Related papers: Saito duality and the classical theory of arithmet…
We study the properties of different type of transforms by means of operational methods and discuss the relevant interplay with many families of special functions. We consider in particular the binomial transform and its generalizations. A…
We develop a theory of Bernstein-Sato polynomials for meromorphic functions and we use it to study the analytic continuation of Archimedian local zeta functions in this setting. We also introduce both an analytic and an algebraic theory of…
We define Bernstein-Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara-Malgrange type theorem on their geometric monodromies, which would be useful also in relation with the…
We discuss a construction of the Fourier-Sato transform for monodromic mixed Hodge modules.
We discuss the duality theorem, which provides a relation between loop integrals and phase space integrals. We rederive the duality relation for the one-loop case and extend it to two and higher-order loops. We explicitly show its…
Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…
Using the simple properties of Riemman integrable functions, Ramanujan's formula for sum of the square roots of first n natural numbers has been generalized to include r'th roots where r is any real number greater than 1.As an application…
We prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ of…
A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent…
A general action is proposed for the fields of $q$-dimensional differential form over the compact Riemannian manifold of arbitrary dimensions. Mathematical tools are based on the well-known de Rham-Kodaira decomposing theorem on harmonic…
This is a sequel to math.AG/0003009. Here we study identities for the Fourier transform of "elementary functions" over finite field containing "exponents" of monomial rational functions. It turns out that these identities are governed by…
Previously, we introduced a duality transformation for Euler $G$--Frobenius algebras. Using this transformation, we prove that the simple $A,D,E$ singularities and Pham singularities of coprime powers are mirror self--dual where the mirror…
We introduce the notion of the automorphic dual of a matrix algebraic group defined over $Q$. This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic functorial properties of this set are derived and used both to…
Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of Riemannian symmetric spaces of compact…
We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any…
We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan's classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd…
We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato--Tate density. Examples of such sequences come from…
It is shown that there exists a duality among fields. If a field is dual to another field, the solution of the field can be obtained from the dual field by the duality transformation. We give a general result on the dual fields. Different…
The special values of multiple polylogarithms, which including multiple zeta values, appear some fields of mathematics and physics. Many kinds of their linear relations are investigated as well as their algebraic relations. From the…
Let $c_q(n)$ be the Ramanujan sums. Many results concerning Ramanujan-Fourier series $f(n)=\sum_{q=1}^\infty a_q c_q (n)$ are obtained by many mathematicians. In this paper we study series of the form $f(q)=\sum_{n=1}^\infty a_n c_q (n)$,…