A seventeenth-order polylogarithm ladder
نویسندگان
چکیده
Cohen, Lewin and Zagier found four ladders that entail the polylogarithms Lin(α −k 1 ) := ∑ r>0 α −kr 1 /r n at order n = 16, with indices k ≤ 360, and α1 being the smallest known Salem number, i.e. the larger real root of Lehmer’s celebrated polynomial α + α − α − α − α − α − α + 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 {Li17(α 1 ) | 0 ≤ k ≤ 630} and {π(logα1) | 0 ≤ j ≤ 8}, 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 − α − α − α + 1, which gives the sole Salem number α < 1.3 with degree d < 12 for which α + α fails to be the largest eigenvalue of the adjacency matrix of a graph. a) This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC03-76SF00098. b) Lawrence Berkeley Laboratory, MS 50B-2239, Berkeley, CA 94720, USA [email protected] c) Open University, Department of Physics, Milton Keynes MK7 6AA, UK [email protected]
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Cohen, Lewin and Zagier found four ladders that entail the polylogarithms Lin( k 1 ) := P r>0 kr 1 =r n at order n = 16, with indices k 360, and 1 being the smallest known Salem number, i.e. the larger real root of Lehmer's celebrated polynomial 10 + 9 7 6 5 4 3 + + 1, with the smallest known non-trivial Mahler measure. By adjoining the index k = 630, we generate a fth ladder at order 16 and a ...
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