums of a system of orthogonal random variables having variance $1$ under additional hypothesis on the moments of these partial sums. In particular, we extend to larger classes of dependencies an old result of Erdős and Hunt on sign changes of partial sums of i.i.d. random variables. In the arithmetic case, the main input in our method is the ``\textit{linearity}'' phase in $1\leq q\leq 1.9$ of the quantity $\log \mathbb{E} |M_f(x)|^q$, provided by the Harper's \textit{better than squareroot cancellation} phenomenon for small moments of $M_f(x)$.