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\f0\fs26 \cf2 Title: Li coefficients for L-functions\
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Abstract:\
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\f1\fs32 \cf2 The Li criterion for the Riemann hypothesis was firstly introduced by Xian-Yin Li in 1997 as a simple positivity criterion for the classical Riemann hypothesis, formulated in terms of the logarithmic derivative of the Riemann zeta function. This criterion was further generalized by Bombieri and Lagarias to a general multiset of complex numbers, and therefore to any L-function.\'a0\
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\cf2 One can further generalize this concept to $tau$-Li coefficients that encode further analytic properties of L-functions. The relevant property of the $tau$-Li coefficients $a_n$ of an L-function is their asymptotic behavior as $n\\to \\infty.$ It has been challenging to compute these coefficients for sufficiently high indices (the range one can handle now is ~15 -- see Mark Coffey's papers).
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\f1\fs32 The goal of this project is to adapt and enhance Dockchitser's L-functions code (that William Stein has implemented in sage already) in order to make it possible to compute high order Li coefficients for as large a class of L-functions as possible.\'a0
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\f1\fs32 \uc0\u8232 Further reading:
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1. T. DoKchitser, Computing special values of motivic L-functions, Exper. Math. 13, No.2 (2004), 137-149\'a0\
2. Documentation for Dokchitser's code ({\field{\*\fldinst{HYPERLINK "http://www.maths.bris.ac.uk/~matyd/computel/index.html"}}{\fldrslt \cf3 \ul \ulc3 http://www.maths.bris.ac.uk/~matyd/computel/index.html}}) and its implementation is sage (I am sure you can find this one)\
3. K. Maslanka, Li's criterion for the Riemann hypothesis \'96 numerical approach, Opusc. Math. 24, No. 1, 103-114 (2004).\
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\f1\fs32 4. X.-J. Li,\'a0The positivity of a sequence of numbers and the Riemann hypothesis,J. Number Theory 65 (1997), 325\'96333.\'a0
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\f1\fs32 5. E. Bombieri and J. C. Lagarias,\'a0Complements to Li\'92s criterion for the Riemann hypothesis,\'a0J. Number Theory 77 (1999), 274\'96287.
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\f1\fs32 \cf2 6. P. Freitas,\'a0A Li-type criterion for zero-free half-planes of Riemann\'92s zeta function,\'a0J. London Math, Soc. 73 (2006), 399-414.\'a0\
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1 and 2 will be used this week\
3 illustrates some of the technical challenges we face\
4-6 are very good (and short!) introduction to Li coefficients. Pay special attention to the formulas in the equivalent definitions of the Li and $\\tau$-Li coefficients}