Mehdi Hassani
In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log n}\right\rfloor\leqπ\big((n+1)^2\big)-π(n^2). $$
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The theorem below gives another way of computing the distribution prime counting function without using recursion and the values of Prime numbers
Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other are prime number indexed basis entities taken from the discrete or continuous Fourier basis.
In this paper we establish a number of new estimates concerning the prime counting function π(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in small intervals.
We provide an elementary proof of an asymptotic formula for prime counting functions. As a minor application we give a new reduction of the proof of Chebotarëv's density theorem to the cyclic case.
In this paper, a new formula for π^(2)(N) is formulated, it is a function that counts the number of semi-primes not exceeding a given number N. A semi-prime is a natural number that is the product of precisely two prime numbers, the two primes in the product may equal each other. Semi-prime numbers are also a case of almost primes. Since a formula for this is already known, a new identity that uses the prime counting function is created by equating the two functions.
We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these approximations are at least as good as the logarithmic integral approximation. As a corollary, we provide some reformulations of the Riemann hypothesis in terms of the prime counting function and the harmonic numbers.
In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin's form: $$ π(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ 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 $\mathcal O(\sqrt x/\log x)$.
In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $π(x)$ holds for every $x \geq N_{\mathcal{R}}$.
In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $π(x)$, which improve the current best estimates for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.
We prove some properties of the sequence $\{a_n\}_{n\ge1}$ defined by $a_n=π(n)-π\bigl(\textstyle\sum_{k=1}^{n-1}a_k\bigr).$
We generalize a formula on the counting of prime geodesics, due to Kuznetsov-Bykovskii, used in the work of Soundararajan-Young on the prime geodesic theorem. The method works over any number field and for any congruence subgroup. We give explicit computation in the cases of principal and Hecke subgroups.
We introduce the sequence $(a_n) \subset (0,1]$ and prove that the asymptotic behaviour of $\sum_{k=1}^n a_k$ is the same than $π(n)$, the prime-counting function. We also obtain that $π(n) \sim n a_n$ and we estimate $\frac{1}{a_n}-\frac{n}{π(n)}$ showing that $\lim_{n \rightarrow \infty} \frac{1}{a_n}-\frac{n}{π(n)}$ is convergent.
A sharp asymptotic formula for the sum of reciprocals of $π(n)$ is derived, where $π(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivić and L. Panaitopol.
The aim of our present work here is to present few results in the theory of Mellin transforms using the method that S. Ramanujan used in proving his Master Theorem. Further applications of our results for some number-theoretic functions such as the prime counting function and the zeta function are established.
Each number field has an associated finite abelian group, the class group, that records certain properties of arithmetic within the ring of integers of the field. The class group is well-studied, yet also still mysterious. A central conjecture of Brumer and Silverman states that for each prime $\ell$, every number field has the property that its class group has very few elements of order $\ell$, where "very few" is measured relative to the absolute discriminant of the field. This paper surveys recent progress toward this conjecture, and outlines its close connections to counting prime numbers,…
The error term in counting prime pairs
本研究建立了Hardy-Littlewood素数对计数猜想公式误差项与von Mangoldt函数构造的素数指数和$L^1$范数的关联，为数论中素数对分布的误差分析提供了新的联系视角。
In this paper we use a theorem first proved by S.W.Golomb and a famous inequality by J.B. Rosser and L.Schoenfeld in order to prove that there exists an exact formula for $π(n)$ which holds infinitely often.
In this paper we give a new semiprimality test and we construct a new formula for $π^{(2)}(N)$, the function that counts the number of semiprimes not exceeding a given number $N$. We also present new formulas to identify the $n^{th}$ semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots of $N : P_{1}, \; P_{2}....P_{π\left( \sqrt[3]{N}\right) }\leq \sqrt[3]{N}$.
We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower bound for $E(Ω,g,x)$.
As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann's seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann's theory with one that is significantly simpler.
Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $π(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that $|π(x)-\text{li}(x)|<\sqrt{x}\log x/8π$ for $2657\leq x\leq 1.101\cdot 10^{26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.
We note that the inequalities $0.92 \frac{x}{\log(x)} <π(x)< 1.11 \frac{x}{\log(x)}$ do not hold for all $x\ge 30$, contrary to some references. These estimates on $π(x)$ came up recently in papers on algebraic number theory.
Let $E$ be an elliptic curve defined over rational field $\mathbb{Q}$ and $N$ be a positive integer. Now, $M_E(N)$ denotes the number of primes $p$, such that the group $E_p(\mathbb{F}_p)$ is of order $N$. We show that $M_E(N)$ follows Poisson distribution when an average is taken over a large class of curves.
