Related papers: An explicit mean-value estimate for the PNT in int…

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindel\"of hypothesis (LH) for general sequences which coincides with the usual Lindel\"of hypothesis for the Riemann zeta function in the case of the sequence of positive…

The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the…

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $\nu$ distinct prime divisors: $\pi_{\nu}(x) \sim x \delta_{\nu}(x)$. Here we consider the restricted…

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…

This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical…

We show the primes have level of distribution $66/107\approx 0.617$ using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level $3/5=0.60$ of Maynard. We…

Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a…

Assume the Riemann hypothesis throughout. We obtain some new estimates for the size of the set of large values of the error term in the prime number theorem. Our argument is based on an analysis of the behavior of zeros of the Riemann zeta…

The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we…

Let $\Sigma_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We show that $\mid\Sigma_{2n}\mid\sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is…

In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.

Let $E_k$ be the set of positive integers having exactly $k$ prime factors. We show that almost all intervals $[x,x+\log^{1+\varepsilon} x]$ contain $E_3$ numbers, and almost all intervals $[x,x+\log^{3.51} x]$ contain $E_2$ numbers. By…

Let $H = N^{\theta}, \theta > 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} \Lambda(n) e(g(n)), \] where $g$ is a polynomial of degree $k$. As a…

Using the fact that the number of combinations $p_{1}$, $p_{2}$, where $p_{1}$ and $p_{2}$ are odd primes, with $p_{1} \leq p_{2}$ and $p_{1} + p_{2} \leq 2N$ is equal to the total number of Goldbach pairs for all the even integers from 6…

Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet $L$-functions to a large prime modulus $q$. As applications, we give alternative proofs of…

Given a finitely generated multiplicative subgroup of rational numbers $\Gamma$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group…

In this paper we first establish new explicit estimates for Chebyshev's $\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime…

In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…