# Irrationality of values of zeta-function

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”2000 Mathematics Subject Classification. Primary 11J72; Secondary 33C60.

## 1. Introduction

The irrationality of values of the zeta-function at odd integers is one of the most attractive problems in number theory. Inspite of a deceptive simplicity and more than two-hundred-year history of the problem, all done in this direction can easily be counted. It was only 1978, when Apéry [A] obtained the irrationality of by a presentation of “nice” rational approximations to this number. During next years the phenomenon of Apéry’s sequence was recomprehended more than once from positions of different analytic methods (see [N2] and the bibliography cited there); new approaches gave rise to improve Apéry’s result quantitatively, i.e., to get a “sharp” irrationality measure of (last stages in this direction are the articles [H2], [RV]). Finally, in 2000 Rivoal [R1] constructed linear forms with rational coefficients involving values of only at odd integers and proved that there exist infinitely many irrational numbers among rm ; more preciselyrm , for the dimension of spaces spanned over by rm , where is oddrm , there holds the estimate

## 2. Main results

In this note we generalize Rivoal’s construction [R1] and prove the following results.

###### Theorem 1

Each of the following collections

contains at least one irrational number.
\normalfont^{\normalfont2}^{\normalfont2}\normalfont2””After finishing this paper the author knew that
Rivoal [R3] had independently obtained the claim of Theorem 1
for the first collection in 1 by another
generalization of his construction from [R1].

###### Theorem 2

For each odd integer the collection

contains at least one irrational number.

###### Theorem 3

There exist odd integers and such that the numbers are linearly independent over .

Theorem 3 improves corresponding result from [R2], where the linear independence of numbers was established for some .

###### Theorem 4

For each odd integer there holds the absolute estimate

We stress that our proofs of Theorems 1–4 exploit calculations via the saddle point method (Section 4) and ideologically leans on the works [N2], [He]. An improvelment of arithmetic estimates (i.e., of denominators of numerical linear forms) in the spirit of [H2], [RV] (Section 3) allows us to sharpen the lower estimate of in Theorems 3, 4 for small values of . In Section 5 we obtain not only an upper bound but also precise asymptotics of coefficients of linear forms. Finally, we prove Theorems 1–4 in Section 6.

The main results of the work were announced in the communication [Z].

The author is grateful to Professor Yu. V. Nesterenko for his permanent attention to the work. This research was carried out with the partial support of the INTAS–RFBR grant no. IR-97-1904.

## 3. Analytic construction

We fix positive odd parameters such that , , and for each positive integer consider the rational function

where the record ‘’ means that the product contains factors corresponding both to a sign ‘’ and to a ‘’. To the function 3 assign the infinite sum

the series on the right-hand side of 4 converges absolutely since as . Decomposing the function 3 in a sum of partial fractions and using its oddness, we deduce that

(see 10 below), where denominators of the rational numbers grow not faster than exponentially (cf. [R1], lemmes 1, 5). By denote the least common multiple of numbers ; the prime number theorem yields

###### Lemma 1

For each odd integer there exists a sequence of integers rm , rm , such that the numbers are integral and the limit relation

holdsrm ; here is Euler’s constant and is the logarithmic derivative of the gamma-function.

###### Demonstration Proof

Let

Then for rational functions

we have the inclusions

(a proof of the inclusions 8 for the function needs a certain generalization of the arithmetic scheme of Nikishin–Rivoal). Now, representing the initial function 3 in the form and applying Leibniz’s rule for the differentiation of a product, by 8 we obtain

These relations yield the desired inclusions since

By 7, for each prime number there holds

where the function is periodic (of period ) with respect to each of its arguments, and is the integral part of a number. Direct calculations show us that

Now, applying the number prime theorem and following arguments from [Ch], Theorem 4.3 and Section 6; [H1], Lemma 3.2, we obtain the limit relation 6. This completes the proof.

It can easily be checked that the value in 6 behaves itself like as .

## 4. Asymptotics of linear forms

Consider the functions

Decomposing in a sum of partial fractions we see that for each integer

in a neighbourhood of .

###### Lemma 2

For the value 4 there holds the integral presentation

where is an arbitrary positive constant from the interval .

###### Demonstration Proof

Consider the integrand in 12 on a rectangle with vertices , , where an integer is sufficiently large, . Expanding the function 3 in Taylor series in a neighbourhood of and using the expansion 11 by Cauchy’s theorem we obtain

On the sides , , and of the rectangle there holds the relation , while the function , which is a polynomial in , is bounded. Hence, tending to in 13 we get the desired presentation 12.

Our next claim follows from Lemma 2 after the change of variables and an application of Stirling’s formula to gamma-factors of the function 3.

###### Lemma 3

For the sum 4 there holds the asymptotic relation

where

and the contour is a vertical line rm , rm , oriented from bottom to top.

We mean the functions and in the complex -plane cut along the rays and , where we choose that branches of the logarithm functions, which we assume to take real values for .

For each the function is a polynomial in with rational coefficients:

this fact immediately follows from the relations

Consequently, the integral 14 can be represented in the form