cohomology over finite fields and

Historically, the polylogarithm has attracted specialists and nonspecialists alike withitslovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.

Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including a longstanding conjec...

Abstract. This is a semi-expository article concerning Langlands functoriality and Deligne’s conjecture on the special values of L-functions. The emphasis is on symmetric power L-functions associated to a holomorphic cusp form. Contents

Abstract The Byzantine Generals Problem requires processes to reach agreement upon a value even though some of them may fad. It is weakened by allowing them to agree upon an "incorrect" value if a failure occurs. The transaction eormmt problem for a distributed database Js a special case

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For each field k, we define a category of rationally decomposed mixed motives with Z-coefficients. When k is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers

s−1 1 1 ∞ x ζ () s = ∑ = dx (Re() s> 1) s 0 x n Γ s e −1 n= 1 s = σ + it (Complex variable)

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the transcendental degrees of the elds