Generalized Artin Primitive Root Conjecture
An asymptotic formula for the number of integers with the primitive root 2, and a generalized Artin primitive root conjecture for composite integers is presented here.
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arXiv subjects
N. A. Carella
Publications and source records attributed to N. A. Carella.
Inequalities For The Primes Counting Function
The prime counting function inequality $π(x+y) < π(x)+π(y)$, which is known as Hardy-Littlewood conjecture, has been established for a variety of cases such as $ δx \leq y \leq x$, where $0< δ\leq 1$, and $x \leq y\leq x \log x \log \log x$ as $ x \to \infty$. The goal in note is to extend the inequality to the new larger ranges $\geq x \log^{-c}x\leq y \leq x$, where $c\geq 0$ is a constant, unconditionally; and for $\geq x^{1/2} \log^3x\leq y \leq x$, conditional on a standard conjecture.
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Irrationality of the Zeta Constants
A general technique for proving the irrationality of the zeta constants $ζ(s)$ 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 $ζ(2n)$, where $n\geq 1$, and $ζ(3)$ are well known, but the results on the irrationality for the zeta constants $ζ(2n+1)$, where $n \geq 2$, are new, and these results seem to confirm that these constants are irrational numbers.
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A Simple Proof Of The Prime Number Theorem
It is shown that the Mean Value Theorem for arithmetic functions, and simple properties of the zeta function are sufficient to assemble proofs of the Prime Number Theorem, and Dirichlet Theorem. These are among the simplest proofs of the asymptotic formulas of the corresponding prime counting functions.
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Results for Wieferich Primes
Let $v\geq 2$ be a fixed integer, and let $x \geq 1$ and $z \geq x$ be large numbers. The exact asymptotic formula for the number of Wieferich primes $p$ such that $ v^{p-1} \equiv 1 \bmod p^2$ in the short interval $[x,x+z]$ is proposed in this note. The search conducted on the last 100 years have produced two primes $p<x=10^{15}$ such that $2^{p-1} -1\equiv 0 \bmod p^2$. The probabilistic and theoretical information within predicts the existence of another base $v=2$ prime on the interval $[10^{15},10^{40}]$. Furthermore, a result for the upper bound on the number of Wieferich primes is used to demontrate that the subset of nonWieferich primes has density 1.
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A Totient Function Inequality
An unconditional inequality of the totient function is contributed to the literature. This result is associated with various problems about the distribution of prime numbers.
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Twin Primes In Quadratic Arithmetic Progressions
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the existence of infinitely many quadratic twin primes $n^{2}+1$ and $n^{2}+3$, $n \geq 1$, are proposed in this note.
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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 $p\geq 2$.
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The Error Term of the Summatory Euler Phi Function
A sharper estimate for the summatory Euler phi function $\sum_{n \leq x} φ(n)$ is presented in this work. It improves the established estimate in the current mathematical literature. In addition, an estimate for its reciprocal $\sum_{n \leq x} 1/φ(n)$ is also determined.
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Asymptotic For Primitive Roots Producing Polynomials
Let $x \geq 1$ be a large number, let $f(x) \in \mathbb{Z}[x]$ be a prime polynomial of degree $\text{deg}(f)=m$, and let $u\ne \pm 1, v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes $p=f(n) \leq x$ with a fixed primitve root $u$ is derived in this note.
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Spectral Methods And Prime Numbers Counting Problems
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof of the more general dePolignac conjecture on the existence of infinitely many primes pairs p and p + 2k, k => 1, is proposed in this note.
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Note On The Maximal Primes Gaps
This note presents a result on the maximal prime gap of the form p_(n+1) - p_n <= C(log p_n)^(1+e), where C > 0 is a constant, for any arbitrarily small real number e > 0, and all sufficiently large integer n > n_0. Equivalently, the result shows that any short interval [x, x + y], y => C(log x)^(1+e), contains prime numbers for all sufficiently large real numbers x => x_0 unconditionally. An application demonstrates that a prime p => x > 2 can be determined in deterministic polynomial time O(log(x)^8).
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Quadratic Primes
The subset of quadratic primes {p = an^2 + bn + c : n => 1} generated by an irreducible polynomial f(x) = ax^2 + bx + c over the integers is widely believed to be an unbounded subset of prime numbers. This note provides the details of a possible proof for some of these quadratic polynomials. In particular, it is shown that the cardinality of the simplest subset of quadratic primes {p = n^2 + 1 : n => 1} is infinite.
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Topics In Primitive Roots
This monograph considers a few topics in the theory of primitive roots g(p) modulo a prime p>=2. A few estimates of the least primitive roots g(p) and the least prime primitive roots g^*(p) modulo p, a large prime, are determined. One of the estimate here seems to sharpen the Burgess estimate g(p) << p^(1/4+e) for arbitrarily small number 3 > 0, to the smaller estimate g(p) <= p^(5/loglog p) uniformly for all large primes p => 2. The expected order of magnitude is g(p) <<(log p)^c, c>1 constant. The corresponding estimates for least prime primitive roots g^*(p) are slightly higher. Anotrher topic deals with an effective lower bound #{p <= x : ord(g)= p-1} >> x/log x for the number of primes p <= x with a fixed primitive root g != -1, b^2 for all large number x >1. The current results in the literature claim the lower bound #{p <= x : ord(g) = p-1} >> x/(log x)^2, and have restrictions on the minimal number of fixed integers to three or more.
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Rapidly Convergent Series of the Divisors Functions
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results of some special values of the sums of divisors functions.
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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.
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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} >> x/log x. The currently known density is #{Tau(n) : n <= x} >> 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.
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Primes Solutions Of Linear Diophantine Equations
Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.
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