Yuri Heymann
The prime-counting function $π(x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $π(x)$ with time complexity $\mathcal{O}(x^{1/2})$ without the need to introduce the non-trivial zeros of the Riemann zeta function.
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Approximating the Prime Counting Function via an Operation on a Unique Prime Number Subsequence
该研究揭示了素数计数函数与对唯一素数子序列进行的运算之间的关联，为素数计数函数的逼近提供了新的研究方向。
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the subset of Germain primes PG = {p and 2p + 1 are primes}, and the subset of quadratic primes Pf = {p = n^2 + 1 primes}. These subsets of primes are widely believed to be infinite subsets of prime numbers.
We present a function that tests for primality, factorizes composites and builds a closed form expression of $π(n^2)$ in terms of $\sum_{3 \leq p \leq n} \frac{1}{p}$ and a weaker version of $ω(n)$.
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.
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.
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.
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…
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.
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.
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)…
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.
The one-sided power spectrum of the fluctuation of Chebyshev's weighted prime counting function is numerically estimated based on samples of the fluctuating function of different sizes. The power spectrum is also estimated analytically for large frequency based on Riemann hypothesis and the exact formula for the fluctuating function in terms of all the non-trivial Riemann zeroes. Our analytical estimate is consistent with our numerical estimate of a 1/f^2 power spectrum.
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…
This short paper presents an exact formula for counting twin prime pairs less than or equal to x in terms of the classical Smarandache Function. An extension of the formula to count prime pairs (p, p+2n), n > 1, is also given.
I define Goldbach counting function with N > 0 and square-free P > 0. Decomposition of this function is discovered and deduction formula is found. I propose a hypothesis on upper bound of Goldbach counting function and prove that Goldbach conjecture is true under this hypothesis. Also, I propose a second hypothesis on upper bound of Goldbach counting function and prove that the twin prime conjecture is true under this hypothesis
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.
We study pairs of consecutive odd numbers through a straightforward indexing. We focus in particular on twin primes and their distribution. With a counting argument, we calculate the limit of an alternating sum that is equal to 1 which means there are few twin primes. Finally, we show how to find the possible congruences for these prime numbers.
We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter $λ_x$ and the estimation of the number of twin primes obtained as a result, is called a fundamental structure of the distribution of twin primes. Combining the latter with an asymptotic bound of $λ_x$, establishes venues, conducive to a discovery of a partial result that can be considered as a suitable variant of the prime number theorem. Furthermore, we obtain an asymptotic bound of the number of twin primes less than $x$.
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