[4] **viXra:1212.0155 [pdf]**
*replaced on 2013-07-30 22:27:36*

**Authors:** Khalid M Ibrahim

**Comments:** 62 Pages.

In this paper, we introduced a method to modify the Dirichlet series over the Mobius function by progressively eliminating the numbers that first have a prime factor 2, then 3, then 5, ..up to the prime pr. The properties of the new series are analyzed as pr approaches infinity and its relationship to the function and the partial Euler product is established and then used to examine the validity of the Riemann Hypothesis.

**Category:** Number Theory

[3] **viXra:1212.0055 [pdf]**
*submitted on 2012-12-08 08:19:07*

**Authors:** Chun-Xuan Jiang

**Comments:** 6 Pages.

Using the complex trigonometric function of order n with (n-1) variables,where n is an odd number.We prove FLT for exponents 3p and p,where p is an odd prime .The proof of FLT must be direct.But indirect proof of FLT is disbelieving

**Category:** Number Theory

[2] **viXra:1212.0053 [pdf]**
*submitted on 2012-12-08 08:26:27*

**Authors:** Chun-Xuan Jiang

**Comments:** 4 Pages.

Using the complex trigonometric functions of order 2n with (2n-1) variables,where n is an odd number.We prove FLT for exponents 6p and 2p,where p is an odd prime.The proof of FLT must be direct.But indirect proof of FLT is disbelieving

**Category:** Number Theory

[1] **viXra:1212.0052 [pdf]**
*submitted on 2012-12-08 08:32:20*

**Authors:** Chun-Xuan Jiang

**Comments:** 4 Pages.

Using the complex trigonometric functions of order 4m with (4m-1) variables,where m=1,2,3,...We prove FLT for exponents 12p and 4p,where p is an odd prime.The proof of FLT must direct.But indirect proof of FLT is disbelieving.

**Category:** Number Theory