Related papers: A note on the Cram\'er-Granville model

For $(M,a)=1$, put \begin{equation*} G(X;M,a)=\sup_{p^\prime_n\leq X}(p^\prime_{n+1}-p^\prime_n), \end{equation*} where $p^\prime_n$ denotes the $n$-th prime that is congruent to $a\pmod{M}$. We show that for any positive $C$, provided $X$…

We prove some results concerning the distribution of primes on the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}{\pi} \sqrt{x} \log x,x]$ for all $x \geq 2$; this improves a…

Motivated by their research on automorphism groups of pseudo-real Riemann surfaces, Bujalance, Cirre and Conder have conjectured that there are infinitely many primes $p$ such that $p+2$ has all its prime factors $q\equiv -1$ mod~$(4)$. We…

We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive…

In this paper we will propose a strategy to prove Goldbach's conjecture: every even integer greater than 2 can be written as the sum of two primes.

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…

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…

Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,2}(N)$ as the number of primes $p$ that such that $N - p$ has at most 2 prime factors. In this paper, we show that $D_{1,2}(N)…

Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this…

In this paper we prove that the binary Goldbach conjecture for sufficiently large even integers would follow under the assumption that both the Elliott-Halberstam conjecture and a variant of the Elliott-Halberstam conjecture twisted by the…

We answer the question positively. In fact, we believe to have proved that every even integer $2N\geq3\times10^{6}$ is the sum of two odd distinct primes. Numerical calculations extend this result for $2N$ in the range $8-3\times10^{6}$.…

In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture for the prime 2.

A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the…

We show that for all large enough $x$ the interval $[x,x+x^{1/2}\log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the…

We show that the splitting conjecture in the hybrid model of Gonek--Hughes--Keating holds to order on the Riemann hypothesis. Our results are valid in a larger range of the parameter $X$ which mediates between the partial Euler and Hadamard…

Let $f(z)=\sum_{n=1}^{\infty} a_f(n)e^{2\pi i n z}$ be a non-CM holomorphic cupsidal newform of trivial nebentypus and even integral level $k\geq 2$. Deligne's proof of the Weil conjectures shows that $|a_f(p)|\leq 2p^{\frac{k-1}{2}}$ for…

A 1976 result from Norton may be used to give an asymptotic (but not explicit) description of the constant in Mertens' second theorem for primes in arithmetic progressions. Assuming the Generalized Riemann Hypothesis, we give an effective…

For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a…

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $\zeta(s)$, that the set of real numbers $x\ge2$ for which $\pi(x)>$ li$(x)$ has a logarithmic…

We exhibit approximately fifty Betti diagrams of free resolutions of rings of smooth, connected canonical curves of genera $9$-$14$ in prime characteristics between $2$ and $11$. Generic Green's conjecture is verified for genera $9$ and…