Power spectrum of the fluctuation of Chebyshev's prime counting function

The Error Term In The Primes Counting Function

This article considers the error term of the primes counting function. It applies some recent results on the densities of prime numbers in short intervals to derive an improvement of the error term from subexponential size to fractional exponential size. The corresponding equivalent results for the zeros of the zeta function and Mertens conjecture are also discussed.

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.

切比雪夫多项式与素数处局部残余/非残余结构的细化

Chebyshev polynomials and a refinement of the local residue/non-residue structure at a prime

本文提出切比雪夫版本的素性检验准则，通过两个二次特征划分素数局部结构，扩展了传统数论中的多项式性质。

Counting Primes Rationally And Irrationally

The recent technique for estimating lower bounds of the prime counting function $π(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $μ(ζ(s)) \geq 2$ of special values of the zeta function claims that $π(x) \gg \log \log x/\log \log \log x$. This note improves the lower bound to $π(x) \gg \log x$, and extends the analysis to the irrationality measures $μ(ζ(s)) \geq 1$ for rational ratios of zeta functions.

An Exact Formula for the Prime Counting Function

This paper discusses a few main topics in Number Theory, such as the Möbius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $π(x)$, and the prime-power counting function, $J(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which implies the Riemann hypothesis, and enabled the creation of a formula for $π(x)$ in the first place), and the realization that sums of divisors and the Möbius function are particular cases of a more general concept. F…

Primes In Cubic Arithmetic Progressions

This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes counting function is established.

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.

整数中的所有信息都蕴含于素数之中

All the information in the integers is in the primes

本文将切比雪夫关于素数加权的渐近公式重新解释为最大熵下典型整数的信息内容，并结合柴廷的算法随机性定理将其推广到算法信息论，给出典型整数算法信息的上界。

New generating and counting Functions of prime numbers applied to approximate Chebyschev 2nd class function and the least action principle applied to find non-trivial roots of the Zeta function and to Riemann Hypothesis

We introduce a new set of prime numbers functions including an exact Generating Function and a Discriminating Function of Prime Numbers neither based on prime number tables nor on algorithms. Instead these functions are defined in terms of ordinary elementary functions, therefore having the advantage of being analytic and readily calculable. Also presented are four applications of our new Prime Numbers Generating Function, namely: obtaining a new analytic formula for counting prime numbers, obtaining an approximant to Euler product function, obtaining an approximant to Riemann Zeta (sigma,tau)…

The Hardy-Littlewood conjectures on the twin primes and the binary Goldbach problem are true

A celebrated conjecture of Hardy and Littlewood provides with an asymptotic formula for the counting function of the twin primes. We give an unconditional proof of such a formula by means of a finite Ramanujan expansion of the counting function expressed in terms of the von Mangoldt function and its incomplete form. In a completely analogous way, we solve the conjugate conjecture on the representations of any even integer as the sum of two prime numbers.

A density-based approach for non-heuristic approximations of prime counting functions

All the known approximations of the number of primes pi(n) not exceeding any given integer n are derived from real-valued functions that are asymptotic to pi(x), such as x/log x, Li(x) and Riemann's function R(x). The degree of approximation for finite values of n is determined only heuristically, by conjecturing upon an error term in the asymptotic relation that can be seen to yield a closer approximation than others to the actual values of pi(n) within a finite range of values of n. By considering the density of each of the set of (i) all integers n, (ii) Dirichlect integers n = a+md, and (i…