A seventeenth-order polylogarithm ladder

TL;DR

The paper develops a computational framework for constructing and verifying high-order polylogarithm ladders in self-reciprocal number fields via cyclotomic relations, applying PSLQ-based integer relation detection to log-based constants. It confirms a unique order-$17$ ladder in Lehmer's number field, arising from a cyclotomic relation with index $k=630$, and provides a highly explicit identity expressing $\zeta(17)$ in terms of ${\rm Li}_{17}(\alpha_1^{-k})$, $\pi^{2j}$, and $(\log\alpha_1)^{17-2j}$, validated to tens of thousands of digits. The work then generalizes the approach to ladder construction across reciprocal fields, including a counting framework relating ladder numbers to field signature, and explores ladders in larger Salem-number bases, particularly those connected to graph-theoretic eigenvalues, to assess the maximal ladder length and potential uniqueness of the Lehmer ladder. Overall, the results illuminate the deep connections between Salem numbers, cyclotomic relations, and polylogarithmic constants, and provide a robust method for discovering and validating high-order ladders with significant numerical verification.

Abstract

Cohen, Lewin and Zagier found four ladders that entail the polylogarithms ${\rm Li}_n(α_1^{-k}):=\sum_{r>0}α_1^{-k r}/r^n$ at order $n=16$, with indices $k\le360$, and $α_1$ being the smallest known Salem number, i.e. the larger real root of Lehmer's celebrated polynomial $α^{10}+α^9-α^7-α^6-α^5-α^4-α^3+α+1$, with the smallest known non-trivial Mahler measure. By adjoining the index $k=630$, we generate a fifth ladder at order 16 and a ladder at order 17 that we presume to be unique. This empirical integer relation, between elements of $\{{\rm Li}_{17}(α_1^{-k})\mid0\le k\le630\}$ and $\{π^{2j}(\logα_1)^{17-2j}\mid 0\le j\le8\}$, entails 125 constants, multiplied by integers with nearly 300 digits. It has been checked to more than 59,000 decimal digits. Among the ladders that we found in other number fields, the longest has order 13 and index 294. It is based on $α^{10}-α^6-α^5-α^4+1$, which gives the sole Salem number $α<1.3$ with degree $d<12$ for which $α^{1/2}+α^{-1/2}$ fails to be the largest eigenvalue of the adjacency matrix of a graph.