We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
Exact summatory functions that count the number of prime $k$-tuples up to some cut-off integer are presented. Related summatory $k$-tuple analogs of the first and second Chebyshev functions are then defined. Using a gamma distribution hypothesis for prime powers, associated average summatory functions are conjectured. With exact and average summatory functions in hand, pertinent $k$-tuple zeta functions can be identified, and Perron's formula allows the formulation of $k$-tuple analogs of explicit formulae. The $k$-tuple zeta functions are then used to make some inferences about $k$-tuple prim…
On arithmetic terms expressing the prime-counting function and the n-th prime
本研究提出首个固定长度的初等闭式算术项，可分别表示素数计数函数$\pi(n)$与第$n$个素数$p(n)$，仅用有限固定次数的初等算术运算，为素数是否存在次序的基础问题提供构造性解答。
Let $r\ge k\ge 2$ be fixed positive integers. Let $\varrho_{r,k}$ denote the characteristic function of the set of $r$-tuples of positive integers with $k$-wise relatively prime components, that is any $k$ of them are relatively prime. We use the convolution method to establish an asymptotic formula for the sum $\sum_{n_1,\ldots,n_r\le x} \varrho_{r,k}(n_1,\ldots,n_r)$ by elementary arguments. Our result improves the error term obtained by J. Hu (2013).
We prove several asymptotic continued fraction expansions of $π(x)$, $Π(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $π(x)$ is the prime counting function, $Π(x) = \sum_{k = 1}^\infty \frac{1}{k}π(\sqrt[k]{x})$ is the Riemann prime counting function, and $\operatorname{Ri}(x) = \sum_{k=1}^\infty \frac{ μ(k)}{k} \operatorname{li}(\sqrt[k]{x})$ is Riemann's approximation to the prime counting function. We also determine asymptotic continued fraction expansions of the function $\sum_{p \leq x} p^s$ for all $s \in \mathbb{C}$ with $\operatorname{Re}(s) > -1$, a…
We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In this paper, we study the common index divisor $I(\mathbb K_m)$ and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.
Counting relatively prime pairs of palindromes
本文针对任意基数下固定奇数长度的回文数对，给出了互素回文数对数量的渐近公式，解决了 Banks 和 Shparlinski 提出的开放问题。
Taking $r>0$, let $π_{2r}(x)$ denote the number of prime pairs $(p, p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $π_{2r}(x)\sim 2C_{2r} {\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders $ω_{2r}(x)=π_{2r}(x)- 2C_{2r} {\rm li}_2(x)$ that corresponds to Riemann's formula for $π(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $ω_{2r}(x)$ which is supported by numerical results.
Zhang has shown there are infinitely many intervals of bounded length containing two primes. It appears that the current techniques cannot prove that there are infinitely many intervals of bounded length containing three primes, even if strong conjectures such as the Elliott-Halberstam conjecture are assumed. We show that there are infinitely many intervals of length at most $10^8$ which contain two primes and a number with at most 31 prime factors.
Prime number theorem asserts that (at large $x$) the prime counting function $π(x)$ is approximately the logarithmic integral $\mbox{li}(x)$. In the intermediate range, Riemann prime counting function $\mbox{Ri}^{(N)}(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{Li}(x^{1/n})$ deviates from $π(x)$ by the asymptotically vanishing sum $\sum_ρ\mbox{Ri}(x^ρ)$ depending on the critical zeros $ρ$ of the Riemann zeta function $ζ(s)$. We find a fit $π(x)\approx \mbox{Ri}^{(3)}[ψ(x)]$ [with three to four new exact digits compared to $\mbox{li}(x)$] by making use of the Von Mangoldt explicit formula for the Cheby…
It is known that all modular forms on SL_2(Z) can be expressed as a rational function in eta(z), eta(2z) and eta(4z). By utilizing known theorems, and calculating the order of vanishing, we can compute the eta-quotients for a given level. Using this count, knowing how many eta-quotients are linearly independent and using the dimension formula, we can figure out a subspace spanned by the eta-quotients. In this paper, we primarily focus on the case where N=p a prime.
We compute the first and second moments of the divisor-counting function for the Euler-Poincaré characteristic and the trace of Frobenius for the reductions modulo $p$ of a rank 2 Drinfeld module with nontrivial endomorphism ring, as the prime $p$ varies over the primes of ordinary reduction of the Drinfeld module. From these moments we derive the normal order of the number of prime divisors of these invariants.
