46:
62:
22:
165:
number of prime factors, then the former set has at least as many members as the latter set. Repeated prime factors are counted repeatedly; for instance, we say that 18 = 2 × 3 × 3 has an odd number of prime factors, while 60 =
77: = 2 × 10. The green spike shows the function itself (not its negative) in the narrow region where the conjecture fails; the blue curve shows the oscillatory contribution of the first Riemann zero.
122:. Though mathematicians typically refer to this statement as the Pólya conjecture, Pólya never actually conjectured that the statement was true; rather, he showed that the truth of the statement would imply the
245:
330:
731:
37: = 10. The (disproved) conjecture states that this function is always negative. The readily visible oscillations are due to the first non-trivial zero of the
582:
133:
is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general, providing an illustration of the
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776:
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754:
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179:
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274:
in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10.
781:
693:
262:
is even, and is negative if it is odd. The big Omega function counts the total number of prime factors of an integer.
134:
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450:
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38:
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388:
271:
119:
796:
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608:
469:
301:
170:
123:
166:
2 × 2 × 3 × 5 has an even number of prime factors.
638:
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99:
709:
61:
21:
560:
497:"A Numerical Investigation on Cumulative Sum of the Liouville Function"
473:
65:
Logarithmic graph of the negative of the summatory
Liouville function
126:. For this reason, it is more accurately called "Pólya's problem".
258:) = (−1) is positive if the number of prime factors of the integer
60:
44:
20:
713:
564:
169:
Equivalently, it can be stated in terms of the summatory
57:) in the region where the Pólya conjecture fails to hold.
328:(1919). "Verschiedene Bemerkungen zur Zahlentheorie".
240:{\displaystyle L(n)=\sum _{k=1}^{n}\lambda (k)\leq 0}
182:
331:
Jahresbericht der
Deutschen Mathematiker-Vereinigung
300:≤ 906,488,079. In this region, the summatory
239:
157:(excluding 0) are partitioned into those with an
292:The conjecture fails to hold for most values of
95:) stated that "most" (i.e., 50% or more) of the
391:(1958). "A disproof of a conjecture of Pólya".
289:= 906,150,257, found by Minoru Tanaka in 1980.
725:
576:
277:A (much smaller) explicit counterexample, of
114:was set forth by the Hungarian mathematician
8:
49:Closeup of the summatory Liouville function
366:. Courier Dover Publications. p. 483.
732:
718:
710:
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161:number of prime factors and those with an
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213:
202:
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145:The Pólya conjecture states that for any
285:in 1960; the smallest counterexample is
317:
270:The Pólya conjecture was disproved by
296:in the region of 906,150,257 ≤
118:in 1919, and proved false in 1958 by
16:Disproved conjecture in number theory
7:
363:Mathematics: The Man-Made Universe
304:reaches a maximum value of 829 at
14:
465:10.1090/S0025-5718-1960-0120198-5
173:, with the conjecture being that
922:Conjectures about prime numbers
228:
222:
192:
186:
1:
25:Summatory Liouville function
502:Tokyo Journal of Mathematics
281:= 906,180,359 was given by
135:strong law of small numbers
938:
451:Mathematics of Computation
149: > 1, if the
745:
599:
446:"On Liouville's function"
407:10.1112/S0025579300001480
129:The size of the smallest
102:any given number have an
740:Prime number conjectures
891:Schinzel's hypothesis H
516:10.3836/tjm/1270216093
444:Lehman, R. S. (1960).
241:
218:
153:less than or equal to
78:
58:
42:
917:Disproved conjectures
896:Waring's prime number
619:Euler's sum of powers
242:
198:
64:
48:
39:Riemann zeta function
24:
180:
861:Legendre's constant
495:Tanaka, M. (1980).
272:C. Brian Haselgrove
120:C. Brian Haselgrove
812:Elliott–Halberstam
797:Chinese hypothesis
609:Chinese hypothesis
547:"Pólya Conjecture"
544:Weisstein, Eric W.
302:Liouville function
237:
171:Liouville function
124:Riemann hypothesis
93:Pólya's conjecture
79:
59:
43:
904:
903:
832:Landau's problems
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389:Haselgrove, C. B.
358:Stein, Sherman K.
283:R. Sherman Lehman
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659:Ono's inequality
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89:Pólya conjecture
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308:= 906,316,571.
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634:Hauptvermutung
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536:External links
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509:(1): 187–189.
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108:prime factors
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116:George Pólya
103:
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50:
34:
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876:Oppermann's
822:Gilbreath's
792:Bunyakovsky
689:Von Neumann
593:conjectures
394:Mathematika
911:Categories
881:Polignac's
854:Twin prime
849:Legendre's
837:Goldbach's
767:Agoh–Giuga
699:Williamson
694:Weyl–Berry
674:Schoen–Yau
591:Disproved
431:0085.27102
344:47.0882.06
312:References
112:conjecture
106:number of
866:Lemoine's
807:Dickson's
787:Brocard's
772:Andrica's
552:MathWorld
415:0025-5793
338:: 31–40.
326:Pólya, G.
232:≤
220:λ
200:∑
141:Statement
100:less than
871:Mersenne
802:Cramér's
669:Ragsdale
649:Keller's
644:Kalman's
604:Borsuk's
360:(2010).
266:Disproof
250:for all
73:) up to
33:) up to
827:Grimm's
777:Artin's
679:Seifert
654:Mertens
525:0584557
482:0120198
474:2003890
423:0104638
684:Tait's
639:Hirsch
614:Connes
523:
480:
472:
429:
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342:
110:. The
87:, the
886:Pólya
664:Pólya
624:Ganea
470:JSTOR
842:weak
411:ISSN
368:ISBN
163:even
91:(or
760:2nd
755:1st
511:doi
460:doi
427:Zbl
403:doi
340:JFM
159:odd
104:odd
83:In
913::
549:.
521:MR
519:.
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499:.
478:MR
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137:.
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306:n
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196:=
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190:n
187:(
184:L
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69:(
67:L
55:n
53:(
51:L
41:.
35:n
31:n
29:(
27:L
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