Pólya’s conjecture
E637315
Pólya’s conjecture is a disproven hypothesis in number theory that proposed a specific long-term sign pattern for the summatory Möbius function, suggesting it would eventually remain nonpositive.
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
disproven conjecture
ⓘ
mathematical conjecture ⓘ number theory conjecture ⓘ |
| assumes | that M(x) does not become positive infinitely often ⓘ |
| contradictedBy | explicit large positive values of M(x) ⓘ |
| disproofMethod | existence of counterexamples for large x ⓘ |
| disproofYear | 1958 ⓘ |
| disprovedBy | C. Brian Haselgrove NERFINISHED ⓘ |
| equivalentlyStates | the summatory Möbius function is eventually nonpositive ⓘ |
| field | number theory ⓘ |
| hasCounterexamples | values of x for which M(x) > 0 ⓘ |
| hasInfluenceOn |
analytic number theory
ⓘ
study of sign changes of M(x) ⓘ subsequent research on the Möbius function ⓘ |
| hasMathematicsSubjectClassification |
11M26
ⓘ
11N37 ⓘ |
| implies | a strong restriction on the sign pattern of M(x) ⓘ |
| involvesConcept |
Riemann hypothesis
NERFINISHED
ⓘ
Riemann zeta function NERFINISHED ⓘ distribution of prime numbers ⓘ sign changes ⓘ |
| involvesFunction |
Möbius function
ⓘ
summatory Möbius function M(x) ⓘ |
| isAbout |
asymptotic behavior of M(x)
ⓘ
long-term sign pattern of the summatory Möbius function ⓘ |
| isWeakerThan | Mertens conjecture ⓘ |
| languageOfOriginalPublication | German ⓘ |
| mainSubject |
Mertens function
NERFINISHED
ⓘ
summatory Möbius function ⓘ |
| motivatedBy | heuristics from the Riemann hypothesis ⓘ |
| namedAfter | George Pólya NERFINISHED ⓘ |
| proposedBy | George Pólya NERFINISHED ⓘ |
| publicationYear | 1919 ⓘ |
| relatedConjecture | Mertens conjecture NERFINISHED ⓘ |
| relatedTo |
oscillation of arithmetic functions
ⓘ
partial sums of multiplicative functions ⓘ |
| statedIn | paper by George Pólya on the zeros of the Riemann zeta function and related functions ⓘ |
| statementForm | M(x) ≤ 0 for all sufficiently large x ⓘ |
| status | disproved ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.