相关论文: A way to prove the irrationality of Zeta(4)

The existence of infinitely many consecutive prime triples $p_n$, $ p_{n+1}$, and $p_{n+2}$ as $n \to \infty$, is sufficient to prove that the Catalan constant $\beta(2)=0.9159655941\ldots $ is an irrational number. This note provides the…

Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain…

We present a new structure called the "conservative matrix field", initially developed to elucidate and provide insight into the methodologies employed by Ap\'ery's in his proof of the irrationality of the Riemann zeta function at 3. This…

The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is…

We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann $\zeta$ function at odd integers are irrational. These generalizations concern multiple series…

In this paper we generalize Nesterenko's criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the…

For a fixed irrational $\theta > 0$ with a prescribed irrationality measure function, we study the correlation $\int_1^X \Delta(x) \Delta(\theta x) dx$, where $\Delta$ is the Dirichlet error term in the divisor problem. When $\theta$ has a…

In this paper we shall prove that every Witten multiple zeta value of weight w>3 attached to sl(4) at nonnegative integer arguments is a finite rational linear combinations of MZVs of the weight w and the depths three or less, except for…

We prove an easy but interesting result about the linear independence of multiple zeta values of different weights.

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective.…

In this note we show that in addition to two integers forming a Pythagorean triple, there also exist two irrational numbers in terms of which this Pythagorean triple can also be obtained. We also put forward a relation between these two…

We compute the weight of a Kontsevich graph in deformation quantization. Up to rationals, the result is Zeta(3)^2/Pi^6.

A new definition for the Riemann zeta function for all positive integer number s > 1 is presented. We discover a most elegant expression and easy method for calculating the Riemann zeta function for small even integer values. Through this…

This note shows that the product $e \pi$ of the natural base $e$ and the circle number $\pi$ is an irrational number.

This article gives an alternative proof of the fact that N_{Q(zeta)/Q}(1-zeta)=p where p is an odd prime number and zeta is a primitive p-th root of unity, and uses it to prove that N_{Q(zeta)/Q}(1+zeta-zeta^2)=L(p) the p-th Lucas number.…

Commenting on an observation of Prof. Edwards, this note presents a method of evaluation of $\zeta(2n)$ that follows easily from Riemann's own representation of the zeta function.

The main goal of this article is to present an elementary proof of Ramanujan's identity for odd zeta values. Our proof solely relies on a Mittag-Leffler type expansion for hyperbolic cotangent function and Euler's identity for even zeta…

We rebut Kowalenko's claims in 2010 that he proved the irrationality of Euler's constant, and that his rational series for it is new.

We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm…

This note describes continued fraction representations for the rational approximations to the zeta function recently found by the author. It is tempting to think that these continued fractions might be analysed using a souped up version of…