The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.
Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that \begin{align}π(x)=Θ(x)+O\bigg(\frac{1}{\log x}\bigg), \nonumber \end{align}where \begin{align}Θ(x)=\frac{θ(x)}{\log x}+\frac{x}{2\log x}-\frac{1}{4}-\frac{\log 2}{\log x}\sum \limits_{\substack{n\leq x\\Ω(n)=k\\k\geq 2\\2\not| n}} \frac{\log (\frac{x}{n})}{\log 2}.\nonumber \end{align}This is an improvement to the estimate \begin{align}π(x)=\frac{θ(x)}{\log x}+O\bigg(\frac{x}{\log^2 x}\bigg)\nonumber \end{align}found in the literature.
We prove the analog of Cramér's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.
Asymptotic counting of integers with prime factors $p_{r^a s^b}$
本文给出具有形如 $p_{r^a s^b}$ 的素数因子的整数的渐近计数公式，误差为 $o(1)$，并解决了 $(2,3)$ 情形下序数编码的计数问题。
Counting and equidistribution over primes in hyperbolic groups
本文研究上半平面中双曲格点角度在素数条件下的等分布问题，扩展Friedlander和Iwaniec的工作，证明完全模群在素数矩阵下等分布成立（在假设下界条件下），并证明特定算术余紧群在素数限制下无条件等分布。
Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.
We define the notion of a weighted discriminant and corresponding counting function for number fields, and what it means for these counting functions to have a mass formula for a set of primes. We extend a result of Kedlaya to show that any proper counting function for a finite group $Γ$ has a mass formula for the set of primes not dividing $|Γ|$. We also prove that if $Γ$ is an $\ell$-group for some prime $\ell$, then there are only finitely many weighted discriminant counting functions for $Γ$-extensions of $\Q$ that have a mass formula for all primes. Finally, we enumerate all such counting…
Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any given length.
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+ε}p_n)$ where $ε> 0$, in the context of the results we have obtained in our paper.
We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.
We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates. Possessing average counting functions along with their corresponding exact counting functions allows to implicitly define pertinent $k$-tuple zeta functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple analogs of explicit formulae. If the zeros of the (implicitly defined) $k$-tuple zeta can be determined, the explicit formulae should yield…
Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones previously known.
Let $p_n$ is the $n$-th prime. With help of the Cramér-like model, we prove that the set of intervals of the form $(2p_n,\enskip2p_{n+1})$ containing at list 3 primes has a positive density with respect to the set of all intervals of such form.
Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.
In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.
An incremental approximation to the prime-counting function via polygonal numbers
提出一种基于广义三角数结构化非均匀划分的素数计数函数π(x)近似算法，具有O(1)均摊更新复杂度和O(√x)总复杂度，通过数值实验修正项达到与经典解析近似相当的精度。
Goldston, Pintz and Yıldırım have shown that if the primes have `level of distribution' $θ$ for some $θ>1/2$ then there exists a constant $C(θ)$, such that there are infinitely many integers $n$ for which the interval $[n,n+C(θ)]$ contains two primes. We show under the same assumption that for any integer $k\ge 1$ there exists constants $D(θ,k)$ and $r(θ,k)$, such that there are infinitely many integers $n$ for which the interval $[n,n+D(θ,k)]$ contains two primes and $k$ almost-primes, with all of the almost-primes having at most $r(θ,k)$ prime factors. If $θ$ can be taken as large as $1-ε$, …
We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the Hölder mean based on the functional equation for the Riemann zeta function. We find a harmonic mean analogue of Chebyshev's inequality for the prime counting function involving the Euler-Mascheroni constant. Furthermore, we define a function taking the Hölder mean of all positive integers up to a given number and investigate its asymptotic behavior, finding two different patterns which are separated by the harmonic mean. Additionally, we discuss the behavior of…
Given good knowledge on the even moments, we derive asymptotic formulas for $λ$-th moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.
Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $δ\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^δ,x]$ contains $\gg_{k} \frac{x^δ}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be re…
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.
Let p be an odd prime, such that p_n<p/2<p_{n+1}, where p_n is the n-th prime. We study the following question: with what probability does there exist a prime in the interval (p, 2p_{n+1})? After the strong definition of the probability with help of the Ramanujan primes ([11], [12])and the introducing pseudo-Ramanujan primes, we show, that if such probability P exists, then P>=0.5. We also study a symmetrical case of the left intervals, which connected with sequence A080359 at OEIS.
Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
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