Related papers: A fast amortized algorithm for computing quadratic…

We study the fourth moment of quadratic Dirichlet $L$-functions at $s= \frac{1}{2}$. We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally. The proofs of these results…

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit…

This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…

We propose the first linear-time algorithm to compute the conjugate of (nonconvex) bivariate piecewise linear-quadratic (PLQ) functions (bivariate quadratic functions defined on a polyhedral subdivision). Our algorithm starts with computing…

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As…

In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the…

In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010)…

We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the…

We establish unconditional sharp upper bounds of the $k$-th moments of the family of quadratic Dirichlet $L$-functions at the central point for $0 \leq k \leq 2$.

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet $L$-functions, using the method of double Dirichlet series. Quantitative non-vanishing result for these…

A sequential quadratic optimization algorithm for minimizing an objective function defined by an expectation subject to nonlinear inequality and equality constraints is proposed, analyzed, and tested. The context of interest is when it is…

Let $L$ be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of H\"ormander, we show that the Dirichlet problem associated with $L$ has a unique smooth…

We present an algorithm which computes the $D^{th}$ term of a sequence satisfying a linear recurrence relation of order $d$ over a field $K$ in $O( \mathsf{M}(\bar d)\log(D) + \mathsf{M}(d)\log(d))$ operations in $K$, where $\bar d \leq d$…

We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.

The Dirichlet divisor problem is used as a model to give a conjecture concerning the conditional convergence of the Dirichlet series of an L-function.

In this article, we investigate conditional large values of quadratic Dirichlet character sums with multiplicative coefficients. We prove some Omega results under the assumption of the generalized Riemann hypothesis.

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of…

Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large…