Stability aspects of arithmetic functions

Arithmetic Aspects of Atomic Structures

Stability and Arithmetic

Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for such studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using semistable parabolic bundles & for p-adic number fields using what we call semi-stable filtered (φ,N;ω)-modules; and non-abelian zeta functions for function fields ove...

Neural Computation of Arithmetic Functions

The basic processing unit of a neural network i s a linear threshold element. I t has been known that neural networks can be much more powerful than traditional logic circuits, assuming that each threshold element can be built at a cost comparable to that o f AND, OR, Nor logic elements. Whereas any logic circuit o f polynomial size (in n) that computes the product of two n-bit numbers requires...

Properties of rational arithmetic functions

Rational arithmetic functions are arithmetic functions of the form g1 ∗···∗ gr ∗ h−1 1 ∗ ···∗h−1 s , where gi, hj are completely multiplicative functions and ∗ denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type ide...

Arithmetic aspects of the theta correspondence

We review some recent results on the arithmetic of the theta correspondence for certain symplectic-orthogonal dual pairs and some applications to periods and congruences of modular forms.