[5] **viXra:1207.0088 [pdf]**
*submitted on 2012-07-24 13:05:32*

**Authors:** S. Maiti

**Comments:** 4 Pages.

In order to find a differential representation of the $n$th partial
sum $\displaystyle \sum_{i=1}^{n}\frac{1}{a+(i-1)d}$ of the general
harmonic series $\displaystyle
\sum_{i=1}^{\infty}\frac{1}{a+(i-1)d}$, a theoretical study has been
performed analytically. Moreover, some special cases of it such as
harmonic number have been discussed.

**Category:** Number Theory

[4] **viXra:1207.0087 [pdf]**
*submitted on 2012-07-24 13:15:05*

**Authors:** S. Maiti

**Comments:** 5 Pages.

The paper is focused to find important relations and identities over
some summations for natural numbers such as $\displaystyle
\sum_{\substack{i,j=1\\(i<j)}}^{n}ij,~\sum_{\substack{i,j,k=1\\(i<j<k)}}^{n}ijk,~\sum_{\substack{i,j,k,l=1
\\(i<j<k<l)}}^{n}ijkl,~\cdots $. These relations are believed to
find applications in the various branches of number theory
particularly in the proposed theorems of Maiti
\cite{Maiti1,Maiti2,Maiti3,Maiti4,Maiti5} which help to represent the
factorial $n!$ entirely new way and also help to exhibit the $n$th
partial sum of the general harmonic series $\displaystyle
\sum_{n=1}^{\infty} \frac{1}{a+(n-1) b}$ and its particular
cases.

**Category:** Number Theory

[3] **viXra:1207.0084 [pdf]**
*replaced on 2012-07-30 16:45:30*

**Authors:** Salvatore Gerard Micheal

**Comments:** 9 Pages. author's email: micheal@msu.edu

Seven related papers develop a novel approach toward elucidating irrational density from a non-standard perspective. The notion of countability is explored in a precise new way. This new definition of countability clearly relates irrational to rational density and sets the stage for a more accessible understanding of the reals. Also, two implications in set theory are discovered. Constructive evaluation, criticism, and collaboration is invited.

**Category:** Number Theory

[2] **viXra:1207.0083 [pdf]**
*submitted on 2012-07-23 15:12:20*

**Authors:** S Maiti

**Comments:** 4 Pages.

New Expression of the factorial of $n$ ($n!$, $n\in N$) is given in
this article. The general expression of it has been proved with help
of the Principle of Mathematical Induction. It is found in the form
\begin{equation}
1+\sum_{i=1}^{n}a_i
+\sum_{\substack{i,j=1 \\(i<j)}}^{n}a_ia_j +
\sum_{\substack{i,j,k=1 \\(i<j<k)}}^{n}a_ia_ja_k +\cdots
+a_1a_2\cdots a_{n},
\label{factorial_expression}
\end{equation}
where $a_i=i-1$ for $i=1,~2,~\cdots ,~n$. More convenient expression
of this form is provided in Appendix.

**Category:** Number Theory

[1] **viXra:1207.0071 [pdf]**
*replaced on 2013-05-18 21:53:37*

**Authors:** Russell Letkeman

**Comments:** 30 Pages. Submitted to Cambridge University Sigma

The sequence of sets of Z_n on multiplication where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A natural reason to study multiplication on these boundaries is a construction exists which evolves these sets from one primorial boundary to the next, via the sieve of Eratosthenes, giving us Just In Time prime sieving. To this we add a parallel study of gap sets of various lengths and their evolution all of which together informs what we call the S model.
We show by construction there exists for each prime number P a local finite probability distribution and it is surprisingly well behaved. That is we show the vacuum; ie the gaps, has deep structure.
We use this framework to prove conjectured distributional properties of the prime numbers by Legendre, Hardy and Littlewood and others. We also demonstrate a novel proof of the Green-Tao theorem. Furthermore we prove the Riemann hypothesis and show the results are perhaps surprising.
We go on to use the S model to predict novel structure within the prime gaps which leads to a new Chebyshev type bias we honorifically name the Chebyshev gap bias. We also probe deeper behavior of the distribution of prime numbers via ultra long scale oscillations about the scale of numbers known as Skewes numbers.

**Category:** Number Theory