Related papers: Computational Number Theory in Relation with L-Fun…

We cover some useful techniques in computational aspects of analytic number theory, with specific emphasis on ideas relevant to the evaluation of L-functions. These techniques overlap considerably with basic methods from analytic number…

We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating…

We address the problem of evaluating an $L$-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that is possible to evaluate the $L$-function more…

In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…

Let us consider a cyclic extension of a function field defined over a finite field. For a character (non-trivial) of this extension, we calculate, as a linear combinations of products of Jacobi sums, the coefficients of the polynomial given…

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…

We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…

We briefly review numerical methods for calculations beyond one loop and then describe new developments within the method of sector decomposition in more detail. We also discuss applications to two-loop integrals involving several mass…

We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.

We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors…

We give a quick survey of the various fixed point theorems in computability theory, partial combinatory algebra, and the theory of numberings, as well as generalizations based on those. We also point out several open problems connected to…

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

This work describes numerical methods that are useful in many areas: examples include statistical modelling (bioinformatics, computational biology), theoretical physics, and even pure mathematics. The methods are primarily useful for the…

Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is…

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier…

This article is the second of a series of three presenting an alternative method to compute the one-loop scalar integrals. It extends the results of the first article to general complex masses. Let us remind the main features enjoyed by…

We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…

Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.

We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of…