An Explicit Formula For The Divisor Function

The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and circle problem.Comment: Twenty Eight Pages. Keywords: Divisor Function, Explicit Formula, Divisor Proble

Note On Prime Gaps And Very Short Intervals

Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <= c1((logx)^2)/loglogx, c1 > 0 constant. Equivalently, it shows that the very short intervals (x, x + y] contain prime numbers for all y > c2((logx)^2)/loglogx, c2 > 0 constant, and sufficiently large x > 0.Comment: 12 Pages, 1 Table, Improve

The Error Term of the Summatory Euler Phi Function

A sharper estimate for the summatory Euler phi function $\sum_{n \leq x} \varphi(n)$ is presented in this work. It improves the established estimate in the current mathematical literature. In addition, an estimate for its reciprocal $\sum_{n \leq x} 1/\varphi(n)$ is also determined.Comment: Sixteen Pages. Keywords: Euler Totient Function, Error Term, Arithmetic Functio

Least Prime Primitive Roots

This note presents an upper bound for the least prime primitive roots $g^*(p)$ modulo $p$, a large prime. The current literature has several estimates of the least prime primitive root $g^*(p)$ modulo a prime $p\geq 2$ such as $g^*(p)\ll p^c, c>2.8$. The estimate provided within seems to sharpen this estimate to the smaller estimate $g^*(p)\ll p^{5/\log \log p}$ uniformly for all large primes $p\geq 2$.Comment: Twelve Pages. Keyword: Prime number; Primitive root; Least primitive root; Prime primitive root; Cyclic group. arXiv admin note: text overlap with arXiv:1405.016

Generalized Fibonacci Primitive Roots

This note generalizes the Fibonacci primitive roots to the set of integers. An asymptotic formula for counting the number of integers with such primitive root is introduced here.Comment: Twelve Pages. Keywords: Primitive root; Fibonacci primitive root; Costas array. arXiv admin note: substantial text overlap with arXiv:1504.00843, arXiv:1405.016

Complexity of Computing Quadratic Nonresidues

This note provides new methods for constructing quadratic nonresidues in finite fields of characteristic p. It will be shown that there is an effective deterministic polynomial time algorithm for constructing quadratic nonresidues in finite fields.Comment: References and Improvement

Note on the Tau Function

This note proposes an improved estimate of the coefficient t(n) of the discriminant modular form using elementary method. It improves a well known estimate of the tau function t(n) by Deligne.Comment: This paper has been withdraw

Irrationality of the Zeta Constants

A general technique for proving the irrationality of the zeta constants $\zeta(s)$ for odd $s = 2n + 1 \geq 3$ from the known irrationality of the beta constants $L(2n+1)$ is developed in this note. The results on the irrationality of the zeta constants $\zeta(2n)$, where $n\geq 1$, and $\zeta(3)$ are well known, but the results on the irrationality for the zeta constants $\zeta(2n+1)$, where $n \geq 2$, are new, and these results seem to confirm that these constants are irrational numbers.Comment: Twenty Seven Pages. Keyword: Irrational number, Transcendental number, Beta constant, Zeta constant, Uniform distributio

Deterministic Integer Factorization Algorithms

This note presents a deterministic integer factorization algorithm of running time complexity O(N^(1/6+e)), e > 0. This improves the current performances of deterministic integer factorization algorithms rated at O(N^(1/4+e)) arithmetic operations. Equivalently, given the least (log N)/6 bits of a factor of N = pq, the algorithm factors the integer in polynomial time O(log(N)^c), c > 0 constant.Comment: Six Pages, Improved Version. arXiv admin note: substantial text overlap with arXiv:1003.326

Density of the Values Set of the Tau Function

It is shown that the density of the values set {Tau(n): n <= x} of the nth coefficients Tau(n) of the discriminant function Delta(z), a cusp form of level N = 1 and weight k = 12, has the lower bound #{Tau(n): n > x/log x. The currently known density is #{Tau(n) : n > x^(1/2+o(1)), and the expected density is #{Tau(n) : n <= x} ~ x. The solutions set of the equation Tau(p) = 0 for all primes p => 2, which arises as a singular case of this analysis, is discussed within.Comment: Eleven Pages. Keywords: Fourier Coefficient, Modular Form, Cusp Form, Tau Function, Lange-Trotter Conjectur