Talk:Pólya conjecture
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Counter example
editAllegedly a smaller counterexample would be 906180359. SyP 17:39, 14 May 2006 (UTC)
I agree. And the smallest counterexample is 906150257.
From The On-Line Encyclopedia of Integer Sequences, sequence id: A002819, in the comment: "... In 1962, R. S. Lehman found that L(906180359) = 1 and in 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257. - Harri Ristiniemi (harri.ristiniemi(AT)nic.fi), Jun 23 2001" --Maybeso 19:44, 2 June 2006 (UTC)
Statement
editThe two definitions of the conjecture do not agree. The one on top of the page states that "most numbers have an odd number of prime factors", while the one in terms of the Liouville theorem states that "no more than half of the natural numbers greater than one but less than any number n have an even number of prime factors". The latter definition is the only one that makes sense, so unless anyone would disagree, I will reword the introduction to give a plain-English definition that agrees with it. --Jeffthompson 01:39, 16 December 2006 (UTC)
- 1 with 0 prime factors should be included in the count, and so should n (k runs from 1 to n in the summation). This is shorter and more precise: "For any integer n > 1, at least half the integers from 1 to n have an odd number of prime factors". PrimeHunter 15:01, 16 December 2006 (UTC)
I think your change must have died. Here is another shot at making this sensible: diff. The problem is that without the "less than any given number" bit, a "counterexample" doesn't seem to make sense. (Actually, I'm not sure that it ever makes sense to talk about most natural numbers, but that is a different matter.)
Apparently impressive bounds
editThe claim that "the extraordinarily high value of this counterexample serves to show the dangers of relying on apparently impressive bounds set by computer searches" (as viewed 2007-01-27T15:00:00 GMT) is flawed because of the existence of much smaller counterexamples. That large counterexamples exist is irrelevant: the reason that apparently impressive bounds should be treated with some caution is that the _smallest_ counterexample could be large. —The preceding unsigned comment was added by 81.102.136.55 (talk) 15:04, 27 January 2007 (UTC).
- I agree and have removed that comment (diff). PrimeHunter 17:25, 27 January 2007 (UTC)
Riemann zeta
editWhat is connection between zeros of Riemann zeta function and Liouville function sum? And why one zero of this function, with imaginary part a bit larger than 14 and real part equal 1/2 can cause oscillations within first 10 000 000 values of this function? Wojowu (talk) 19:56, 4 February 2012 (UTC)
Counterexamples
editThere is a mentioned section where the conjecture tends to fail. Is there another? 02:33, 10 December 2016 (UTC)02:33, 10 December 2016 (UTC)02:33, 10 December 2016 (UTC)02:33, 10 December 2016 (UTC)02:33, 10 December 2016 (UTC)32ieww (talk) 32ieww (talk) 02:33, 10 December 2016 (UTC)
Yes, it fails many times. However, it is unknown if it changes sign finite or infinite amount of times.
Inverse Polya conjecture
editSo, with Polya conjecture being disproved by both theoretical and actual counter examples, is there an inverse conjecture proposed, that is that the value can actually go arbitrarily high above the zero, and crosses zero infinitely many times? 2A02:168:2000:5B:37F1:215E:C4AA:E2F7 (talk) 15:36, 27 February 2021 (UTC)