An estimate for the number of reducible Bessel polynomials of bounded degree
An "exact" formula for the 2n-th Bernoulli number
An "exact" formula for the m-th Bernoulli number
An example for Gelfand's theory of commutative Banach algebras
An example in Beurling's theory of generalised primes
We prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes. The example has its generalised Chebyshev function given by [x]-1, and associated zeta function ζ₀(s) given via , where ζ is Riemann’s zeta function. We study the behaviour of the corresponding Beurling integer counting function N(x), producing O- and Ω- results for the ’error’ term. These are strongly influenced by the size of ζ(s) near...
An example in Beurling's theory of primes
An example of an arithmetic Fuchsian group.
An example of Fourier transforms of orbital integrals and their endoscopic transfer.
An exercise on Fibonacci representations
We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.
An explicit 2nth reciprocity law.
An explicit algebraic family of genus-one curves violating the Hasse principle
We prove that for any , the curvein is a genus curve violating the Hasse principle. An explicit Weierstrass model for its jacobian is given. The Shafarevich-Tate group of each contains a subgroup isomorphic to .
An explicit bound for Iwasawa's λ-invariant
An explicit Brauer formula for local Galois characters.
An explicit computation of -stabilized vectors
In this paper, we give a concrete method to compute -stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over -adic fields. An application to the global setting is also discussed. In particular, we give an explicit -stabilized form of a Saito-Kurokawa lift.
An explicit description of the set of all normal bases generators of a finite field
An explicit estimate for zeros of Dedekind zeta functions
An explicit evaluation of the Gosper sum.
An explicit factorisation of the zeta functions of Dwork hypersurfaces
An explicit formula for the Hilbert symbol of a formal group
A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ()-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ()-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the...
An explicit formula for the Mahler measure of a family of -variable polynomials
An explicit formula for the Mahler measure of the -variable Laurent polynomial is given, in terms of dilogarithms and trilogarithms.