Apéry’s famous proof [Reference van der Poorten10] of the irrationality of $\zeta (3)$ displayed a particular phenomenon (which could certainly have been dismissed if discussed only in the arithmetic context). One considers the recurrence equation

and its two *rational* solutions
$u_n$
and
$v_n$
, where
$n\ge 0$
, originating from the initial data
$u_0=0$
,
$u_1=6$
and
$v_0=1$
,
$v_1=5$
. Then
$v_n$
are, in fact, integral for any
$n\ge 0$
and the denominators of
$u_n$
have a moderate growth as
$n\to \infty $
, and are certainly not like
$n!^3$
, as suggested by the recursion, but are
$O(C^n)$
for some
$C>1$
. In fact,
$D_n^3u_n\in \mathbb Z$
for all
$n\ge 1$
, where
$D_n$
denotes the least common multiple of
$1,2,\dots ,n$
; the limit
$D_n^{1/n}\to e$
as
$n\to \infty $
is a consequence of the prime number theorem. An important additional property is that the quotient
$u_n/v_n\to \zeta (3)$
as
$n\to \infty $
(and also
$u_n/v_n\ne \zeta (3)$
for *all* *n*). Even sharper,
$v_n\zeta (3)-u_n\to 0$
as
$n\to \infty $
, and, at the highest level of sharpness,
$D_n^3(v_n\zeta (3)-u_n)\to 0$
as
$n\to \infty $
. It is the latter sharpest form that leads to the conclusion
$\zeta (3)\notin \mathbb Q$
. But already the arithmetic properties of
$u_n,v_n$
coupled with the ‘irrational’ limit relation
$u_n/v_n\to \zeta (3)$
as
$n\to \infty $
are phenomenal.

One way of proving all the above claims in one go is to recast the sequence $I_n=v_n\zeta (3)-u_n$ as the Beukers triple integral [Reference Beukers4]

A routine use of creative telescoping machinery, based on the Almkvist–Zeilberger algorithm [Reference Almkvist and Zeilberger2] (in fact, its multivariable version [Reference Apagodu and Zeilberger3]), then shows that
$I_n$
indeed satisfies (1), while the evaluations
$I_0=\zeta (3)$
and
$I_1=5\zeta (3)-6$
are straightforward. The arithmetic and analytic properties follow from the analysis of the integrals
$I_n$
performed in [Reference Beukers4]; more *practically*, they can be predicted and checked numerically based on the recurrence equation (1).

A common belief is that we have a better understanding of the phenomenon these days. Namely, we possess some (highly nonsystematic!) recipes and strategies (see, for example, [Reference Almkvist, van Straten, Zudilin, Yui, Verrill and Doran1, Reference Chamberland and Straub6, Reference Dougherty-Bliss, Koutschan and Zeilberger7, Reference Zagier and Harnad13, Reference Zudilin15, Reference Zudilin16]) for getting other meaningful constants *c* as *Apéry limits*. In other words, there are (irreducible) recurrence equations with coefficients from
$\mathbb Z[n]$
such that, for two *rational* solutions
$u_n,v_n$
, we have
$u_n/v_n\to c$
as
$n\to \infty $
and the denominators of
$u_n,v_n$
are growing at most exponentially in *n*. (We may also consider *weak* Apéry limits when the latter condition on the growth of denominators is dropped.) Although one would definitely like to draw some conclusions about the irrationality of those constants *c*, this constraint for the arithmetic to be in the sharpest form would severely shorten the existing list of known Apéry limits; for example, it would throw out Catalan’s constant from the list. A very basic question is then as follows.

Question 1. What real numbers can be realised as Apéry limits?

Without going into this in any detail, we present here a (‘weak’) construction of Apéry limits which are related to the *L*-values of elliptic curves (or of weight two modular forms). The construction emanates from identities, most of which remain conjectural, between the *L*-values and Mahler measures.

