Related papers: The Digamma function, Euler-Lehmer constants and t…
We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erd\H os-Hooley $\Delta$-function, we derive lower bounds for the…
We present an abstract functional analytic formulation of the celebrated $\dive$-$\curl$ lemma found by F.~Murat and L.~Tartar. The viewpoint in this note relies on sequences for operators in Hilbert spaces. Hence, we draw the functional…
We offer some new applications of an extension of Abel's lemma, as well as its more general form established by Andrews and Freitas. A nice connection is established between this lemma and series involving the Riemann zeta function.
In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator $H^\sigma=(-\Delta+|x|^2)^\sigma$, $0<\sigma<1$. More precisely, a new class of smooth functions $C^{k,\alpha}_H$ is defined, in which we study…
We prove two transformations for the $p$-adic hypergeometric functions which can be described as $p$-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them…
This paper concerns elliptic systems of $p$-Laplace type with complex valued coefficient and source term. We extend the real valued theory of the elliptic $p$-Laplace equation to the complex valued case. We establish the existence and…
Euler's Gamma function $\Gamma$ either increases or decreases on intervals between two consequtive critical points. The inverse of $\Gamma$ on intervals of increase is shown to have an extension to a Pick-function and similar results are…
This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that…
Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and…
The aim of this paper is to show further results following those published in [5], and to relate the Riemann zeta function to the relativistic cosmology.
We introduce a one-parameter family of series associated to the Riemann $\zeta$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the…
The question was asked: Is it possible to express the function \begin{equation} \tag{1.1} h(a)\equiv\,{_4F_3}(a,a,a,a;2a,a+1,a+1;1) \label{question} \end{equation} in closed form? After considerable analysis, the answer appears to be "no",…
Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+\delta\leq\sigma<1$, fixed $\delta\in(0,1/2)$ and for functions in…
Some properties of $m$-density points and density-degree functions are studied. Moreover the following main results are provided: \vskip2mm \begin{itemize} \item {\it Let $\lambda$ be a continuous differential form of degree $h$ in…
This thesis contains some new results on the uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sums
We present analytic properties and extensions of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal…
We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of…
We prove the $\boldsymbol{p}$-adic duality theorem for the finite star-multiple polylogarithms. That is a generalization of Hoffman's duality theorem for the finite multiple zeta-star values.
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
We study $p$-adic Euler's series $E_p(t) = \sum_{k=0}^{\infty}k!t^k$ at a point $p^a$, $a \in \mathbb{Z}_{\ge 1}$, and use Pad\'e approximations to prove a lower bound for the $p$-adic absolute value of the expression $cE_p\left(\pm…