Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant $γ$. The proof is by reduction to known irrationality criteria for $γ$ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, $γ$, and logarithms from Nesterenko-type series of rational functions. In the Appendix, Sergey Zlobin gives a change-of-variables proof that the series and the double integral are equal.
We prove the new upper bound 5.095412 for the irrationality exponent of $ζ(2)=π^2/6$; the earlier record bound 5.441243 was established in 1996 by G. Rhin and C. Viola.
In this paper, we simplify and improve the constant, $c$, that appears in effective irrationality measures, $|(a/b)^{m/n}-p/q|>c|q|^{-(κ+1)}$, obtained from the hypergeometric method for $a/b$ near $1$. The dependence of $c$ on $|a|$ in our result is best possible (as is the dependence on $n$ in many cases). For some applications, the dependence of this constant on $|a|$ becomes important. We also establish some new inequalities for hypergeometric functions that are useful in other diophantine settings.
We present a brief survey of the methods used in deducing upper estimates for irrationality measures of the logarithm values. We particularly expose the best known estimates for $\log2$ (due to E. Rukhadze), $π$ (due to M. Hata) and $\log3$ (due to G. Rhin).
We present a new proof of the irrationality of values of the series $T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2}$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $T_q(z)$.
We give a new hypergeometric construction of rational approximations to $ζ(4)$, which absorbs the earlier one from 2003 based on Bailey's ${}_9F_8$ hypergeometric integrals. With the novel ingredients we are able to get a better control of arithmetic and produce a record irrationality measure for $ζ(4)$.
We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $π^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.
We show how one can obtain rational approximants for $q$-extensions of the harmonic series and the logarithm (and many other similar quantities) by Padé approximation using little $q$-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for the harmonic series and a better upper bound than the one given by Matala-aho and Väänänen for the logarithm.
For integer $p$, $|p|>1$, and generic rational $x$ and $z$, we establish the irrationality of the series $$\ell_p(x,z)=x\sum_{n=1}^\infty\frac{z^n}{p^n-x}.$$ It is a symmetric ($\ell_p(x,z)=\ell_p(z,x)$) generalization of the $q$-logarithmic function ($x=1$ and $p=1/q$ where $|q|<1$), which in turn generalizes the $q$-harmonic series ($x=z=1$). Our proof makes use of the Hankel determinants built on the Padé approximations to $\ell_p(x,z)$.
We give three elementary proofs of a nice equality of definite integrals, which arises from the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generalization together with an equally elementary proof and discuss some consequences.
We present a new `elementary' proof of the irrationality of $ζ(3)$ based on some recent `hypergeometric' ideas of Yu.Nesterenko, T.Rivoal, and K.Ball, and on Zeilberger's algorithm of creative telescoping.
We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under the assumption that $a$ is $p$-adically very close to $b$, that is, that a large power of $p$ divides $a-b$. As an application, we solve completely certain families of Thue-Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.
Following earlier results of Sondow, we propose another criterion of irrationality for Euler's constant $γ$. It involves similar linear combinations of logarithm numbers $L\_{n,m}$. To prove that $γ$ is irrational, it suffices to prove that, for some fixed $m$, the distance of $d\_n L\_{n,m}$ ($d\_n$ is the least common multiple of the $n$ first integers) to the set of integers $\mathbf{Z}$ does not converge to 0. A similar result is obtained by replacing logarithms numbers by rational numbers: it gives a sufficient condition involving only rational numbers. Unfortunately, the chaotic behavior…
In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions. Improved bounds for $θ(k,l;x)$ and $ψ(k,l;x)$ for $k=1,3,4,6$ are also presented.
The irrationality exponent $μ(t)$ of an irrational number t, defined using the irrationality measure $1/q^μ$, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure $1/β^q$, we define the "irrationality base" $β(t)$, which distinguishes among Liouville numbers and is 1 for non-Liouville numbers. We give some properties and examples. Assuming a condition on certain linear forms in logarithms, for which we present numerical evidence supplied by P. Sebah, we prove an upper bound on the irrationality base of Euler's constant, $γ$. If $γ$ is…
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $ζ(2)$ and $ζ(3)$, as well as to explain Rivoal's "infinitely-many" result (math.NT/0008051) and to prove that at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$, and $ζ(11)$ is irrational.
