十七阶多对数阶梯
A seventeenth-order polylogarithm ladder
AI总结：
本文通过加入指标630，在已知四个16阶多对数阶梯基础上，构造了唯一的17阶阶梯，并验证了其涉及125个常数、近300位整数系数的整数关系，精度超过59000位小数。
摘要：
Cohen、Lewin和Zagier发现了四个涉及多对数${\rm Li}_n(α_1^{-k}):=\sum_{r>0}α_1^{-k r}/r^n$的阶梯，阶数$n=16$，指标$k\le360$，其中$α_1$是已知最小的Salem数，即Lehmer著名多项式$α^{10}+α^9-α^7-α^6-α^5-α^4-α^3+α+1$的较大实根，该多项式具有已知最小的非平凡Mahler测度。通过加入指标$k=630$，我们生成了阶数16的第五个阶梯以及阶数17的一个阶梯，我们推测后者是唯一的。这个经验整数关系，涉及集合$\{{\rm Li}_{17}(α_1^{-k})\mid0\le k\le630\}$与$\{π^{2j}(\logα_1)^{17-2j}\mid 0\le j\le8\}$的元素，包含125个常数，乘以具有近300位数字的整数。该关系已通过超过59,000位小数的验证。在我们于其他数域中发现的阶梯中，最长的阶数为13，指标为294。它基于$α^{10}-α^6-α^5-α^4+1$，该多项式给出了唯一的Salem数$α<1.3$，次数$d<12$，且对于该数，$α^{1/2}+α^{-1/2}$不是图的邻接矩阵的最大特征值。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
