HEXAGONbeta

Monotonicity of Erdős sums

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Jaideep Padhi ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

For k≥1k \geq 1 let fk=∑Ω(n)=k1/(nlog⁡n)f_k = \sum_{\Omega(n)=k} 1/(n\log n), where Ω\Omega counts prime factors with multiplicity, and let gkg_k be the same sum restricted to odd nn. Lichtman showed that fk→1f_k \to 1, with a global minimum at k=6k=6, and Gorodetsky, Lichtman and Wong that fk=1−2−k(a1k2+O(klog⁡(k+1)))f_k = 1 - 2^{-k}(a_1 k^2 + O(k \log(k+1))). We prove that fk<fk+1f_k < f_{k+1} for every k≥6k \geq 6, and that gk>gk+1g_k > g_{k+1} for every k≥1k \geq 1. The two statements come from one analytic distinction. Writing ∑kPk(s)zk=ζ(s)zH(s,z)\sum_k P_k(s) z^k = \zeta(s)^z H(s,z), at s=1s=1 the point z=2z=2 lies on the first singular circle of HH, and strictly inside the first singular circle of the product with the prime 22 removed. We show that this distinction decides the direction in general. If the complement of a set of primes P\mathcal{P} is thin and its least element is at least 33, the corresponding sums eventually decrease, so the second theorem is one case of a general phenomenon. The inequalities used at the last step of the first theorem are certified by interval arithmetic.

Provenance statement

The mathematical content originates with the author; the manuscript text was drafted and revised using Claude Opus 5 (Anthropic) under the author's direction, as described in the paper's Statement on the use of AI. All numerical claims are certified in ball arithmetic using Arb via python-flint, and the scripts, their logs of record and the resulting certificate are included as ancillary files and in the repository cited in Section 8. Four verification scripts check the manuscript's printed numerals against those logs rather than against the manuscript itself; this was added after three transcription errors were found that no earlier check could catch, including a bound stated for the wrong function. The author is responsible for the content.

Tools used

Anthropic
Claude OpusVersion 5

References

  1. W. D. Banks and G. Martin, Optimal primitive sets with restricted primes , Integers 13 (2013), #A69, 10 pp.; arXiv:1301.0948.arXiv
  2. J. D. Lichtman, Almost primes and the Banks--Martin conjecture , J. Number Theory 211 (2020), 513--529; arXiv:1909.00804.arXiv
  3. O. Gorodetsky, J. D. Lichtman and M. W. Wong, On Erdős sums of almost primes , C. R. Math. Acad. Sci. Paris 362 (2024), no. G12, 1571--1596; arXiv:2303.08277 .arXiv
  4. P. Erdős, Note on sequences of integers no one of which is divisible by any other , J. London Math. Soc. 10 (1935), no. 2, 126--128.
  5. L. N. Trefethen, Is Gauss quadrature better than Clenshaw--Curtis? , SIAM Rev. 50 (2008), 67--87.
  6. L. N. Trefethen, Approximation Theory and Approximation Practice , SIAM, Philadelphia, 2013.
  7. H. Cohen, High precision computation of Hardy--Littlewood constants , preprint (1991), https://www.math.u-bordeaux.fr/\textasciitilde hecohen/ .link
  8. H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I: Classical Theory , Cambridge Univ. Press, 2006.
  9. F. W. J. Olver et al. (eds.), NIST Digital Library of Mathematical Functions , https://dlmf.nist.gov/ , Release 1.2.1 (2024).link
  10. Z. Zhang, On a problem of Erdős concerning primitive sequences , Math. Comp. 60 (1993), no. 202, 827--834.
Show all 13 referencesShow fewer references
  1. J. Bayless, P. Kinlaw and D. Klyve, Sums over primitive sets with a fixed number of prime factors , Math. Comp. 88 (2019), 3063--3077.
  2. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers , Illinois J. Math. 6 (1962), 64--94.
  3. F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic , IEEE Trans. Comput. 66 (2017), 1281--1292.

Version history

  1. v1 · 28 pagesSubmitted by Jaideep PadhiInitial depositCurrentNov 08, 2026