Reciprocal Mersenne Subseries

Averaging antecedents.

Both averaging proofs in this section end by choosing a term no larger than a finite mean. Duverney and Tachiya choose their index in the same way in their refinement of the Chowla–Erdős method: they average a local coefficient mass over an arithmetic progression built by the Chinese remainder theorem, and a term attaining the minimum supplies the bounds that control the tail beyond it (Section 2, (2.3)–(2.9), pp. 5–6 of the author preprint). Kaneko, Suzuki and Tachiya measure the average decay of scaled tails by $R_c(q,x,z)$, their (1.7) on p. 3 of arXiv v1. Under a counting condition on the support and a bound on $R_c$, most scaled tails in a range are small, and arbitrarily small nonzero scaled tails give irrationality (Lemmas 1 and 2, pp. 6–8). Here sparsity concerns the positions of nonzero power-series coefficients, not the set of selected Lambert denominators. The divisor-incidence coefficients $\mathrm{\#}\{a\in A:a\mid n\}$ are positive on every multiple of $minA$. For any splitting into two sequences the two coefficient supports therefore cover a set of positive lower density, which is incompatible with the support-counting conditions [6] requiring both counts to be $o(x_n/z_n)$ with $z_n\ge 1$. Their remote-tail quantity $R_c(q,x,z)$ sums only offsets $j\ge z$, whereas the full scaled tail

is at least $X_A(b)$, which gives no lower bound for $R_c$. The proofs above average the displacement instead, along multiples of a modulus that freezes a finite part of $A$; the weighted proof adds the dyadic average (6).