Lindelöf hypothesis
E518478
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Lindelöf hypothesis (background and consequences) | 1 |
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in analytic number theory
ⓘ
mathematical conjecture ⓘ unproven conjecture ⓘ |
| concerns |
Riemann zeta function
NERFINISHED
ⓘ
asymptotic behavior of the Riemann zeta function ⓘ growth of the Riemann zeta function on the critical line ⓘ |
| concernsProperty | order of growth of |ζ(1/2+it)| ⓘ |
| domain | critical line Re(s) = 1/2 of the complex plane ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| hasConsequence |
bounds for error terms in prime-counting functions
ⓘ
results on the distribution of primes in arithmetic progressions ⓘ results on the distribution of primes in short intervals ⓘ |
| hasGeneralization |
Lindelöf hypothesis for Dirichlet L-functions
ⓘ
Lindelöf hypothesis for automorphic L-functions NERFINISHED ⓘ |
| hasHeuristicSupportFrom | random matrix theory models of the zeta function ⓘ |
| hasImplicationFor |
distribution of zeros of the Riemann zeta function
ⓘ
error term in the prime number theorem ⓘ size of gaps between primes ⓘ |
| hasOpenStatusIn | Clay Millennium Problem context (via relation to Riemann Hypothesis) ⓘ |
| implies |
strong bounds on the growth of the Riemann zeta function on the critical line
ⓘ
subpolynomial growth of the Riemann zeta function on the critical line ⓘ upper bounds for moments of the Riemann zeta function on the critical line ⓘ |
| isConnectedTo |
moment conjectures for the Riemann zeta function
ⓘ
subconvexity problems for L-functions ⓘ |
| isDiscussedIn |
analytic number theory textbooks
ⓘ
research literature on the Riemann zeta function ⓘ |
| isNotKnownToBeEquivalentTo | Riemann Hypothesis NERFINISHED ⓘ |
| isPartOf | classical problems about the Riemann zeta function ⓘ |
| isStrongerThan | trivial bounds for |ζ(1/2+it)| ⓘ |
| isSubjectOf | ongoing research in analytic number theory ⓘ |
| isWeakerThan | Riemann Hypothesis NERFINISHED ⓘ |
| mathematicalObject | statement about complex-valued function growth ⓘ |
| motivatedBy | understanding fine distribution of primes ⓘ |
| namedAfter | Ernst Leonard Lindelöf NERFINISHED ⓘ |
| relatedTo |
Dirichlet L-functions
NERFINISHED
ⓘ
Riemann Hypothesis NERFINISHED ⓘ distribution of prime numbers ⓘ generalized Lindelöf hypothesis ⓘ |
| statedIn | early 20th century ⓘ |
| status | open problem ⓘ |
| typeOfBound | upper bound on |ζ(1/2+it)| as t → ∞ ⓘ |
| usedIn |
analytic estimates in prime number theory
ⓘ
study of zeros of L-functions ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lindelöf hypothesis (background and consequences)