906,150,257 and the Pólya conjecture (MegaFavNumbers)

#MegaFavNumbers

"Most numbers have an odd number of prime factors!"

...or do they...?

Ben chats about the large counterexample to Polya's conjecture, for Matt Parker and James Grime's MegaFavNumbers project.

Ben is @SparksMaths on twitter and at http://www.bensparks.co.uk on the web

Polya's conjecture on Wikipedia: https://en.m.wikipedia.org/wiki/P%C3%....

EDIT:
At around 10 mins I suggest that no-one knows whether the summatory Liouville function crosses between negative and positive infinitely often. Tanaka claimed this in 1980 in his paper https://doi.org/10.3836/tjm/1270216093 - *but he was in fact wrong*. Haselgrove had actually proved this was true, in order to make his 1958 argument. Haselgrove's paper is here: https://londmathsoc.onlinelibrary.wil... (but you may need to pay to access). Lehman in 1960 clearly recognised this too (https://www.ams.org/journals/mcom/196...) so I'm a bit puzzled why Tanaka claimed it was still an open question.
(For full disclosure, Mathworld also made this error on their page on the Polya Conjecture, though it has now been pointed out, and will hopefully be corrected soon).

I'm grateful to an eagle eyed viewer for pointing this out to me! Thanks - you know who you are.


This video is part of the MegaFavNumbers project. Maths YouTubers have come together to make videos about their favourite numbers bigger than one million, which we are calling #MegaFavNumbers.

We want you (yes you humble viewer) to join in! Make your own video about your favourite mega-number. You can think of a cool big number, or think of a cool topic first and hang a mega-number on it. Upload your videos to YouTube with the hashtag #MegaFavNumbers and with MegaFavNumbers in the title, and your video will be added to the megafavnumbers playlist.

Submit your videos anytime before Wednesday 2nd September 2020 to be added to the MegaFavNumbers playlist! MegaFavNumbers Playlist:    • MegaFavNumbers