arXiv subjects
Publications and source records attributed to Cezar Lupu.
A famous theorem of Zudilin states that at least one of the Riemann zeta values $ζ(5), ζ(7), ζ(9), ζ(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result, it is proved that for any prime number $p \geqslant 5$ there exists an odd integer $i$ in the interval $[3,p+p/\log p+5]$ such that the $p$-adic zeta value $ζ_p(i)$ is irrational.
In this note, we give an exact formula for a general family of rational zeta series involving the coefficient $ζ(2n)$ in terms of Hurwitz zeta values. This formula generalizes two formulas from a previous paper of the first author. Our method will involve derivatives polynomials for the cotangent function.
In this paper, we give elementary proofs of Zagier's formula for multiple zeta values involving Hoffman element and its odd variant due to Murakami. Zagier's formula was a key ingredient in the proof of Hoffman's conjecture. Moreover, using the same approach, we prove Murakami's formula for multiple $t$-values. This formula is essential in proving a Brown type result which asserts that each multiple zeta value is a $\mathbb{Q}$-linear combination of multiple $t$-values of the same weight involving $2$'s and $3$'s.
In this paper, we give an elementary account into Zagier's formula for multiple zeta values involving Hoffman elements. Our approach allows us to obtain direct proof in a special case via rational zeta series involving the coefficient $ζ(2n)$. This formula plays an important role in proving Hoffman's conjecture which asserts that every multiple zeta value of weight $k$ can be expressed as a $\mathbb{Q}$-linear combinations of multiple zeta values of the same weight involving $2$'s and $3$'s. Also, using a similar hypergeometric argument via rational zeta series, we produce a new Zagier-type formula for the multiple special Hurwitz zeta values.
In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $ζ(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new representations of Apery's constant $ζ(3)$. In particular cases, we recover some well-known series representations of $π$.
Using the KKM technique, we establish some existence results for variational-hemivariational inequalities involving monotone set valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive several sufficient conditions for the existence of solutions in the case of unbounded subsets.
In this paper we shall prove a sharpened version of the Finsler-Hadwiger inequality which is a strong generalization of Weitzenbock inequality. After that we give another refinement of this inequality and in the final part we provide some basic applications.
In this note we shall give a new proof to a quadrature formulae due to Newton.