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.
An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial
We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.
An explicit Mertens' type inequality for arithmetic progressions.
An explicit seven cube theorem
An explicit upper bound for the number of solutions of the S-unit equation.
An explicit version of Birch's Theorem
An explicit version of Faltings' Product Theorem and an improvement of Roth's lemma
An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers
A generalization of the well-known Fibonacci sequence given by F₀ = 0, F₁ = 1 and for all n ≥ 0 is the k-generalized Fibonacci sequence whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This generalizes...
An exponential polynomial formed with the Legendre symbol
An extension of a result of C. J. Smyth to polynomials in several variables
An extension of a theorem of Duffin and Schaeffer in Diophantine approximation
Duffin and Schaeffer have generalized the classical theorem of Khintchine in metric Diophantine approximation in the case of any error function under the assumption that all the rational approximants are irreducible. This result is extended to the case where the numerators and the denominators of the rational approximants are related by a congruential constraint stronger than coprimality.
An extension of a theorem of Euler
An extension of a theorem of Nielsen
An extension of Bourgain and Garaev's sum-product estimates
An extension of Daykin's generalized Möbius function to unitary divisors.