[7] **viXra:1712.0539 [pdf]**
*submitted on 2017-12-20 06:47:39*

**Authors:** Martin Nicholson

**Comments:** 8 Pages.

We study several integrals that contain the infinite product ${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation $\displaystyle\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$ and to some other formulas.

**Category:** Functions and Analysis

[6] **viXra:1712.0519 [pdf]**
*submitted on 2017-12-19 19:49:52*

**Authors:** Seong Won Cha

**Comments:** 11 Pages.

We show that some interesting properties of the bilateral Laplace transform of even and positive functions both on the line z=x+iy0 and on a circle. We also show the Riemann hypothesis is true using these properties.

**Category:** Functions and Analysis

[5] **viXra:1712.0478 [pdf]**
*submitted on 2017-12-15 08:30:49*

**Authors:** Martin Nicholson

**Comments:** 10 Pages.

Several Fourier transformations of functions of one and two variables are evaluated and
then used to derive some integral and series identities. It is shown that certain two-
dimensional Mordell integrals factorize into product of two integrals and that the square
of the absolute value of the Mordell integral can be reduced to a single one-dimensional
integral. Some connections to elliptic functions and lattice sums are discussed.

**Category:** Functions and Analysis

[4] **viXra:1712.0463 [pdf]**
*submitted on 2017-12-16 01:01:52*

**Authors:** Carl Wigert, Quincy-Howard Xavier

**Comments:** 1 Page.

In this paper, we define very small numbers and very very small numbers and use them to construct derivatives as ratios of real numbers. We then use that result to rigorously prove that the chain rule treats derivatives as fractions being multiplied.

**Category:** Functions and Analysis

[3] **viXra:1712.0355 [pdf]**
*submitted on 2017-12-08 19:58:12*

**Authors:** Seong Won Cha

**Comments:** 9 Pages.

This is a brief report before writing a full paper.
We proved the Riemann hypothesis using the properties of the bilateral Laplace transform.

**Category:** Functions and Analysis

[2] **viXra:1712.0113 [pdf]**
*submitted on 2017-12-04 21:50:14*

**Authors:** D Williams

**Comments:** 8 Pages.

An overview of some types of multiplicative infinitesimal calculi is given. Analogs of standard results ("Simpson's" Product, "Maclurin's" Product, fundamental theorems, etc) are shown. An area that deserves more attention.

**Category:** Functions and Analysis

[1] **viXra:1712.0019 [pdf]**
*submitted on 2017-12-02 12:52:22*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

It is showed that a large class of functions defined by integrals verify the Riemann Hypothesis.

**Category:** Functions and Analysis