arXiv subjects
Publications and source records attributed to F. M. S. Lima.
In this note, I develop step-by-step proofs of irrationality for $\,ζ{(2)}\,$ and $\,ζ{(3)}$. Though the proofs follow closely those based upon unit-square integrals proposed originally by Beukers, I introduce some modifications which certainly will be useful for those interested in understanding this kind of proof and/or trying to extend it to higher zeta values, Catalan's constant, or other related numbers.
It is well-known that separation of variables in 2nd order partial differential equations (PDEs) for physical problems with spherical symmetry usually leads to Cauchy's differential equation for the radial coordinate and Legendre's differential equation for the polar angle $θ$. For eigenvalues of the form $\,n\,(n+1)$, $n \ge 0\,$ being an integer, Legendre's equation admits certain polynomials $P_n(\cosθ)$ as solutions, which form a complete set of continuous orthogonal functions for all $θ\in [0,π]$. This allows us to take the polynomials $P_n(x)$, where $x = \cosθ$, as a basis for the Fourier-Legendre series expansion of any function $f(x)$ continuous by parts over $\,x \in [-1,1]$. These lecture notes correspond to the end of my course on Mathematical Methods for Physics, when I did derive the differential equations and solutions for physical problems with spherical symmetry. For those interested in Number Theory, I have included an application of shifted Legendre polynomials in \emph{irrationality proofs}, following a method introduced by Beukers to show that $ζ{(2)}$ and $ζ{(3)}$ are irrational numbers.
In a recent work, Farhi developed a Fourier series expansion for the function $\,\ln{Γ(x)}\,$ on the interval $(0,1)$, which allowed him to derive a nice formula for the constant $\,η:= 2 \int_0^1{\ln{Γ(x)} \, \sin{(2 πx)} \, dx}$. At the end of that paper, he asks whether $η$ could be written in terms of other known mathematical constants. Here in this work, after deriving a simple closed-form expression for $η$, I show how it can be used for evaluating other related integrals, as well as certain logarithmic series, which allows for a generalization in the form of a continuous function $η(x)$, $x \in [0,1]$. Finally, from the Fourier series expansion of $\,\ln{Γ(x)}$, $x \in (0,1)$, I make use of Parseval's theorem to derive a closed-form expression for $\,\int_0^1{\ln^2{Γ(x)}~dx}$.
In a recent work, Gun and co-workers have proposed that $\,\sum_{n=-\infty}^{\infty}{(n+α)^{-k}}\,$ is a transcendental number for all integer $\,k$, $k > 1$, and $\,α\in \mathbb{Q} \backslash \mathbb{Z}$. Here in this work, this proposition is shown to be \emph{false} whenever $\,k\,$ is odd and $\,α\,$ is a half-integer. It is also shown that these are the only counterexamples, which allows for a correct reformulation of the original proposition. This leads to a theorem yielding a closed-form expression for the summation of that series, which determines its arithmetic nature. The result is then extended to a sum of polygamma functions and some related zeta series. In view of the recurrent appearance of these series and functions in different areas of mathematics and applications, the closed-form results put forward here could well be included in modern computer algebra systems (CAS).
In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Apéry-like irrationality proof could be found for this constant.
The arithmetic nature of values of some functions of a single variable, particularly, $\sin{z}$, $\cos{z}$, $\sinh{z}$, $\cosh{z}$, $e^z$, and $\ln{z}$, is a relevant topic in number theory. For instance, all those functions return transcendental values for all non-zero algebraic values of $z$ ($z \ne 1$ in the case of $\ln{z}$). On the other hand, not even an irrationality proof is known for some numbers like $\,e^e$, $\,π^e$, $\,π^π$, $\,\lnπ$, $\,π+ e\,$ and $\,π\, e$, though it is well-known that at least one of the last two numbers is irrational. In this note, I first derive a more general form of this last result, showing that at least one of the sum and product of any two transcendental numbers is transcendental. I then use this to show that, given any complex number $\,t \ne 0, 1/e$, at least two of the numbers $\,\ln{t}$, $\,t + e\,$ and $\,t \, e\,$ are transcendental. I also show that $\,\cosh{z}$, $\sinh{z}\,$ and $\,\tanh{z}\,$ return transcendental values for all $\,z = r \, \ln{t}$, $\,r \in \mathbb{Q}$, $r \ne 0$. Finally, I use a recent algebraic independence result by Nesterenko to show that, for all integer $\,n > 0$, $\,\lnπ\,$ and $\,\sqrt{n} \, π\,$ are linearly independent over $\mathbb{Q}$.