Consider the family of double integrals

Thanks to the nice hypergeometric representation, a recurrence equation satisfied by the double integral can be computed using Zeilberger’s fast summation algorithm [Reference Apagodu and Zeilberger3, Reference Zeilberger14], which is based on the method of creative telescoping. It leads to the third-order recurrence equation:

Furthermore, if we take

then $J_0(z)=\lambda (z)$ ,

In other words, each $J_n(z)$ is a $\mathbb Q(z)$ -linear combination of $\lambda (z),\rho _1(z),\rho _2(z)$ . For $z^{-1}\in \mathbb Z\setminus \{\pm 1\}$ , we find experimentally that the coefficients $a_n,b_n,c_n$ (depending, of course, on this $z^{-1}$ ) in the representation

satisfy

Now observe that

for $n=0,1,2,\dots .$ The sequences

satisfy the following third-order (again!) recurrence equation which is the exterior square of the recurrence for $J_n$ :

where

and

and

Furthermore, by construction,

and, still only experimentally and for $z^{-1}\in \mathbb Z\setminus \{\pm 1\}$ ,

for
$n=0,1,2,\ldots .$
In other words, the number
$\lambda /\rho _1$
(but also the quotients
$\lambda /\rho _2$
and
$\rho _1/\rho _2$
) are (weak) Apéry limits for the values of *z* under consideration.

For real $k>0$ with $k^2\in \mathbb Z\setminus \{0,16\}$ , the Mahler measure

is expected to be rationally proportional to the *L*-value

of the elliptic curve $E=E_k:X+X^{-1}+Y+Y^{-1}+k=0$ of conductor $N=N_k=N(E_k)$ . This is actually proven [Reference Brunault and Zudilin5] when $k=1$ , $\sqrt 2$ , $2$ , $2\sqrt 2$ and $3$ for the corresponding elliptic curves 15a8, 56a1, 24a4, 32a1 and 21a4 labelled in accordance with the database [9]; the first number in the label indicates the conductor.

For the range $0<k<4$ ,

which thus links
$\mu (k)$
to
$z^{-1/2}\lambda (z)/\pi $
at
$z=k^2/16$
. Furthermore, the quantity
$z^{-1/2}\rho _1(z)$
in this case is rationally proportional to the imaginary part of the nonreal period of the same curve, while
$z^{-1/2}\rho _2(z)$
is a
$\mathbb Q$
-linear combination of the imaginary parts of the nonreal period and the corresponding quasiperiod. This means that, in many cases, we can record
$z^{-1/2}\rho _1(z)$
as a rational multiple of the central *L*-value of a quadratic twist of the curve *E*. For example, when
$k=2\sqrt 2$
(and hence
$z=1/2$
) the quadratic twist of the elliptic curve of conductor 32 coincides with itself and

From this, we see that the last recursion above with the choice $z=1/2$ realises the quotient $L(E,2)/(\pi L(E,1))$ as an Apéry limit for an elliptic curve of conductor 32. When $k=1$ ,

for the twist of the elliptic curve by the quadratic character $\chi _{-4}=\big(\frac{-4}{\cdot}\big)$ ; this means that the quotient $L(E,2)/(\pi L(E,\chi _{-4},1))$ for an elliptic curve of conductor 15 is realised as an Apéry limit.

Clearly, the range
$0<k<4$
has a limited supply of elliptic *L*-values. When
$k>4$
, one can write

where

with $Z=z^{-1}>1$ . At this point, we see that the integrals resemble the integrals

where $h,j,k,l,m$ are nonnegative integers, appearing in the linear independence results for the dilogarithm [Reference Rhin and Viola11, Reference Viola and Zudilin12]. This similarity suggests looking at the family

where
$Z=z^{-1}$
is a large (positive) integer. We tackle this double integral by iterated applications of creative telescoping: while the first integration (regardless of whether one starts with *x* or with *y*) can be done with the Almkvist–Zeilberger algorithm, the second one requires more general holonomic methods, since the integrand is no longer hyperexponential. Using the Mathematica package HolonomicFunctions [Reference Koutschan8], where these algorithms are implemented, we find that the integral
$L_n(Z)$
satisfies a lengthy fourth-order recurrence equation. Moreover, it turns out that
$L_n(Z)$
is a
$\mathbb Q(Z)$
-linear combination of
$\rho _1=\rho _1(1/Z)$
,
$\rho _2=\rho _2(1/Z)$
,
$\sigma _1=L_0(Z)$
and

One can produce a recurrence equation out of the one for $L_n(Z)$ to cast, for example, $\sigma _1/\rho _1$ as an Apéry limit. Because this finding does not meet any reasonable aesthetic requirements and does not imply anything (to be claimed) irrational, we do not include it in this article.