Related papers: An effective analytic recurrence for prime numbers

We construct a smooth real-valued function P(n) in [0,1], defined via a triple integral with a periodic kernel, that approximates the characteristic function of prime numbers. The function is built to suppress when n is divisible by some m…

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…

We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a…

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L^2-L^p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential…

We introduce a notion of capacity for high dimensional critical percolation by showing that for any finite set $A$, the suitably rescaled probability that the cluster of $z$ intersects $A$ converges as $\|z\|\to\infty$. This can be viewed…

In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number…

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…

A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to…

In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as…

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ is a sufficiently small constant. Let $k(p,\,n)$ denote the least positive integer $k$ such that for…

For a Carmichael number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-1,\cdots,p_m-1],$$ and let $C_\nu(X)$ denote the number of Carmichael numbers up to $X$ such that $K=\nu$. Assuming a strong conjecture on the first prime…

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael…

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, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…

The unconditional, i.e. without assuming validity of RH, sharp limit relationship (as p tends to infinity) is found between the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals and maximal values of…

This open problem, first posed by Erd{\omicron}s, was further explored by Terence Tao. Tao work shows that the series can converge conditionally, but only under a sufficiently strong form of the Hardy-Littlewood conjecture for k-primary…

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…

Let $(a(n) : n \in \mathbb{N})$ denote a sequence of nonnegative integers. Let $0.a(1)a(2)...$ denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of $(a(n) : n \in \mathbb{N})$.…

For a fixed exponent $0 < \theta \leq 1$, it is expected that we have the prime number theorem in short intervals $\sum_{x \leq n < x+x^\theta} \Lambda(n) \sim x^\theta$ as $x \to \infty$. From the recent zero density estimates of Guth and…

Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, we denote by $R_\lambda (n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We…

Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <=…