The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures
We study the asymptotic behaviour of higher order correlations En≤X/d g1(n + ah1)· · · gk (n + ahk ) as a function of the parameters a and d, where g1, . . . , gk are bounded multiplicative functions, h1, . . . , hk are integer shifts, and X is large. Our main structural result asserts, roughly spea...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Mathematical Sciences Publishers
2019
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| _version_ | 1797113003870519296 |
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| author | Tao, T Teräväinen, J |
| author_facet | Tao, T Teräväinen, J |
| author_sort | Tao, T |
| collection | OXFORD |
| description | We study the asymptotic behaviour of higher order correlations
En≤X/d g1(n + ah1)· · · gk (n + ahk )
as a function of the parameters a and d, where g1, . . . , gk are bounded multiplicative functions, h1, . . . , hk
are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations
asymptotically vanish for almost all X if g1 · · · gk does not (weakly) pretend to be a twisted Dirichlet
character n 7→ χ (n)n
it, and behave asymptotically like a multiple of d
− itχ (a) otherwise. This extends
our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is
averaged out and one can set t = 0. Among other things, the result enables us to establish special cases
of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we
establish the k-point Chowla conjecture En≤X λ(n + h1)· · · λ(n + hk ) = o(1) for k odd or equal to 2 for
all scales X outside of a set of zero logarithmic density |
| first_indexed | 2024-03-06T20:44:40Z |
| format | Journal article |
| id | oxford-uuid:35799e5e-eeb0-407c-a0ba-e226c7f3092a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-04-09T03:56:43Z |
| publishDate | 2019 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | oxford-uuid:35799e5e-eeb0-407c-a0ba-e226c7f3092a2024-03-22T10:37:23ZThe structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjecturesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:35799e5e-eeb0-407c-a0ba-e226c7f3092aEnglishSymplectic Elements at OxfordMathematical Sciences Publishers2019Tao, TTeräväinen, JWe study the asymptotic behaviour of higher order correlations En≤X/d g1(n + ah1)· · · gk (n + ahk ) as a function of the parameters a and d, where g1, . . . , gk are bounded multiplicative functions, h1, . . . , hk are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all X if g1 · · · gk does not (weakly) pretend to be a twisted Dirichlet character n 7→ χ (n)n it, and behave asymptotically like a multiple of d − itχ (a) otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is averaged out and one can set t = 0. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the k-point Chowla conjecture En≤X λ(n + h1)· · · λ(n + hk ) = o(1) for k odd or equal to 2 for all scales X outside of a set of zero logarithmic density |
| spellingShingle | Tao, T Teräväinen, J The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures |
| title | The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures |
| title_full | The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures |
| title_fullStr | The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures |
| title_full_unstemmed | The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures |
| title_short | The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures |
| title_sort | structure of correlations of multiplicative functions at almost all scales with applications to the chowla and elliott conjectures |
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