Related papers: A naive integral

We present a set of lectures on topics of advanced calculus in one real and complex variable with several new results and proofs on the subject, specially with detailed proof-always missing in the literature - of the Cissoti explicitly…

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this article, we express divided differences of $y$ in terms of bivariate divided differences of $g$, generalizing a recent result…

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…

Let $(\Sigma,g)$ be a closed Riemannian surface, $W^{1,2}(\Sigma,g)$ be the usual Sobolev space, $\textbf{G}$ be a finite isometric group acting on $(\Sigma,g)$, and $\mathscr{H}_\textbf{G}$ be a function space including all functions $u\in…

We study inequalities of the form \begin{equation*} \rho ( \lvert \hat{f} \rvert) \leq C \sigma(f) < \infty, \end{equation*} with $f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on $\mathbb{R}^n$ and \begin{equation*}…

We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to…

A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developing…

Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire $L$-function of real archimedian type has a zero in the interval $\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\approx 14.13$ corresponds to the first…

In this paper, given a topological space $X$, an interval $I\subseteq {\bf R}$ and five continuous functions $\varphi, \psi, \omega :X\to {\bf R}$, $\alpha, \beta:I\to {\bf R}$, we are interested in the infimum of the function $\Phi:X\to…

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the…

In the present paper it is shown that real function $g(x)=\int_{0}^{x}f(t)dt$ is a linear-space computable real function on interval $[0,1]$ if $f$ is a linear-space computable $C^2[0,1]$ real function on interval $[0,1]$, and this result…

Let $\lambda$ be a positive number, and let $(x_j:j\in\mathbb Z)\subset\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb R\ni t\mapsto e^{ix_jt}:j\in\mathbb Z\}$ is a…

The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An…

One of the basics of calculus is the following proposition: If $F$ and $G$ are antiderivatives of two real functions $f$ and $g$ resp., then an antiderivative of $\lambda f+g$ is $\lambda F + G$. It may be surprising that such systematic an…

In our last work, we formulate a Fourier transformation on the infinite-dimensional space of functionals. Here we first calculate the Fourier transformation of infinite-dimensional Gaussian distribution $\exp(-\pi…

The Aviles-Giga functional $I_{\epsilon}(u)=\int_{\Omega} \frac{\left|1-\left|\nabla u\right|^2\right|^2}{\epsilon}+\epsilon \left|\nabla^2 u\right|^2 \, dx$ is a well known second order functional that models phenomena from blistering to…

The Bessel-Neumann expansion (of integer order) of a function $g:\mathbb{C}\rightarrow\mathbb{C}$ corresponds to representing $g$ as a linear combination of basis functions $\phi_0,\phi_1,\ldots$, i.e., $g(z)=\sum_{\ell = 0}^\infty w_\ell…

Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function on $M$. It is well known the functional $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8\pi\int_M udv_g-8\pi\log\int_Mhe^{u}dv_g$$ achieves its minimum under…

In 1859 Riemann (1826-1866) published his only paper on number theory. In this eight-page paper he obtained a formula for the number of primes less than or equal to a real number x, and revealed the deep connection between the distribution…

We improve the lower bound for $V(T)$, the number of sign changes of the error term $\psi(x)-x$ in the Prime Number Theorem in the interval $[1,T]$ for large $T$. We show that \[ \liminf_{T\to\infty}\frac{V(T)}{\log…