Abstract
We prove that at least one of the six numbers \(\beta (2i)\) for \(i=1,\ldots ,6\) is irrational. Here \(\beta (s)=\sum _{k=0}^{\infty }(-1)^k(2k+1)^{-s}\) denotes Dirichlet’s beta function, so that \(\beta (2)\) is Catalan’s constant.
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1 Introduction
In this note we discuss arithmetic properties of the values of Dirichlet’s beta function
at positive even integers s. The very first such beta value \(\beta (2)\) is famously known as Catalan’s constant; its irrationality remains an open problem, though we expect the number to be irrational and transcendental. The best known results in this direction were given by T. Rivoal and this author in [4]. Namely, we showed that at least one of the seven numbers \(\beta (2),\beta (4),\ldots ,\beta (14)\) is irrational, and that there are infinitely many irrational numbers among the even beta values \(\beta (2),\beta (4),\beta (6),\ldots \) . Here we use a variant of the method from [3, 8] to improve slightly on the former achievement; a significant strengthening towards the infinitude result, based on a further development of the ideas in [2, 6], is a subject of the recent preprint [1] of S. Fischler.
Theorem 1
At least one of the six numbers
is irrational.
In Sect. 2 we illustrate principal ingredients of the method in a particularly simple situation; this leads to a weaker version of Theorem 1, namely, to the irrationality of at least one number \(\beta (2i)\) for \(i=1,\ldots ,8\). The details about the general construction of approximating forms to even beta values and our proof of Theorem 1 are given in Sect. 3.
2 Outline of the construction
For an odd integer \(s\ge 3\) (which we eventually set to be 17) and even \(n>0\), define the rational function
and assign to it the related sequence of quantities
The sums \(r_n\) are instances of generalized hypergeometric functions, for which we can use some standard integral representations to write
(details are given in Lemma 2 below). This form clearly implies that \(r_n>0\) and also gives access to the asymptotics
An important ingredient of the construction is the following decomposition of the quantities \(r_n\).
Lemma 1
For odd s and even n as above,
where \(a_i=a_{i,n}\) satisfy the inclusions \(\Phi _n^{-1}d_n^{s-i}a_i\in \mathbb {Z}\) for \(i=0,1,\ldots ,s\) even. Here \(d_n\) denotes the least common multiple of the numbers \(1,2,\ldots ,n\), and
the product taken over primes.
Note that from the prime number theorem we deduce the asymptotics
where
the function \(\psi (x)\) denotes the logarithmic derivative of the gamma function.
Remark 1
The analogous construction in [4] makes use of a slightly different rational function than (1), namely, of
so that
The analogous decomposition of a related quantity \(\widetilde{r}_n\) assumes the form
in which the rational coefficients \(\widetilde{a}_i=\widetilde{a}_{i,n}\) satisfy \(\Phi _n^{-1}d_n^{s-i}\widetilde{a}_i\in \mathbb {Z}\) for \(i=1,\ldots ,s\) even, but \(\Phi _n^{-1}d_{2n}^s\widetilde{a}_0\in \mathbb {Z}\). The appearance of \(d_{2n}^s\) as the common denominator in place of \(d_n^s\) changes the scene drastically and leads to weaker arithmetic applications.
Proof of Lemma 1
Following the strategy in [5, 8] we can write the function (1) as sum of partial fractions,
in which \(\Phi _n^{-1}d_n^{s-i}a_{i,k}\in \mathbb {Z}\) for all i and k. Indeed, the rational function (1) is a product of simpler ones
and \(2t+n\); the inclusions \(d_n^{s-i}a_{i,k}\in \mathbb {Z}\) then follow from [8, Lemma 1]. The cancellation by the factor \(\Phi _n\) originates from the p-adic analysis of the binomial factors entering
and the estimate \(\hbox {ord}_pa_{i,k}\ge -(s-i) +\hbox {ord}_pa_{s,k}\ge -(s-i)+\varphi (n/p,k/p)\) for primes in the range \(2\sqrt{n}<p\le n\), where
is a periodic function of period 1 in both x and y, and from the inequality
the details can be borrowed from [4, Sect. 7]. Furthermore, the property \(R_n(-t-n)=R_n(t)\) derived from (1) implies \(a_{i,k}=(-1)^ia_{i,n-k}\) for \(i=1,\ldots ,s\) and \(k=0,1,\ldots ,n\).
Recall that n is even, so that \(n/2=m\) is a positive integer. The summation over \(\nu \) in (2) can also start from \(-m-1\) (rather than 1 or \(n+1\)), because the function \(R_n(t)\) vanishes at all half-integers between \(-2n\) and n. Therefore,
where the rules
were applied. Thus, the rational numbers
satisfy \(\Phi _n^{-1}d_n^{s-i}a_i\in \mathbb {Z}\), while for the quantity
the inclusion \(\Phi _n^{-1}d_n^sa_0\in \mathbb {Z}\) follows from noticing that
Finally,
so that \(a_i\) vanish for odd i. \(\square \)
Set now \(s=17\), in which case we compute from (3) that
hence the linear forms
are positive and tend to 0 as \(n\rightarrow \infty \). This implies that the eight numbers \(\beta (2),\beta (4),\ldots ,\beta (16)\) cannot be all rational.
