Ramanujan's formulas for L-functions

Explicit Formulas for Dirichlet and Hecke L-functions

In 1997, the author proved that the Riemann hypothesis holds if and only if λn = ∑ [1−(1−1/ρ)n] > 0 for all positive integers n, where the sum is over all complex zeros of the Riemann zeta function. In 1999, E. Bombieri and J. Lagarias generalized this result and obtained a remarkable general theorem about the location of zeros. They also gave an arithmetic interpretation for the numbers λn. In...

Optimal Stochastic Quadrature Formulas For Convex Functions

We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deterministic methods we prove that adaptive Monte Carlo methods are much better. Abstract. We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deter-ministic ...

Some Asymptotic Formulas for Multiplicative Functions

Generating Kummer Type Formulas for Hypergeometric Functions

Kummer type formulas are identities of hypergeometric series. A symmetry by the permutations of n-letters yields these formulas. We will present an algorithmic method to derive known and new Kummer type formulas. The algorithm utilizes several algorithms in algebra and geometry for generating Kummer type formulas.

Summation formulas for Schur functions involving hyperdeterminants

We give general integral formulas involving for hyperdeterminants or hyperpfaffians. In the applications, we obtain several summation formulas for the products of Schur functions, which are generalizations of Cauchy’s determinant. Further, we study Toeplitz hyperdeterminants by using the theory of Jack polynomials and give a hyperdeterminant version of a strong Szegö limit theorem. MSC-class: p...