In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{Γ(x)} + \log{Γ(1-x)}\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \emph{one} possible exception, but the proof presented there in that work is \emph{incorrect}. Here in this paper, I point out the mistake they committed and I present a theorem that establishes the transcendence of those numbers with at most \emph{two} possible exceptions. As a consequence, I make use of the reflection property of this function to establish a criteria for the transcendence of $\,\logπ$, a number whose irrationality is not proved yet. This has an interesting consequence for the transcendence of the product $\,π\cdot e$, another number whose irrationality remains unproven.
In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $β{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Apéry constant, $ζ(3)$, and the Catalan constant, $G = β(2)$. Interestingly, our series for $\,ζ(3)\,$ converges faster than that used by Apéry in his irrationality proof (1978). Also, our series for $\,G\,$ converges faster than a celebrated one discovered by Ramanujan (1915). At last, I present a simpler, more direct proof for a recent theorem by Katsurada which generalizes the above results.
In this shortnote, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values $\:ζ{(2 k +1)}$, $ζ{(s)}$ being the Riemann zeta function and $k$ a positive integer, is modified in a manner to furnish the even zeta values $ ζ{(2k)}$. As a result, I find an elementary proof of $\sum_{n=1}^\infty{1/{n^2}} = {π^2/6}$, as well as a recurrence formula for $ζ{(2k)}$ from which it follows that the ratio ${ζ{(2k)} / π^{2k}}$ is a rational number, without making use of Euler's formula and Bernoulli numbers.
In this note, I present a simple PSLQ code for finding null linear combinations, with the best rational coefficients, of mathematical constants, within some prescribed precision. As an example, I explore approximate expressions for the Apéry's constant $\,ζ{(3)} = \sum_{n\ge1}{\,1/n^3}$, an irrational number to which no exact, finite closed-form expression is known. % For this, I choose a suitable search basis composed by numbers which seem to be closely related to $\,ζ{(3)}$, namely $\,π$, $\,\ln{2}\,$, $\,\ln{(1+\sqrt{2}\,)}$, and $G$ (the Catalan's constant). On taking into account a suitable search basis, I have found a simple expression for $\,ζ{(3)}\,$ accurate to 21 decimal places, which is triply more accurate than the best previous one. As the short \emph{Maple}$^\mathrm{TM}$ code presented here can be easily adapted to study other constants, I decided to supply it to encourage the readers to conduct their own computational experiments, as well as to adopt it in projects of numerical analysis, number theory, or linear algebra.
A transcendental function usually returns transcendental values at algebraic points. The (algebraic) exceptions form the so-called \emph{exceptional set}, as for instance the unitary set $\{0\}$ for the function $f(z) = e^z \,$, according to the Hermite-Lindemann theorem. In this note, we give some explicit examples of transcendental entire functions whose exceptional set are empty.
In this short paper, I introduce an elementary method for exactly evaluating the definite integrals $\, \int_0^π{\ln{(\sinθ)}\,dθ}$, $\int_0^{π/2}{\ln{(\sinθ)}\,dθ}$, $\int_0^{π/2}{\ln{(\cosθ)}\,dθ}$, and $\int_0^{π/2}{\ln{(\tanθ)}\,dθ} \,$ in finite terms. The method consists in to manipulate the sums obtained from the logarithm of certain products of trigonometric functions at rational multiples of $π$, putting them in the form of Riemann sums. As this method does not involve any search for primitives, it represents a good alternative to more involved integration techniques. As a bonus, I show how to apply the method for easily evaluating $\,\int_0^1{\ln{Γ(x)} \, d x}$.
