We study a local multiplicity problem related to so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan L-packet and are related to local exterior square L-functions.

8(1):187-187

The study of the present paper is inspired by Gangl, Kaneko and Zagier's result of the connection with double zeta values and modular forms. We introduce double Eisenstein series $E_{r,s}$ in positive characteristic with double zeta values $\zeta_A(r,s)$ as their constant term and compute the t-expansions of the double Eisenstein series. Moreover, we derive the shuffle relations of double Eisenstein series which match the shuffle relations of double zeta values in \cite{Huei2015}.

The estimate of integral points in an n-dimensional polyhedron has many applications in singularity theory, number theory and toric geometry. The third named author formulated a conjecture (i.e., Yau Number Theoretic Conjecture) which gives a sharp polynomial upper estimate on the number of positive integral points in n-dimensional (n  3) real right-angled simplices. The previous results on the conjecture in low dimension cases (n  6) have been proved by using the sharp GLY conjecture. However, it is only valid in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new method in [22]. In this paper, on the one hand, the similar method has been applied to prove the conjecture for n = 8, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in seven-dimensional simplex over the cross sections of eight-dimensional simplex. This is a signicant progress since it sheds light on proving the Yau Number Theoretic Conjecture in full generality. On the other hand, we give a new sharper estimate of the Dickman-De Bruijn function (x; y) for 5  y < 23, compared with the result obtained by Ennola.

In the classical theory of multiple zeta values (MZV's), Furusho proposed a conjecture asserting that the p-adic MZV's satisfy the same Q-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho's conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations.