Explicit Interval Estimates for Prime Numbers

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta, x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - \Delta^{-1}), x]$.Comment: 15 pages, 3 tables, 1 figur

Problems on Twin Primes, Goldbach's Conjecture, the Riemann Hypothesis and zeros of LL-functions in Number Theory (Women in Mathematics)

Number Theory has a very long history that dates back thousands of years. The main goal of this study is to understand properties of numbers which essentially can be reduced to understanding prime numbers. Although we have the outstanding Prime Number Theorem, more precise information about the distribution of prime numbers is mostly unknown. For example, it is also not known if there are infinitely many pairs of prime numbers having difference 2, the so-called twin prime pairs. Recent breakthroughs in Analytic Number Theory have succeeded in showing the infinitude of prime pairs with small gaps, which is the contribution of Yitang Zhang, one of this year's Fields medalists, James Maynard, and also Terrence Tao. The 280-year-old Goldbach's conjecture and the Riemann hypothesis which is now over 160 years old are also among the most famous yet important unsolved problems in Analytic Number Theory. The Riemann Hypothesis is a conjecture about the location of zeros of the Riemann zeta function. The importance of this problem not only in Number Theory but also many other areas of Mathematics and even Physics is reflected in many known equivalent statements. In Analytic Number Theory alone, we know the equivalence between the Riemann Hypothesis and many prime distribution related problems. Its equivalence to Goldbach related problems is also known. It is important to note that Goldbach's conjecture itself is an independent problem to the Riemann Hypothesis and neither is stronger than the other. In this talk, I would like to introduce a few interesting recent results in this direction

On irregular prime power divisors of the Bernoulli numbers

Let $B_n$ ($n = 0, 1, 2, ...$) denote the usual $n$-th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with B_l/l \not\equiv B_{l+p-1}/(l+p-1) \modp{p^2}. We show that for every $r \geq 1$ the congruence B_{m_r}/m_r \equiv 0 \modp{p^r} has a unique solution $m_r$ where m_r \equiv l \modp{p-1} and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$ defines a $p$-adic integer $\chi_{(p, l)}$ which is a zero of a certain $p$-adic zeta function $\zeta_{p, l}$ originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) $p$-adic expansion of $\chi_{(p, l)}$ for irregular pairs $(p,l)$ with $p$ below 1000.Comment: 42 pages; final accepted paper, slightly revised and extended, to appear in Math. Com

Note on Prime Gaps and Zero Spacings

The article focuses on the problems of prime gaps and zero spacings. Possible solutions of several related problems such as the greatest lower bound, the least upper bound of the zero spacings, and the least upper bound of the prime gaps are considered.Comment: Keywords: Prime gaps, Zero spacings, Zeta function, Pair correlation conjecture, 16 Pages. Refinements of the proof

Two-point correlation function for Dirichlet L-functions

The two-point correlation function for the zeros of Dirichlet L-functions at a height E on the critical line is calculated heuristically using a generalization of the Hardy-Littlewood conjecture for pairs of primes in arithmetic progression. The result matches the conjectured Random-Matrix form in the limit as $E\rightarrow\infty$ and, importantly, includes finite-E corrections. These finite-E corrections differ from those in the case of the Riemann zeta-function, obtained in (1996 Phys. Rev. Lett. 77 1472), by certain finite products of primes which divide the modulus of the primitive character used to construct the L-function in question.Comment: 10 page