Related papers: Pair correlation and twin primes revisited
New formulas for approximation of zeta-constants were derived on the basis of a number-theoretic approach constructed for the irrationality proof of certain classical constants. Using these formulas it's possible to approximate certain…
A celebrated conjecture of Hardy and Littlewood provides with an asymptotic formula for the counting function of the twin primes. We give an unconditional proof of such a formula by means of a finite Ramanujan expansion of the counting…
We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery's function $F(\alpha, T)$ in bounded intervals, the second is an integral introduced by Selberg…
An easy generalization of Beukers' integrals allows us to conjecture a double integral formula involving the zeta and the gamma functions. A special case of this formula is Sondow's double integral formula for Euler's constant gamma.
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…
We examine values of certain Tornheim's type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_2$,…
Associate a unique numerical sequence called the modular signature with each positive integer, using modular residues of each integer under the prime numbers, and distinguishing between the core seed primes and non-core seed primes used to…
In this paper, we investigate new relationships for bilateral series related to two-parameter mock theta functions, which lead to many identities concerning the bilateral mock theta functions. In addition, interesting relations between the…
The accepted resolution of the twin paradox in relativity states that the age of the inertial twin `jumps' when the traveling twin undergoes his turn-around acceleration. This resolution is based on the use of the equivalent gravitational…
We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is…
We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that…
In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the $(2k-2h)^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative on the critical line in the…
We introduce the unified double zeta function of Mordell--Tornheim type and compute its values at non-positive integer points. We then discuss a possible generalization of the Kaneko--Zagier conjecture for all integer points.
In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…
Shuffle algebra has been employed to give a proof of the duality theorem for multiple zeta-star values of height one.
We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these…
Recursion formulae are derived for the calculation of two centre matrix elements of a radial function in relativistic quantum mechanics. The recursions are obtained between not necessarily diagonal radial eigensates using arbitrary radial…
The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.
We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken…
These notes are based on my four lectures given at the Newton Institute in April 2004 during the Recent Perspectives in Random Matrix Theory and Number Theory Workshop. Their purpose is to introduce the reader to the analytic number theory…