In a recent work, Dancs and He found new `Euler-type' formulas for $\,\ln{2}\,$ and $\,ζ{(2\,n+1)}$, $\,n\,$ being a positive integer, each containing a series that apparently can not be evaluated in closed form, distinctly from $\,ζ{(2\,n)}$, for which the Euler's formula allows us to write it as a rational multiple of $\,π^{2n}$. There in that work, however, the formulas are derived through certain series manipulations, by following Tsumura's strategy, which makes it \emph{curious} --- in the words of those authors themselves --- the appearance of the numbers $\,\ln{2}\,$ and $\,ζ{(2\,n+1)}$. In this short paper, I show how some known zeta-series can be used to derive the Dancs-He series in an alternative manner.
A mathematical derivation of the force exerted by an \emph{inhomogeneous} (i.e., compressible) fluid on the surface of an \emph{arbitrarily-shaped} body immersed in it is not found in literature, which may be attributed to our trust on Archimedes' law of buoyancy. However, this law, also known as Archimedes' principle (AP), does not yield the force observed when the body is in contact to the container walls, as is more evident in the case of a block immersed in a liquid and in contact to the bottom, in which a \emph{downward} force that \emph{increases with depth} is observed. In this work, by taking into account the surface integral of the pressure force exerted by a fluid over the surface of a body, the general validity of AP is checked. For a body fully surrounded by a fluid, homogeneous or not, a gradient version of the divergence theorem applies, yielding a volume integral that simplifies to an upward force which agrees to the force predicted by AP, as long as the fluid density is a \emph{continuous function of depth}. For the bottom case, this approach yields a downward force that increases with depth, which contrasts to AP but is in agreement to experiments. It also yields a formula for this force which shows that it increases with the area of contact.
In a recent work, Dancs and He found an Euler-type formula for $\,ζ{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in comparison to $ζ(2n)$, which is a rational multiple of $π^{2n}$. For the Dirichlet beta function, the things are `inverse': $β(2n+1)$ is a rational multiple of $π^{2n+1}$ and no closed-form expression is known for $β(2n)$. Here in this work, I modify the Dancs-He approach in order to derive an Euler-type formula for $\,β{(2n)}$, including $\,β{(2)} = G$, the Catalan's constant. I also convert the resulting series into zeta series, which yields new exact closed-form expressions for a class of zeta series involving $\,β{(2n)}$ and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured.
In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers $B_{2 n}$, $n$ being any positive integer. This new recurrence seems advantageous in comparison to other known formulae since it allows the computation of both $B_{4 n}$ and $B_{4 n +2}$ from only $B_0, B_2,..., B_{2n}$.
Among the several proofs known for $\sum_{n=1}^\infty{1/n^2} = {π^2/6}$, the one by Beukers, Calabi, and Kolk involves the evaluation of $\,\int_0^1 {\int_0^1{1/(1-x^2 y^2) \, dx} \, dy}$. It starts by showing that this double integral is equivalent to $\frac34 \sum_{n=1}^\infty{1/n^2}$, and then a non-trivial \emph{trigonometric} change of variables is applied which transforms that integral into $\,{\int \int}_T \: 1 \; du \, dv$, where $T$ is a triangular domain whose area is simply ${π^2/8}$. Here in this note, I introduce a hyperbolic version of this change of variables and, by applying it to the above integral, I find exact closed-form expressions for $\int_0^\infty{[\sinh^{-1}{(\cosh{u})}-u] d u}$, $\,\int_α^\infty{[u-\cosh^{-1}{(\sinh{u})}] d u}$, and $\,\int_{\,α/2}^\infty{\ln{(\tanh{u})} \: d u}$, where $α= \sinh^{-1}(1)$. From the latter integral, I also derive a two-term dilogarithm identity.
A simple setup was assembled to study the motion of an object while it falls. The setup was used to determine the instantaneous velocity, terminal velocity and acceleration due to gravity. Also, since the whole project was done within $20 it can easily be popularized.