We obtain the last of standard Kuznetsov formulas for $SL(3,\Bbb{Z})$. In previous paper, we were able to exploit relationship between positive-sign Bessel function and Whittaker apply Wallach's expansion; now demonstrate expansion functions into all four signs, generalizing theorem $SL(3)$. As applications, again consider Kloosterman zeta smooth sums sums. The new pose same difficulties as saw...

We give a new geometric construction of the big projective module in the principal block of the BGG category O, or rather the corresponding D-module on the flag variety. Namely, given a one-parameter family of nondegenerate additive characters of the unipotent radical of a Borel subgroup which degenerate to the trivial character, there is a corresponding one-parameter family of Whittaker sheave...

Despite much impressive work (e.g. [1], [3], [4], [5], [7]) since the introduction of the Shannon capacity in [8], many natural questions regarding this parameter remain widely open (see [2], [6] for surveys). Let G1 + G2 denote the disjoint union of the graphs G1 and G2. It is easy to see that c(G1 + G2) ≥ c(G1) + c(G2). Shannon [8] conjectured that c(G1 + G2) = c(G1) + c(G2), but this was dis...

11 The Zagier–Broadhurst formula Theorem 41. For any n ≥ 1, ζ ï¿¿ {3, 1} n ï¿¿ = 1 2 2n ζ ï¿¿ {4} n ï¿¿ .

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results...

Complex analysis texts written in the early 1900’s, mostly by British authors such as E. T. Copson [2] and E. T. Whittaker and G. N. Watson [4], devote considerable space in later chapters to what we call special functions. These include the gamma, the hypergeometric, the elliptic, the modular, and the zeta functions. Paul Halmos [3] wrote an interesting discussion illustrating the neglect of t...

Generalizing work of Gross–Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to quadratic spaces over an arbitrary totally real base field. As a special case, we obtain an explicit formula for the n...