SOLVED - $500

If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that
\[\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?\]

The Erdős discrepancy problem. This is true, and was proved by Tao [Ta16], who also proved the more general case when $f$ takes values on the unit sphere.

In [Er81] it is further conjectured that \[\max_{md\leq x}\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert \gg \log x.\]

In [Er85c] Erdős also asks about the special case when $f$ is multiplicative.