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Author: William A. Stein
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Yves explained to me the following idea, which he attributes to Sunseri.
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In his project, he explicitly computes a power series a_0+a_1T+a_2T^2+...
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representing the "component" of the 37-adic zeta function with the zero.
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By "explicitly computes", he means that he considered the first 1000 or so
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a_i, and computed the first 1000 terms of each of their 37-adic
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expansions. He then computes the zero using Hensel's lemma. However, he
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says that if he had not bothered computing the a_i explicitly, and just
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used the formula for L_p(chi,s) in Washington's book on Cyclotomic
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Fields to evaluate L_p(T) (and its derivative) for certain *explicit* T
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in Z_p, he would still have had enough data to use Hensel's lemma to
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find the root, and that the computation would have been an order of
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magnitude faster. Hence he estimates that one could easily compute
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zeros in this way, to thousands and thousands of significant figures,
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nowadays (Aug 2001). Yves tells me that this is what Sunseri did
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originally, but his work was not published either.
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