Related papers: The Euler Legacy to Modern Physics
This is a broad-brush review of how string theory addresses several important questions of gravitational physics. The problem of non-renormalizability is first reviewed, followed by introduction of string theory as an ultraviolet-finite…
In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the…
A perturbative description of Large Scale Structure is a cornerstone of our understanding of the observed distribution of matter in the universe. Renormalization is an essential and defining step to make this description physical and…
In this paper, we show that the regularized determinants of some Dirichlet series are multiplicative. As an application, we give generalizations of Lerch's formula for the classical gamma function and we determine the sum of some Dirichlet…
In this paper, we give a purely algebraic proof of an identity coming directly from Euler's reflection formula for the gamma function. Our proof uses Hoffman's harmonic algebra and some binomial identities.
These notes were written from a series of lectures given in March 2010 at the Universidad Complutense of Madrid and then in Barcelona for the centennial anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an…
A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker…
Euler defines a function f(x) somehow as an infinite product and a generalization of [x], where [x] ist, what we now call following Legendre the Gamma-Funktion. He gets some recursive relationships for f(x), by applying some very nice…
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic…
Euler stressed the importance of hypotheses, which he thought were the only means of arriving at a certain knowledge of the physical causes, essential to establish the laws of physics. This thought was communicated to Emilie du Chatelet in…
We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of…
This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently [2] and developed in [4], [5] and [6]. Their peculiarity is…
In this paper we present a method to deal with divergences in perturbation theory using the method of the Zeta regularization, first of all we use the Euler-Mc Laurin Sum formula to associate the divergent integral to a divergent sum in the…
A well-known result, due to Dirichlet and later generalized by de la Vallee-Poussin, expresses a relationship between the sum of fractional parts and the Euler-Mascheroni constant. In this paper, we prove an asymptotic relationship between…
In this note, we extend Euler's transformation formula from the alternating series to more general series. Then we give new expressions for the Riemann zeta function $\zeta(s)$ by the generalized difference operator $\Delta_{c}$, which…
We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.
We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of…
Back in 1755, Euler explored an interesting array of numbers that now frequently appears in polynomial identities, combinatorial problems, and finite calculus, among other places. These numbers share a strong connection with well-known…
This paper is mostly a collection of ideas already published by various authors, some of them even a long time ago. Its intention is to bring the reader to know some rather unknown papers of different fields that merit interest and to show…