Editing Entropy (information theory) (section)

==Use in number theory==
[[Terence Tao]] used entropy to make a useful connection trying to solve the [[Erdős discrepancy problem]].<ref>{{Cite web |last=Klarreich |first=Erica |author-link=Erica Klarreich |date=1 October 2015 |title=A Magical Answer to an 80-Year-Old Puzzle |url=https://www.quantamagazine.org/a-magical-answer-to-an-80-year-old-puzzle-20151001/ |access-date=18 August 2014 |website=[[Quanta Magazine]]}}</ref><ref>{{Cite journal |last=Tao |first=Terence |date=2016-02-28 |title=The Erdős discrepancy problem |url=https://discreteanalysisjournal.com/article/609 |journal=Discrete Analysis |language=en |arxiv=1509.05363v6 |doi=10.19086/da.609 |s2cid=59361755 |access-date=20 September 2023 |archive-date=25 September 2023 |archive-url=https://web.archive.org/web/20230925184904/https://discreteanalysisjournal.com/article/609 |url-status=live }}</ref>

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)  {{math|''X''<sub>''H''</sub>}} {{=}} <math>\lambda(n+H)</math>. And in an interval [n, n+H] the sum over that interval could become arbitrary large.  For example, a sequence of +1's (which are values of {{math|''X''<sub>''H''</sub>}} 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 it used the [[Liouville function]] along with averages of modulated multiplicative functions<ref>https://arxiv.org/pdf/1502.02374.pdf {{Webarchive|url=https://web.archive.org/web/20231028111132/https://arxiv.org/pdf/1502.02374.pdf |date=28 October 2023 }}</ref> in short intervals. Proving it also broke the "parity barrier"<ref>https://terrytao.wordpress.com/2007/06/05/open-question-the-parity-problem-in-sieve-theory/  {{Webarchive|url=https://web.archive.org/web/20230807211237/https://terrytao.wordpress.com/2007/06/05/open-question-the-parity-problem-in-sieve-theory/ |date=7 August 2023 }}</ref> 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.