Related papers: The error term in counting prime pairs

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…

A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (1981) established an asymptotic formula for the number…

Let $\Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} \Lambda\Big(\Big[\frac{x}{n}\Big]\Big) =…

The method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the…

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…

We prove precise conditional estimates for the third moment of the logarithm of the Riemann zeta function, refining what is implied by the Selberg central limit theorem, both for the real and imaginary parts. These estimates match…

Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? The author and Pieter Moree, building on work of Zhang, Cobeli, and Zaharescu, gave heuristics for…

Lavrik and the author gave uniform bounds of the error term in the approximate functional equation for the derivatives of the Hardy's Z-function. We obtain a new bound of this error term which is much better for high order derivatives.

While the sequence of primes is very well distributed in the reduced residue classes (mod $q$), the distribution of pairs of consecutive primes among the permissible $\phi(q)^2$ pairs of reduced residue classes (mod $q$) is surprisingly…

In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (1998).

We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers…

We examine the prime gaps using a statistical approach. It is first shown that the Andrica's conjecture is true for half or more cases. Using the arguments of averages, it is further shown that Andrica's conjecture is true. We further…

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…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…

We present an algorithm analogous to the sieve of Eratosthenes that produces the list of twin primes. Next, we count the number of twin primes resulting from the construction with two different heuristic arguments. The first method is…

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that…

We establish quantitative bounds on the $U^k[N]$ Gowers norms of the M\"obius function $\mu$ and the von Mangoldt function $\Lambda$ for all $k$, with error terms of shape $O((\log\log N)^{-c})$. As a consequence, we obtain quantitative…

We use classical methods from analytic number theory to resolve the lattice point counting problem on the first Heisenberg group, in the case where the gauge function is taken to be the Cygan-Kor$\acute{a}$nyi Heisenberg-norm…

n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for…