Related papers: Theta operators, refined Delta conjectures, and co…

For $f_1,...,f_r\in \mathbb C[z_1,...,z_n]\setminus \mathbb C$, we introduce the variation of archimedean zeta function. As an application, we show that the $n/d$-conjecture, proposed by Budur, Musta\c{t}\u{a}, and Teitler, holds for…

Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly-graded S_n module can be expressed using the Frobenius characteristic map as \nabla e_n, where e_n is…

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators"…

This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes).…

Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let…

Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential…

We consider the Toeplitz operators on the weighted Bergman spaces over the unit ball $\mathbb{B}^n$ and their analytic continuation. We proved the commutativity of the $C^*-$algebras generated by the analytic continuation of Toeplitz…

A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $\Xi$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives…

In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, \emph{Derivation and double shuffle relations for multiple zeta…

We apply differential operators to modular forms on orthogonal groups $\mathrm{O}(2, \ell)$ to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are…

We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms…

We study families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their fundamental connections to combinatorial q-series and automorphic forms. In particular, we show that the generating functions…

In this paper we prove a new version of Kransoselskii's fixed-point theorem under a ($\psi, \theta, \varphi$)-weak contraction condition. The theoretical result is applied to prove the existence of a solution of the following fractional…

Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbol\Theta_n$-contraction is a commuting tuple of operators on a Hilbert space having…

A convex lattice polygon Delta determines a pair (S,L) of a toric surface together with an ample toric line bundle on S. The Severi degree N^{Delta,delta} is the number of delta-nodal curves in the complete linear system |L| passing through…

The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves.…

By finding orthogonal representation for a family of simple connected called $\delta$-graphs it is possible to show that $\delta$-graphs satisfy delta conjecture. An extension of the argument to graphs of the form…

Let $R$ be a commutative $\mathbb{Z}[1/p]$-algebra, let $m \leq n$ be positive integers, and let $G_n=\text{GL}_n(F)$ and $G_m=\text{GL}_m(F)$ where $F$ is a $p$-adic field. The Weil representation is the smooth $R[G_n\times G_m]$-module…

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $ \exp (C |s|^\delta) $ where $ \delta $…

By applying the inverse Mellin transform to some simple closed form identities, a number of relationships are established that connect integrals containing Riemann's and Hurwitz' zeta functions ($\zeta(s)$ and $\zeta(s,a)$) and their…