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The Exact Second Order Term In Ramanujan's Bound For σ(n)/n\sigma(n)/n

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Jacob Crim ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

Let σ(n)\sigma(n) be the sum of the divisors of nn, and assume the Riemann hypothesis. Ramanujan showed that σ(n)/n≤eγ(log⁡log⁡n−2(2−1)/log⁡n+S1(log⁡n)+O(1/(log⁡nlog⁡log⁡n)))\sigma(n)/n\le e^\gamma(\log\log n-2(\sqrt2-1)/\sqrt{\log n}+S_1(\log n)+O(1/(\sqrt{\log n}\log\log n))), where S1S_1 is a sum over the zeros of ζ\zeta. Nicolas made this effective, placed the coefficient of 1/(log⁡nlog⁡log⁡n)1/(\sqrt{\log n}\log\log n) in [0.9567,3.789][0.9567, 3.789] at colossally abundant numbers, and asked for its exact asymptotic value. We show that this coefficient is κ+W2(log⁡n)\kappa+W_2(\log n) with κ=2(2+log⁡2)−2=1.80868…\kappa=\sqrt2(2+\log2)-2=1.80868\ldots, where W2(x)=−∑ρxiℑρ/(1−ρ)2W_2(x)=-\sum_\rho x^{i\Im\rho}/(1-\rho)^2 oscillates in [−β,β][-\beta,\beta], β=2+γ−log⁡4π\beta=2+\gamma-\log4\pi. The next coefficient is the constant c0=8−2(8+4log⁡2+34log⁡22)=−7.7443…c_0=8-\sqrt2(8+4\log2+\frac34\log^2 2)=-7.7443\ldots, up to an oscillation of amplitude at most 0.00660.0066. In particular, the rescaled residual studied numerically by Leshchiner, Musin and Thompson does not converge, and it exceeds 3/23/2 at every colossally abundant number with parameter ξ≥4.4⋅1015\xi\ge4.4\cdot10^{15}; this answers their Questions 3 and 4, the second in the negative. We also prove explicit two-sided bounds for the remainder at every colossally abundant number above Nicolas's threshold N(0)N^{(0)}, where log⁡N(0)≈109\log N^{(0)}\approx10^9. This part is computer-assisted: rigorous interval arithmetic covers every real parameter ξ∈[109,e300]\xi\in[10^9,e^{300}], and an analytic argument covers ξ>e300\xi>e^{300}. As corollaries, σ(n)/n≤eγ(log⁡log⁡n−0.699/log⁡n)\sigma(n)/n\le e^\gamma(\log\log n-0.699/\sqrt{\log n}) for all n≥N(0)n\ge N^{(0)}, improving Nicolas's constant 0.582; Ramanujan's limsup is approached from above at every colossally abundant number with ξ≥109\xi\ge10^9; and an odd-integer analogue holds with an essentially sharp constant. The proof reads colossally abundant numbers as ground states of a free gas of prime modes, in which κ\kappa splits into a prime-square term, a saturation-deficit term and a shell-energy term. All enclosures are checked against exact values computed from a sieve to 101010^{10}.

Provenance statement

This paper was produced with substantial assistance from Claude, an AI model by Anthropic. I was investigating a separate research question involving Robin's inequality, when Claude remarked that it was possible to fill in small gap in literature related to it. Claude drafted the text, derived the explicit enclosures, wrote and ran all verification code (a segmented sieve to 10^10, the computation of zeta zeros, the interval-arithmetic covering of the parameter range, and the symbolic checks), and carried out the literature checks against the cited sources. A separate instance of the same model, working only from the written appendix and the code, independently re-derived the decomposition, checked every enclosure and compared the code with the appendix line by line; its corrections are incorporated. No human referee has checked the proofs. The contributor (Jacob Crim) takes responsibility for the submission. All code, the per-cell output and the validation logs are included as ancillary files so that every computational claim can be reproduced.

References

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  1. L. C. Washington and A. Yang, Analogues of the Robin--Lagarias criteria for the Riemann hypothesis , arXiv:2008.04787.arXiv

Version history

  1. v1 · 14 pagesSubmitted by Jacob CrimRevised after moderator-requested changesCurrentNov 10, 2026