Conjectures on the frequency and sign distribution of extreme values

Let $T$ be a height parameter, and let the two functions discussed in the source have extreme values indexed by integers $n$. Consider the extreme values up to height $T$, and the sets of indices for which these functions are respectively positive or negative.

Conjectures on extreme values. One may conjecture that the number of extreme values up to height $T$ is less than $C\log T$ for some constant $C$, and that the sets of indices $n$ for which the respective functions are positive or negative have natural density zero.

These are presented as possible conjectures of different strengths based on numerical data in a table. The supplied context does not specify the functions or define the extreme-value indexing sufficiently to assess the claims more precisely, and no resolution status is given; they remain open.