Halász theorem
E637298
Halász theorem is a fundamental result in analytic number theory that provides sharp bounds on the mean values of multiplicative functions, playing a key role in understanding their average behavior.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | theorem in analytic number theory ⓘ |
| appearsIn | research on mean values of arithmetic functions ⓘ |
| appliesTo |
bounded multiplicative functions
ⓘ
complex-valued multiplicative functions ⓘ |
| assumes |
growth conditions on the multiplicative function
ⓘ
multiplicativity of the function ⓘ |
| characterizes |
distance of a multiplicative function from characters or n^{it}
ⓘ
when a multiplicative function has large mean value ⓘ |
| concerns |
average behavior of multiplicative functions
ⓘ
partial sums of multiplicative functions over n ≤ x ⓘ |
| context |
distribution of additive arithmetic functions via multiplicative methods
ⓘ
probabilistic number theory ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| hasConsequence |
logarithmic density results for multiplicative functions
ⓘ
mean value estimates uniform in the range of summation ⓘ |
| hasVariant | Halász inequality for multiplicative functions NERFINISHED ⓘ |
| implies |
bounds for partial sums of multiplicative functions
ⓘ
cancellation in sums of non-pretentious multiplicative functions ⓘ |
| influenced |
modern theory of multiplicative functions
ⓘ
pretentious analytic number theory ⓘ |
| mainTopic |
mean values of multiplicative functions
ⓘ
multiplicative functions ⓘ |
| mathematicalDomain | additive and multiplicative number theory ⓘ |
| namedAfter | Gábor Halász NERFINISHED ⓘ |
| provedBy | Gábor Halász NERFINISHED ⓘ |
| provides | sharp bounds on mean values of multiplicative functions ⓘ |
| relatedTo |
Delange theorem
NERFINISHED
ⓘ
Dirichlet characters modulo q NERFINISHED ⓘ Erdős–Wintner theorem NERFINISHED ⓘ Granville–Soundararajan theory of pretentious multiplicative functions NERFINISHED ⓘ large sieve methods ⓘ mean values over initial intervals of the integers ⓘ pretentious approach to multiplicative functions ⓘ |
| timePeriod | 20th century mathematics ⓘ |
| typeOfBound | asymptotically sharp ⓘ |
| typicalForm | upper bound for |∑_{n≤x} f(n)| in terms of a distance parameter ⓘ |
| usedFor |
bounding correlations of multiplicative functions
ⓘ
estimating mean values of Dirichlet characters ⓘ estimating mean values of the Liouville function ⓘ estimating mean values of the Möbius function ⓘ proving results on sign changes of multiplicative functions ⓘ studying distribution of values of multiplicative functions ⓘ |
| usesConcept |
Dirichlet series
NERFINISHED
ⓘ
Fourier analysis on the unit circle ⓘ Halász–Montgomery inequality NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.