CoCalc Public Fileswww / talks / bernoulli / buzzard-project_improve.txt
Author: William A. Stein
1Yves explained to me the following idea, which he attributes to Sunseri.
2In his project, he explicitly computes a power series a_0+a_1T+a_2T^2+...
3representing the "component" of the 37-adic zeta function with the zero.
4By "explicitly computes", he means that he considered the first 1000 or so
5a_i, and computed the first 1000 terms of each of their 37-adic
6expansions. He then computes the zero using Hensel's lemma. However, he
7says that if he had not bothered computing the a_i explicitly, and just
8used the formula for L_p(chi,s) in Washington's book on Cyclotomic
9Fields to evaluate L_p(T) (and its derivative) for certain *explicit* T
10in Z_p, he would still have had enough data to use Hensel's lemma to
11find the root, and that the computation would have been an order of
12magnitude faster. Hence he estimates that one could easily compute
13zeros in this way, to thousands and thousands of significant figures,
14nowadays (Aug 2001). Yves tells me that this is what Sunseri did
15originally, but his work was not published either.
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