Prime Counting Research Articles

On positive sequences of reals whose block sequence has an asymptotic distribution function

In this paper we study the properties of the unbounded sequence $0 < y_1 \le y_2 \le y_3 \le \cdots$ of positive reals having asymptotic distribution function of the form $x^\lambda$. As a consequence, we immediately get information on the asymptotic behavior of the power means of order $r>0$ of function values of some arithmetic functions, e.g., the first $n$ prime numbers or the values of the prime counting function.

Twin, cousin, and sexy prime counting function. Explicit formulas

UDC 511 We give explicit formulas for the twin-prime and cousin-prime counting functions. We propose a formula that computes the number of primes less than or equal to n whose difference is m ⩾ 6. We also present a characterization specifying when two numbers whose difference is n ⩾ 2 are primes.

The Riemann's Hypothesis, the Prime Numbers Theorem (PNT), and the Error

In this simple paper, a small refinement to the Prime Number Theorem (PNT) is proposed, which allows us to limit the error with which said theorem predicts the value of the Prime-counting function π(x); and, in this way, endorse the veracity of the Riemann Hypothesis. Many people know that the Riemann Hypothesis is a difficult mathematical problem - even to understand - without a certain background in mathematics. Many techniques have been used, for more than 150 years, to try to solve it. Among them is the one that establishes that, if the Riemann hypothesis is true, then the error term that appears in the prime number theorem can be bounded in the best possible way. Specifically, Helge von Koch demonstrated in 1901 that it should be:

On Prime Counting Functions Using Odd $K$-Almost Primes

This work takes an interesting diversion, revealing the extraordinary capacity to determine the precise number of primes in a space tripled over another. Exploring the domain of K-almost prime numbers, this paper provides a clear explanation of the complex idea. In addition to outlining the conditions under which odd K-almost prime numbers must exist, it presents a novel method for figuring out how often odd numbers are as 2-almost prime, 3-almost prime, 4-almost prime, and so on, up to a specified limit n. The work goes one step further and offers useful advice on how to use these approaches to precisely calculate the prime counting function, π(n). Essentially, it offers a comprehensive exploration of the mathematical fabric, where primes reveal their mysteries in both large and small spaces.

Sparse sets that satisfy the prime number theorem

For arbitrary real t>1 we examine the set {⌊x/nt⌋:n≤x}. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set.

On certain inequalities for the prime counting function – Part III

As a continuation of [10] and [11], we offer some new inequalities for the prime counting function $\pi (x).$ Particularly, a multiplicative analogue of the Hardy–Littlewood conjecture is provided. Improvements of the converse of Landau's inequality are given. Some results on $\pi (p_n^2)$ are offered, $p_n$ denoting the $n$-th prime number. Results on $\pi (\pi (x))$ are also considered.

New estimates for some integrals of functions defined over primes

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $\pi(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $\zeta(s)$.

Sharper bounds for the Chebyshev function ψ(x)

We improve the unconditional explicit bounds for the error term in the prime counting function ψ(x). In particular, we prove that, for all x>2, we have|ψ(x)−x|<9.22022x(log⁡x)3/2exp⁡(−0.8476836log⁡x), and that, for all x≥exp⁡(3000),|ψ(x)−x|<4.9678⋅10−15x. This compares to results of Platt and Trudgian (2021) who obtained 4.51⋅10−13x. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning π(x) will appear in the follow up work [11].

Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence

In mathematics, the prime counting function $\pi(x)$ is defined as the function yielding the number of primes less than or equal to a given number $x$. In this paper, we prove that the asymptotic limit of a summation operation performed on a unique subsequence of the prime numbers yields the prime number counting function $\pi(x)$ as $x$ approaches $\infty$. We also show that the prime number count $\pi(n)$ can be estimated with a notable degree of accuracy by performing the summation operation on the subsquence up to a limit $n$.

Counting prime numbers in arithmetic sequences

This paper presents the theoretical elements that support the calculation of the prime number counting function, π(x), based on properties of the sequences (6n-1) and (6n+1), n ≥ 1. As a result, Sufficient primality criteria are exposed for the terms of both sequences that support a deterministic computational algorithm that reduces the number of operations in calculating the function π(x) by exonerating all multiples of 3 from the analysis. In analyzing the primality of a particular term, the divisions by factor 3 are excluded. Such an algorithm can be applied to the search for prime numbers in each sequence separately, a possibility that allows approximately half the time of number search in practical applications.

Quantum and classical study of prime numbers, prime gaps and their dynamics

A wave function constructed from prime counting function is employed to study the properties of primes using quantum dynamics. The prime gaps are calculated from the expectation values of position and a formula for maximal gaps is proposed. In an analogous nonlinear system, the trajectories, associated nodes with their stability condition and the bifurcation dynamics are studied using classical dynamics. It is interesting to note that the Lambert W functions appear as a natural solution for the fixed points as functions of energy. The derived potential with the divergence resembles the effective potential experienced by a particle near a massive spherical object in general theory of relativity. The coordinate time and proper time corresponding to a black hole serendipitously find their analogy in the solution of the nonlinear dynamics representing primes. The stereographic projection obtained from quantum dynamics on unit circle in the $$(\theta ,p_{\theta })$$ phase space of the real numbers present along x-axis in general and prime numbers in particular provides a simple way to calculate a formula for upper bounds on the prime gaps. The estimated prime gaps is found to be significantly better than that of Cramer’s predicted values.

Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

Using a recent verification of the Riemann hypothesis up to height 3cdot 10^{12}, we provide strong estimates on pi (x) and other prime counting functions for finite ranges of x. In particular, we get that |pi (x)-li(x)|<sqrt{x}log x/8pi for 2657le xle 1.101cdot 10^{26}. We also provide weaker bounds that hold for a wider range of x, and an application to an inequality of Ramanujan.

A Direct Proof of the Prime Number Theorem using Riemann’s Prime-counting Function

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin’s form: . Instead of performing asymptotic expansion on Chebyshev functions as in conventional analytic methods, this new approach uses contour-integration method to analyze Riemann’s prime counting function J(x), which only differs from π(x) by .

Error estimates for a class of continuous Bonse-type inequalities

Let p n p_n be the n n th prime number. In 2000, Papaitopol proved that the inequality p 1 ⋯ p n &gt; p n + 1 n − π ( n ) p_1\cdots p_n&gt;p_{n+1}^{n-\pi (n)} holds, for all n ≥ 2 n\geq 2 , where π ( x ) \pi (x) is the prime counting function. In 2021, Yang and Liao tried to sharpen this inequality by replacing n − π ( n ) n-\pi (n) by n − π ( n ) + π ( n ) / π ( log ⁡ n ) − 2 π ( π ( n ) ) n-\pi (n)+\pi (n)/\pi (\log n)-2\pi (\pi (n)) , however there is a small mistake in their argument. In this paper, we exploit properties of the logarithm error term in inequalities of the type p 1 ⋯ p n &gt; p n + 1 k ( n , x ) p_1\cdots p_n&gt;p_{n+1}^{k(n,x)} , where k ( n , x ) = n − π ( n ) + π ( n ) / π ( log ⁡ n ) − x π ( π ( n ) ) k(n,x)=n-\pi (n)+\pi (n)/\pi (\log n)-x\pi (\pi (n)) . In particular, we improve Yang and Liao estimate, by showing that the previous inequality at x = 1.4 x=1.4 holds for all n ≥ 21 n\geq 21 .

Inequalities involving $\pi(x)$

We present several inequalities involving the prime-counting function \pi(x) . Here, we give two examples of our results. We show that \frac{16}{9} \pi(x)\pi(y)\leq \pi^2(x+y) is valid for all integers x,y\geq 2 . The constant factor 16/9 is the best possible. The special case x=y leads to \frac{4}{3}\leq \frac{\pi(2x)}{\pi(x)} \quad (x=2,3,\ldots ), where the lower bound 4/3 is sharp. This complements Landau’s well-known inequality \frac{\pi(2x)}{\pi(x)}\leq 2 \quad (x=2,3,\ldots ). Moreover, we prove that the inequality \Bigl(2\frac{\pi(x+y)}{x+y}\Bigr)^s \leq \Bigl( \frac{\pi(x)}{x}\Bigr)^s + \Bigl(\frac{\pi(y)}{y}\Bigr)^s \quad (0&lt; s\in \mathbb{R}) holds for all integers x,y\geq 2 if and only if s\leq s_0=0.94745\ldots. Here, s_0 is the only positive solution of \Bigl( \frac{16}{7}\Bigr)^t -\Bigl( \frac{6}{5}\Bigr)^t=1.

On the average value of

AbstractWe prove that the Riemann hypothesis is equivalent to the condition $\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t&lt;0$ for all $x&gt;2$ . Here, $\pi (t)$ is the prime-counting function and $\operatorname {\textrm {li}}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function $\theta (t)$ and discuss the extent to which one can make related claims unconditionally.

On certain inequalities for the prime counting function – Part II

As a continuation of [6], we deduce some inequalities of a new type for the prime counting function π(x).

A new quantum mechanical pseudo material: Application in cryptography

Recently relativistic Dirac material such as graphene and the topological material as topological insulators have played prominent role in providing some of the materials which incorporate more mathematical and relativistic physical nature in general. Here we suggest a material which is quantum mechanical in nature but is a pseudo material. An interacting quantum many-particle system in one dimension representing the prime numbers on the x-axis through the asymptotic form of the prime counting function is formulated. The effective quantum and classical mechanics of a single particle of the system are studied by deriving the potential function that a single particle experiences in this system. We express our classical Hamiltonian in terms of the angle coordinate to illustrate certain interesting mathematical properties related to the prime numbers. Since it is a mathematical material having prime numbers as the atoms interacting with each other like ordinary mater, this material can have real and important application in cryptography. The elliptic curves that appear in the study of this system are related to primality test and are heavily used in communication cryptography. This analysis opens a new way of looking at materials which shows that simulating a material through a potential is a productive one.

On Ramanujan's prime counting inequality

On Ramanujan's prime counting inequality

Remarks on Ramanujan’s inequality concerning the prime counting function

Abstract In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x 2 ) &lt; e x log x π ( x e ) \pi \left( {{x^2}} \right) &lt; {{ex} \over {\log x}}\pi \left( {{x \over e}} \right) for x sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor x log x {x \over {\log x}} on its right hand side by the factor x log x - h {x \over {\log x - h}} for a given h, and by replacing the numerical factor e by a given positive α. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.