Closed-Form Expressions for Odd-Order Riemann Zeta Functions and Their Rigorous Proofs (Part II)

This paper, as the second part of a series, systematically develops multiple closed-form expressions for odd-order Riemann zeta functions ζ(2m + 1) based on the fundamental theorem established in the first part. We first establish families of integral closed-form expressions, sine integral closed-form expressions, polynomial-sine integral closed-form expressions, and polylog arithmic closed-form expressions. Subsequently, we explore in depth the profound connections between ζ(2m + 1) and Catalan’s constant G and the Dirichlet beta function β(s), providing elegant expressions in terms of β(2k+1). Finally, we propose and partially prove the “Rational Combination Conjecture”, demonstrating that under normalization, ζ(2m+1) can be expressed as the ratio of two polynomials with bounded degrees (≤ 4), with the optimal degree being 4.All results are accompanied by rigorous mathematical derivations and numerical validation. We strive to maintain mathematical rigor while showcasing the inherent beauty and practical value of the theory.