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Truncated pretentious distances of multiplicative arithmetic functions

  • Thomas Renard
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Abstract

Let f:ℕ→𝕌 be a non-pretentious multiplicative function in the sense of Granville and Soundararajan. We study the convergence of the autocorrelation averages of f. In particular, we investigate the relationship between the local pretentious behaviour of f on intervals of the form [xε,x] and its deviation from $1$ on the primes. As an application, we exhibit a class of non-pretentious multiplicative functions for which Elliott's conjecture in its original formulation holds, following up on a result by Klurman, Mangerel and Teräväinen.

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Cite

BibTeXRIS
BibTeX
@misc{renard2026truncated,
  title = {{Truncated pretentious distances of multiplicative arithmetic functions}},
  author = {Thomas Renard},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37332},
  eprint = {2609.37332},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37332},
}
RIS
TY  - UNPB
TI  - Truncated pretentious distances of multiplicative arithmetic functions
AU  - Thomas Renard
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37332
UR  - https://arxiv.org/abs/2609.37332
AB  - Let f:ℕ→𝕌 be a non-pretentious multiplicative function in the sense of Granville and Soundararajan. We study the convergence of the autocorrelation averages of f. In particular, we investigate the relationship between the local pretentious behaviour of f on intervals of the form [x^ε,x] and its deviation from $1$ on the primes. As an application, we exhibit a class of non-pretentious multiplicative functions for which Elliott's conjecture in its original formulation holds, following up on a result by Klurman, Mangerel and Teräväinen.
ER  -