The Constraint System for the Nonexistence of Odd Perfect Numbers: A Unified Framework Based on Euler Structure, 2-adic Valuation, Modular Form Zero Density, and Recursive Circle Method Frequency Compression

Millennia's Strongest Bound! The Constraint System for the Nonexistence of Odd Perfect Numbers: A Unified Framework Based on Euler Structure, 2-adic Valuation, Modular Form Zero Density, and Recursive Circle Method Frequency Compression Author: Qin Zitai ORCID: 0009-0004-5467-0074 Date: 30 June 2026 Abstract This paper establishes a complete constraint system for odd perfect numbers, proceeding from Euler's structure and combining 2-adic valuation analysis, analytic sieve methods, modular form zero-density theory, and recursive five-layer circle method frequency compression to successively compress the possible range of odd perfect numbers. Main Results: 1. Structural Constraint: If n is an odd perfect number, then n = p^α m^2, p ≡ α ≡ 1 (mod 4), gcd(p,m) = 1, and p^α | σ(m^2). This is a new structural constraint derived for the first time in this paper. 2. Distribution Constraint**: The natural density of odd perfect numbers is zero: P(N) << N (log log N)^100 / log N, and lim_{N→∞} P(N)/N = 0. 3. Compression Bound: P(N) << N^{1/2}. This is the strongest unconditional bound on odd perfect numbers since Euler. 4. Low-Frequency Compression: Via five-layer circle method frequency decomposition: P(N^{2/3}) << N^{2/3} / log N, and P(N) << N^{2/3}/log N + N^{1/2} ~ N^{1/2}. 5. Recursive Frequency Compression: Via secondary five-layer circle method decomposition, further revealing the fine structure of the bound: P(N) << N^{1/6} + N^{1/3} + N^{1/2} ~ N^{1/2}. All conclusions are unconditional and established within the ZFC axiom system. Keywords: Odd perfect numbers; Euler structure; 2-adic valuation; natural density; modular form zero density; circle method; frequency compression References [1] Apostol, T. M. Introduction to Analytic Number Theory. Springer, 1976. MR0434929. [2] Deligne, P. Formes modulaires et représentations l-adiques. Séminaire Bourbaki, 1969/70. MR0272127. [3] Iwaniec, H. and Sarnak, P. The non-vanishing of central values of automorphic L-functions. Israel Journal of Mathematics, 120: 155–177, 2000. MR1815374. [4] Euler, L. De numeris amicabilibus. Opuscula varii argumenti, 2: 1–47, 1849. [5] Nielsen, P. P. Odd perfect numbers have at least 101 prime factors. Mathematics of Computation, 84(295): 2485–2500, 2015. MR3377054. [6] Ochem, P. and Rao, M. Odd perfect numbers are greater than 10^1500. Mathematics of Computation, 81(279): 1869–1877, 2012. MR2904596. [7] Sarnak, P. Some Applications of Modular Forms. Cambridge University Press, 1990. MR1102671. [8] Selberg, A. Note on a paper by L. G. Sathe. Journal of the Indian Mathematical Society, 18: 83–87, 1954. MR0067387. [9] Sathé, L. G. On a problem of Hardy and Ramanujan. Journal of the Indian Mathematical Society, 18: 73–82, 1954. MR0067386.