Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$

Abstract

Let $p_1, \dotsc, p_k$ be primes not exceeding $x$ ($k \geqslant 2$), and define the additive Mertens sum \[ S_k(x) = \sum_{p_1 \leqslant x} \cdots \sum_{p_k \leqslant x} \frac{1}{p_1 + \dotsm + p_k}. \] In contrast to Tenenbaum's generalized (multiplicative) Mertens sum, whose leading term has order $(\log \log x)^k$, the sum $S_k(x)$ has leading term of order $x^{k-1}/\log^k x$. We establish the complete asymptotic expansion \[ S_k(x) = \frac{x^{k-1}}{\log^k x} \sum_{n=0}^{N} \frac{E_{k,n}}{\log^n x} + O\left(\frac{x^{k-1}}{\log^{k+N+1} x}\right) \quad (\forall\, N \geqslant 0), \] where the coefficients are given by absolutely convergent multiple logarithmic integrals \[ E_{k,n} = (-1)^n \int_{(0,1]^k} \frac{h_n(\log t_1, \dotsc, \log t_k)}{t_1 + \dotsm + t_k}\, \mathrm{d}\mathbf{t}, \] with $h_n$ the complete homogeneous symmetric polynomial of degree $n$. We give closed-form expressions for the first two coefficients $E_{k,0}$ and $E_{k,1}$ for all $k$, and obtain the closed form for the diagonal part of the third coefficient $E_{k,2}$ (with $E_{k,2}$ fully explicit for $k \leqslant 4$); consequently, the first three terms of the expansions of $S_2(x)$ and $S_3(x)$ are fully explicit. For $k = 2$, we further obtain a closed-form expression for the entire sequence $\{E_{2,n}\}_{n \geqslant 0}$, whose values are explicit $\mathbb{Q}$-linear combinations of $\log 2$ and zeta values $\zeta(j)$. The proofs rely on the real-variable form of the prime number theorem, variable rescaling, and multivariate Taylor remainder estimates.