Multiple Zeta Values
Investigating the Algebraic and Combinatorial Aspects of Multiple Zeta Values
by Neel Kumari*,
- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659
Volume 3, Issue No. 6, Aug 2012, Pages 0 - 0 (0)
Published by: Ignited Minds Journals
ABSTRACT
Tounderstand the arithmetic nature of special values of the Riemann zetafunction, it has become increasingly clear that multiple zeta values (MZV’s forshort) must be studied. These are defined as follows:
KEYWORD
Multiple Zeta Values, arithmetic nature, special values, Riemann zeta function, MZV
d1 = 1 + 0 = 1. Is this true? Let’s adapt Euler’s technique to evaluate . As noted in the introduction, (1 - x/r1) (1 - x/r2) ... (1 - x/rn) has roots equal to r1,r2,...,rn. When we expand the polynomial, the coeffcient of x is -(1/r1 + 1/r2 + ... + 1/rn). The coefficient of x2 is With this observation, we see from the product expansion that the coefficient of x4 is preciesely. An easy computation shows that It is now clear that this method can be used to evaluate (say). By comparing the coefficient of x2m in our expansion of f(x), we obtain that We could have also evaluated 2,2using the identity but we had opted to the method above to indicate its generalization which allows us to also evaluate. What about This is a bit more difficult and will not come out of out earlier work. In 1998, Borwein, Bradley, Brodhurst and Lisonek showed that. What about With some work, one can show that this is equal to. Thus, we conclude that d4=1 as predicted by Zagier. What about d5? With more work, we can show that This proves that. Zagier conjectures that d5 = 2. In other words, d5 = 2 if and only if is irrational. Can we prove Zagier’s conjecture? To this date, not a single example is known for which. If we write then it is easy to see that Zagier’s conjecture is equivalent to the assertion that dn = Dn for all Deligne and Goncharov and (independently) Terasoma showed that.
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