Analytic Number Theory Research

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker c…

• CEM YALÇIN YILDIRIM

No abstract yet.

• SERKAN ARACI

We consider weighted q‐Genocchi numbers and polynomials. We investigated some interesting properties of the weighted q‐Genocchi numbers related to weighted q‐Bernstein polynomials by using fermionic p‐adic integrals on ℤp.

• SERKAN ARACI

Abstract We introduce a family of algebraic curves over 𝔽 q 2n (for an odd n) and show that they are maximal. When n = 3, our curve coincides with the 𝔽 q 6 -maximal curve that has been found by Giulietti and Korchmáros recently. Their curve (i.e., the case n = 3) is the first example of a maximal curve proven not to…

We prove that $ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $where pn denotes the nth prime. Since on average pn+1−pn is asymptotically log n, this shows that we can always find pairs of primes much closer together than the average…

• CEM YALÇIN YILDIRIM

This paper presents an improved Plantard’s modular arithmetic (Plantard arithmetic) tailored for Lattice-Based Cryptography (LBC). Based on the improved Plantard arithmetic, we present faster implementations of two LBC schemes, Kyber and NTTRU, running on Cortex-M4. The intrinsic advantage of Plantard arithmetic is th…

• ÇETİN KAYA KOÇ

In the preprint [3], Goldston, Pintz, and Yıldırım established, among other things, \begin{equation} \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0, \end{equation} with $p_n$ the $n$th prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3].…

• CEM YALÇIN YILDIRIM

No abstract yet.

Analytical expressions through the binomial coefficients and recursive relations are derived for the expansion coefficients of overlap integrals in terms of a product of well-known auxiliary functions Ak and Bk. These formulas are especially useful for the calculation of overlap integrals for large quantum numbers. Ac…

• ÜLFET ATAV

Let $p_n$ denote the $n^{\textrm {th}}$ prime. Goldston, Pintz, and Yıldırım recently proved that \begin{equation*} \liminf _{n\to \infty } \frac {(p_{n+1}-p_n)}{\log p_n} =0. \end{equation*} We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $…

• CEM YALÇIN YILDIRIM

The essential aim of this paper is to introduce novel identities forq-Genocchi numbers and polynomials by using the method by T. Kim et al. (article in press). We show that these polynomials are related top-adic analogue of Bernstein polynomials. Also, we derive relations betweenq-Genocchi andq-Bernoulli numbers.

• SERKAN ARACI

The complete description of the Alexander polynomial of the complement of an irreducible sextic in [Formula: see text] is given. Some general results about Alexander polynomials of algebraic curves are also obtained.