The aim of the paper is to relate computational and arithmetic questions about Euler's constant $γ$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical formula for $γ$ due to Ramanujan, together with Vacca's and Gosper's series for $γ$, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler's constant. The main tools are Euler-type integrals and hypergeometric series.
We present a hypergeometric construction of rational approximations to $ζ(2)$ and $ζ(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, $ζ(2)$ and $ζ(3)$ with rational coefficients. A new notion of (simultaneous) diophantine exponent is introduced to formalise the arithmetic structure of these specific linear forms. Finally, the properties of this newer concept are studied and linked to the classical irrationality exponent and its generalisations given recently by S. Fischler.
A convolution approach leading to an explicit computation of a value of a 4F3 function is outlined. We also investigate about the role of the dilogarithm reflection formula, leading to a remarkable consequence: in some cases, values of 4F3 are given by linear combinations of a squared arctangent and a squared logarithm.
We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apery-like (polynomial) recursions.
In the joint work of T.Rivoal and the author, a hypergeometric construction was proposed for studing arithmetic properties of the values of Dirichlet's beta function $β(s)$ at even positive integers. The construction gives some bonuses for Catalan's constant $G=β(2)$, such as a second-order Apery-like recursion and a permutation group in the sense of G.Rhin and C.Viola. Here we prove expected integrality properties of solutions to the above recursion as well as suggest a simpler (also second-order and Apery-like) one for $G$. We "enlarge" the permutation group of our previous (joint with T.Riv…
This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of $ζ(2)$ and $ζ(3)$, the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ and $ζ(11)$ is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the general…
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.
For certain class of hypergeometric functions ${}_3F_2$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators.
By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we derive criteria for irrationality of Euler's constant, gamma. For n > 0, we define a double integral I(n) and a positive integer S(n), and prove that if d(n) = LCM(1,...,n), then the fractional part of logS(n) is given by {logS(n)} = d(2n)I(n), for all n sufficiently large, if and only if gamma is a rational number. A corollary is that if {logS(n)} > 1/2^n infinitely often, then gamma is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 0 < n < 2500 and n = 10000) and gamma …
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic $\mathcal D$-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points …
The Stieltjes constants $γ_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the p…
We show that the integral $J(t) = (1/π^3) \int_0^π\int_0^π\int_0^πdx dy dz \log(t - \cos{x} - \cos{y} - \cos{z} + \cos{x}\cos{y}\cos{z})$, can be expressed in terms of ${_5F_4}$ hypergeometric functions. The integral arises in the solution by Baxter and Bazhanov of the free-energy of the $sl(n)$ Potts model, which includes the term $J(2)$. Our result immediately gives the logarithmic Mahler measure of the Laurent polynomial $k - (x+1/x) - (y+1/y) - (z+1/z) + 1/4(x+1/x) (y+1/y) (z+1/z)$ in terms of the same hypergeometric functions.
The modular $j$-function is a bijective map from $X_0(1) \setminus \{\infty\}$ to $\mathbb{C}$. A natural question is to describe the inverse map. Gauss offered a solution to the inverse problem in terms of the arithmetic-geometric mean. This method relies on an elliptic curve model and the Gaussian hypergeometric series. Here we use the theory of polar harmonic Maass forms to solve the inverse problem by directly examining the Fourier expansion of the weight $2$ polar harmonic Maass form obtained by specializing the logarithmic derivative of the denominator formula for the Monster Lie algebra…
We initiate the study of spectral zeta functions $ζ_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $ζ(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of …
In 2007, A.I.Aptekarev and his collaborators discovered a sequence of rational approximations to Euler's constant $γ$ defined by a linear recurrence. In this paper, we generalize this result and present an explicit construction of rational approximations for the numbers $\ln(b)-ψ(a+1),$ $a, b\in {\mathbb Q},$ $b>0, a>-1,$ where $ψ$ defines the logarithmic derivative of the Euler gamma function. We prove exact formulas for denominators and numerators of the approximations in terms of hypergeometric sums. As a consequence, we get rational approximations for the numbers $π/2\pmγ.$ We compare the …