3 General settings
A natural way to generalize the construction in Sect. 2 follows the recipe of [4] and [7].
For an odd integer \(s\ge 5\), consider a collection \(\varvec{\eta }=(\eta _0,\eta _1,\ldots ,\eta _s)\) of integral parameters satisfying the conditions
to which we assign, for each positive integer n, the collection
In what follows, we assume that \(h_0-1=\eta _0n\) is even—the condition that is automatically achieved when \(\eta _0\in 2\mathbb {Z}\), otherwise by restricting to even n.
Define the rational function
where
and the (very-well-poised) hypergeometric sum
Then [4, Lemma 1] implies the following Euler-type integral representation of \(r_n\) (see also [4, Lemma 3]).
Lemma 2
The formula
is valid. In particular, \(r_n>0\) and
Computation of the latter maximum is performed in [4, Sect. 4, Remark], and the result is as follows.
Lemma 3
Assume that \(x_0\) is a unique zero of the polynomial
in the interval \(0<x<1\), and set
Then
Arithmetic ingredients of the construction are in line with the strategy used in the proof of Lemma 1. For simplicity we split the corresponding statement into two parts. Define
and notice that the poles of the rational function (4) are located at the points \(t=-k-\frac{1}{2}\) for integers k in the range \(N\le k\le h_0-N-1\).
Lemma 4
The coefficients in the partial-fraction decomposition
of (4) satisfy
and
for \(i=1,\ldots ,s\) and \(N\le k\le h_0-N-1\), where the product over primes
is defined through the 1-periodic functions
and
Proof
For this, we write the function \(R_n(t-\tfrac{1}{2})\) as the product of \(2t+h_0-1\), the three integer-valued polynomials
where \(h_1^*=h_1-\frac{1}{2}=\eta _1n\), and the rational functions
Then [4, Lemmas 4, 5, 10, 11] and the Leibniz rule for differentiating a product imply the inclusions \(d_M^{s-i}a_{i,k}\in \mathbb {Z}\) and estimates
for the p-adic order of the coefficients. These are combined to conclude with (7).
The property (6) follows from the symmetry of the rational function (4). \(\square \)
Lemma 5
The decomposition
takes place, where \(\Phi _n^{-1}d_M^{s-i}a_i\in \mathbb {Z}\) for \(i=0,1,\ldots ,s\) even, and \(\Phi _n\) is defined in Lemma 4.
Proof
Since the function (4) vanishes at \(t=-1,-2,\ldots ,-h_0+2\), we can shift the summation in (5):
Now proceeding as in the proof of Lemma 1 we arrive at the desired decomposition (8) with
and
with \(a_i\) vanishing for i odd in view of the property (6). The inclusions for the coefficients in (8) therefore follow from Lemma 4 and
\(\square \)
Proof of Theorem 1
Take \(s=13\) and
hence \(M=11n\). Then
and
so that
This means that the positive linear forms
tend to 0 as \(n\rightarrow \infty \). Thus, at least one of the even beta values in consideration must be irrational. \(\square \)
References
Fischler, S.: Irrationality of values of \(L\)-functions of Dirichlet characters. Preprint arXiv:1904.02402 [math.NT] (2019)
Fischler, S., Sprang, J., Zudilin, W.: Many odd zeta values are irrational. Compos. Math. 155(5), 938–952 (2019)
Krattenthaler, C., Zudilin, W.: Hypergeometry inspired by irrationality questions. Kyushu J. Math. 73(1), 189–203 (2019)
Rivoal, T., Zudilin, W.: Diophantine properties of numbers related to Catalan’s constant. Math. Ann. 326(4), 705–721 (2003)
Rivoal, T., Zudilin, W.: A note on odd zeta values. Preprint arXiv:1803.03160 [math.NT] (2018)
Sprang, J.: Infinitely many odd zeta values are irrational. By elementary means. Preprint arXiv:1802.09410 [math.NT] (2018)
Zudilin, W.: Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux 16(1), 251–291 (2004)
Zudilin, W.: One of the odd zeta values from \(\zeta (5)\) to \(\zeta (25)\) is irrational. By elementary means. SIGMA 14, 028 (2018)
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Communicated by Jens Funke.
To Peter Bundschuh, with many irrational wishes, on the occasion of his 80th birthday.
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Zudilin, W. Arithmetic of Catalan’s constant and its relatives. Abh. Math. Semin. Univ. Hambg. 89, 45–53 (2019). https://doi.org/10.1007/s12188-019-00203-w
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DOI: https://doi.org/10.1007/s12188-019-00203-w