linguistic:paraphrasing
Use in number theory

Terence Tao used entropy to make a useful connection trying to solve the Erdős discrepancy problem.

Intuitively the idea behind the proof was if there is low information in terms of the Shannon entropy between consecutive random variables (here the random variable is defined using the Liouville function (which is a useful mathematical function for studying distribution of primes) X = λ (n + H) {\displaystyle \lambda (n+H)}. And in an interval the sum over that interval could become arbitrary large. For example, a sequence of +1's (which are values of X could take) have trivially low entropy and their sum would become big. But the key insight was showing a reduction in entropy by non negligible amounts as one expands H leading inturn to unbounded growth of a mathematical object over this random variable is equivalent to showing the unbounded growth per the Erdős discrepancy problem.

The proof is quite involved and it brought together breakthroughs not just in novel use of Shannon Entropy, but also its used the Liouville function along with averages of modulated multiplicative functions Archived 28 October 2023 at the Wayback Machine in short intervals. Proving it also broke the "parity barrier" Archived 7 August 2023 at the Wayback Machine for this specific problem.

While the use of Shannon Entropy in the proof is novel it is likely to open new research in this direction.