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PreprintCC BY 4.0HAL

On the Hardy--Littlewood--Chowla conjecture

  • Guangshi Lü
  • Zhiwei Wang
  • Jie Wu
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Abstract

An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the M\"obius function exhibits cancellation on shifted primes; that is, ∑p≤ xμ(p+h)=o(π(x)) for any fixed shift h>0. This is also a well-known model case for the parity problem in sieve theory. In this paper, we prove this conjecture. As an application, we show that each of the values $1$ and −1 of μ(p+h) occur with asymptotic density 1/2: limx→∞ (#{p≤ x: μ(p+h)=κ})/(#{p≤ x: μ2(p+h)=1}) =1/2, κ∈{−1,1}.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026hardylittlewoodchowla,
  title = {{On the Hardy--Littlewood--Chowla conjecture}},
  author = {Guangshi Lü and Zhiwei Wang and Jie Wu},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20242},
  eprint = {2610.20242},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05793028v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Hardy--Littlewood--Chowla conjecture
AU  - Guangshi Lü
AU  - Zhiwei Wang
AU  - Jie Wu
PY  - 2026
DA  - 2026/10/11
PB  - HAL
DO  - 10.48550/arxiv.2610.20242
UR  - https://hal.science/hal-05793028v1
AB  - An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the M\"obius function exhibits cancellation on shifted primes; that is, ∑_p≤ xμ(p+h)=o(π(x)) for any fixed shift h>0. This is also a well-known model case for the parity problem in sieve theory. In this paper, we prove this conjecture. As an application, we show that each of the values $1$ and −1 of μ(p+h) occur with asymptotic density 1/2: lim_x→∞ (#{p≤ x: μ(p+h)=κ})/(#{p≤ x: μ^2(p+h)=1}) =1/2, κ∈{−1,1}.
ER  -