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PreprintCC BY-NC-ND 4.0HAL

ON INTEGERS WHOSE FRIABLE COMPONENT IS LARGE

  • Gérald Tenenbaum
  • Xiangyu Wang
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Abstract

We estimate the number of integers up to x whose y-friable component exceeds z. Uniformly for max{2, log z} ⩽ y ⩽ z ⩽ x/2, we determine the logarithm of this counting function up to an error O(log z/ log y). This sharpens an estimate arising from a problem in Ford's sieve methods lecture notes. The proof combines classical friablenumber estimates with an elementary argument for the lower bound.

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Cite

BibTeXRIS
BibTeX
@misc{tenenbaum2026integers,
  title = {{ON INTEGERS WHOSE FRIABLE COMPONENT IS LARGE}},
  author = {Gérald Tenenbaum and Xiangyu Wang},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.13140/rg.2.2.13436.50563},
  url = {https://hal.science/hal-05792809v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - ON INTEGERS WHOSE FRIABLE COMPONENT IS LARGE
AU  - Gérald Tenenbaum
AU  - Xiangyu Wang
PY  - 2026
DA  - 2026/10/11
PB  - HAL
DO  - 10.13140/rg.2.2.13436.50563
UR  - https://hal.science/hal-05792809v1
AB  - We estimate the number of integers up to x whose y-friable component exceeds z. Uniformly for max{2, log z} ⩽ y ⩽ z ⩽ x/2, we determine the logarithm of this counting function up to an error O(log z/ log y). This sharpens an estimate arising from a problem in Ford's sieve methods lecture notes. The proof combines classical friablenumber estimates with an elementary argument for the lower bound.
ER  -