$\lambda_\infty$: A New Mathematical Constant from the Spectral Theory of the Prime LCM Matrix

Abstract

We introduce a new mathematical constant $\lambda_\infty = 0.674036183193696139936660007576508455780\ldots$ (OEIS A396695), defined as the unique solution in $(1/4,+\infty)$ of $h(x) := \sum_{p \text{ prime}} 1/(xp^2-p+1) = 1$. This equation arises as the $n\to\infty$ limit of the secular equation of rank-$n$ truncations of the infinite prime LCM matrix $\mathcal{L}[p_i,p_j]=1/\mathrm{lcm}(p_i,p_j)$, where $\mathcal{L}=D+vv^T$. Viewed as a compact self-adjoint operator on $\ell^2(\mathcal{P})$, $\mathcal{L}$ has spectral radius $\rho(\mathcal{L})=\lambda_\infty$ (Theorem 5.4). The corresponding integer LCM matrix satisfies $\lambda_{\max}(W_N^*)/N\to\lambda_\infty$ (Theorem 5.8), a prime-indexed counterpart to $\lambda_{\max}(M_N)/N\to\zeta(2)=\pi^2/6$ for the integer divisor matrix. We prove that $h$ is real-analytic, strictly decreasing and strictly convex on $(1/4,+\infty)$, ensuring existence and uniqueness of $\lambda_\infty$. We compute 500 decimal digits of $\lambda_\infty$, certified by rigorous error bounds and independently verified by six computational runs, including a fully independent recomputation (Run E, PARI-GP) and a machine-verifiable Arb interval certificate (Run F) covering 505 digits. Extensive PSLQ and LLL searches find no minimal polynomial of degree $\le 8$ satisfied by $\lambda_\infty$ (at 560 decimal digits), and no integer relation against catalogs of up to 31 classical constants. The arithmetic nature of $\lambda_\infty$ remains open.