Complete Asymptotic Expansion of the Additive Mertens Sum Sk(x)
Abstract
Let p1, dotsc, pk be primes not exceeding x (k geqslant 2), and define the additive Mertens sum Sk(x) = ∑p1 leqslant x ⋯ ∑pk leqslant x 1/(p1 + dotsm + pk). In contrast to Tenenbaum's generalized (multiplicative) Mertens sum, whose leading term has order (log log x)k, the sum Sk(x) has leading term of order xk−1/logk x. We establish the complete asymptotic expansion Sk(x) = (xk−1)/(logk x) ∑n=0N Ek,n/(logn x) + Ok,N((xk−1)/(logk+N+1 x)) (∀ N geqslant 0), where the coefficients are given by absolutely convergent multiple logarithmic integrals Ek,n = (−1)n ∫(0,1]k (hn(log t1, dotsc, log tk))/(t1 + dotsm + tk) dt, with hn the complete homogeneous symmetric polynomial of degree n. We derive closed-form expressions for the first two coefficients Ek,0 and Ek,1 for all k, and obtain the closed form for the diagonal part of the third coefficient Ek,2 (with Ek,2 fully explicit for k leqslant 7), together with a uniform closed form for the coefficients of the new constants G(−1/m), where G(z)=∫0z Li2(t)/(1+t) dt, whose genuinely new instances first occur for k geqslant 5 (Proposition 5.4); consequently, the first three terms of the expansions of S2(x) and S3(x) are fully explicit. For k = 2, we further obtain a closed-form expression for the entire sequence {E2,n}n geqslant 0, whose values are explicit ℚ-linear combinations of log 2 and zeta values ζ(j). The proofs rely on the real-variable form of the prime number theorem, variable rescaling, and multivariate Taylor remainder estimates.
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BibTeX
@misc{peng2026complete,
title = {{Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$}},
author = {Daoyi Peng and Hao Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04366},
eprint = {2607.04366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04366},
}
RIS
TY - UNPB
TI - Complete Asymptotic Expansion of the Additive Mertens Sum S_k(x)
AU - Daoyi Peng
AU - Hao Liu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2607.04366
UR - https://arxiv.org/abs/2607.04366
AB - Let p_1, dotsc, p_k be primes not exceeding x (k geqslant 2), and define the additive Mertens sum S_k(x) = ∑_p_1 leqslant x ⋯ ∑_p_k leqslant x 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) = (x^k−1)/(log^k x) ∑_n=0^N E_k,n/(log^n x) + O_k,N((x^k−1)/(log^k+N+1 x)) (∀ N geqslant 0), where the coefficients are given by absolutely convergent multiple logarithmic integrals E_k,n = (−1)^n ∫_(0,1]^k (h_n(log t_1, dotsc, log t_k))/(t_1 + dotsm + t_k) dt, with h_n the complete homogeneous symmetric polynomial of degree n. We derive 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 7), together with a uniform closed form for the coefficients of the new constants G(−1/m), where G(z)=∫_0^z Li_2(t)/(1+t) dt, whose genuinely new instances first occur for k geqslant 5 (Proposition 5.4); 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 ℚ-linear combinations of log 2 and zeta values ζ(j). The proofs rely on the real-variable form of the prime number theorem, variable rescaling, and multivariate Taylor remainder estimates.
ER -