An Equivalence Reduction of the Riemann Hypothesis via Log-Heat Kernel Spectral Measure Theory (Chinese and English Versions)

本文给出了黎曼猜想(RH)的一个精确双向等价归约。我们修正了既往版本中因使用不完整模变换而导致的积分核渐近失配,通过引入对数坐标u=logx将问题置于热核空间。我们构造了Riemann Ξ 函数余弦积分核的精确Stieltjes 积分表示,证明了存在唯一的谱测度dµ(λ) 使得 Φ(u)= ∞0 dµ(λ)u2+λ2。该测度的非负性等价于一个显式 Dirichlet级数的虚部非负性。利用Jacobi 三重积恒等式,我们将该条件转化为无穷乘积的辐角分析,最终将RH等价地归约为一个具体的无穷振荡级数不等式(见第5节猜想5.1)。本文不声称证明RH,而是提供一个逻辑上闭合的等价归约框架:若该不等式成立,则RH 成立;反之,RH也蕴含该不等式。全部推导均为双向等价,且核心步骤可Lean4形式化验证(参见第7节的设计草案说明)。 This paper presents a precise bidirectional equivalence reduction of the Riemann Hypothesis (RH). We correct the asymptotic mismatch of the integral kernel caused by theuse of an incomplete modular transformation in previous versions, by introducing thelogarithmic coordinate u = log x and placing the problem in the heat-kernel space. Weconstruct an exact Stieltjes integral representation for the cosine integral kernel of theRiemann Ξ function, and prove the existence of a unique spectral measure dµ(λ) suchthat Φ(u) = ∞0 dµ(λ)u2+λ2. The non-negativity of this measure is shown to be equivalent tothe non-negativity of the imaginary part of an explicit Dirichlet series. Using the Jacobitriple product identity, we transform this criterion into an argument analysis of an infiniteproduct, finally reducing RH equivalently to a concrete oscillatory series inequality (seeConjecture 5.1). This paper does not claim to prove RH; rather, it provides a logicallyclosed equivalence reduction: if the inequality holds, then RH holds; conversely, RH implies the inequality. All implications are bidirectional, and the core steps are amenableto Lean4 formal verification (see the design draft in Section 7)