None of the identities in this section seems to express γ as a value of an arithmetic special function.

IS EULER'S CONSTANT A VALUE OF AN ARITHMETIC SPECIAL FUNCTION?

TANGUY RIVOAL Abstract. Euler's constant γ is one of the mathematical constants with the most dierent analytic representations, probably on par with π. Yet, none of these representations proves that γ is a value of an E-function, a G-function or an M -function at an algebraic point. In fact, it is plausible that no such representation of γ exists with these three arithmetic special functions, and thus the arithmetic nature of γ might not be determined by the powerful Diophantine theorems of Siegel-Shidlovsky, Chudnovsky and Nishioka. Nonetheless, we explain here why certain of these representations show that γ is not far from being a special value of these functions. We also present a new family of series summing to γ, which generalize an identity of Vacca.

1. Representations of Euler's constant.

The usual denition of Euler's constant is

γ := lim k→+∞ ( k ∑ n=1 1 n -log(k) ) = ∞ ∑ n=1 ( 1
n -log ( 1 + 1 n ) ) , (1)
where the second equality is a trivial reformulation of the rst one. There exist a lot of seemingly dierent analytic representations of γ. We list below thirty three expressions, all equal to γ. See [START_REF] Allouche | Summation of series dened by counting blocks of digits[END_REF][START_REF] Appell | Sur les séries numériques convergentes à termes positifs dont la somme est rationnelle[END_REF][START_REF] Berndt | Ramanujan's short unpublished manuscript on integrals and series related to Euler's constant[END_REF][START_REF] Catalan | Sur la constante d'Euler et la fonction de Binet[END_REF][START_REF] Finch | Mathematical constants, Encyclopedia of Mathematics and Its Applications[END_REF][START_REF] Glaisher | On Dr. Vacca's series for γ[END_REF][START_REF] Hardy | Note on Dr. Vacca's series for γ[END_REF][START_REF] Havil | Gamma: Exploring Euler's constant[END_REF][START_REF] Lagarias | Euler's constant: Euler's work and modern developments[END_REF][START_REF] Nimbran | Euler's constant and Gamma function: a survey and some new results[END_REF][START_REF] Pilehrood | Vacca-type series for values of the generalized Euler constant function and its derivative[END_REF][START_REF] Krämer | Die Eulersche Konstante γ und verwandte Zahlen[END_REF][START_REF] Sondow | Double integrals for Euler's constant and ln(4/π) and an analog of Hadjicostas's formula[END_REF][START_REF] Sondow | New Vacca-type rational series for Euler's constant γ and its alternating analog ln(4/π)[END_REF][START_REF] Sondow | Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper[END_REF][START_REF] Vacca | A new series for the Eulerian constant γ = 0 • 577[END_REF][START_REF] Vacca | Nuova serie per la costante di Eulero C = 0, 577[END_REF] for proofs, other identities and references.

∫ ∞ 1 ( 1
⌊x⌋ - 1 x ) dx, 1 - ∫ ∞ 1 x -⌊x⌋ x 2 dx, ∫ ∞ 0 log(1 + x) x 2 (log(x) 2 + π 2 ) dx, (2)
-

∫ ∞ 0 log(x)e -x dx, - ∫ ∞ 1 log log(x) x 2 dx, ∫ 1 0 ( 1 log(x) + 1 1 -x ) dx, (3)
∫ 1 0 ∫ 1 0 x -1 (1 -xy) log(xy) dxdy, ∫ ∞ 0 1 x ( 1 1 + x 2 -cos(x) ) dx, ab a -b ∫ ∞ 0 e -x a -e x b
x dx, (4)

1 2 + ∫ ∞ 0 2xdx (1 + x 2 )(e 2πx -1)
,

∫ 1 0 1 -e x -e -1/x x dx, ∫ ∞ 0 ( 1
e x -1 - 1 xe x ) dx, (5) ∫ ∞ 0 1 x ( 1 1 + x -e -x ) dx, ∫ 1 0 1 x(1 + x) ∞ ∑ n=1 x 2 n dx, ∫ 1 0 2 + x x(1 + x + x 2 ) ∞ ∑ n=1 x 3 n dx, (6) ∞ ∑ n=1 | ∫ 1 0 ( t n ) dt| n , ∞ ∑ n=1 (-1) n-1 n!n - 1 e ∫ ∞ 0 e -x 1 + x dx, ∞ ∑ n=2 1 n 2 ( n ∑ k=1 (-1) k ( n k ) k log(k)
) , [START_REF] Andrews | Special functions, Encyclopedia of Mathematics and Its Applications %[END_REF] Date : December 20, 2017.

∞ ∑ n=1 (-1) n ⌊log 2 (n)⌋ n , 1 - ∞ ∑ n=2 (-1) n ⌊log 2 (n)⌋ n + 1 , 2 ∞ ∑ n=1 cos( 2πn 3 )⌊log 3 (n)⌋ n , (8) 1 2 + ∞ ∑ n=1 ⌊log 2 (2n)⌋ 2n(2n + 1)(2n + 2) , 1 2 + ∞ ∑ n=1 (12n + 6)⌊log 3 (3n)⌋ 3n(3n + 1)(3n + 2)(3n + 3) , ( 9
) log(π) -4 log Γ ( 3
4 ) + 4 π ∞ ∑ k=1 (-1) k+1 log(2k + 1) 2k + 1 , 1 2 + ∞ ∑ n=2 1 2 n+1 n-1 ∑ k=0 1 ( 2 n-k +k k ) , (10) ∞ ∑ n=1 N 0 (n) + N 1 (n) 2n(2n + 1) , N j (n) = number of j's in n written in base 2, ( 11
) ∞ ∑ n=1 ( 1
⌊ √ n⌋ 2 - 1 n - 1 n 2 ) , 1 s -1 ∞ ∑ n=1 ( s n - (n+1) s -1 ∑ j=n s 1 j ) , s ≥ 2, (12)
∞ ∑ n=2 (-1) n ζ(n) n , 1 + ∞ ∑ n=2 1 -ζ(n) n , 1 + log ( 2 3
)

+ ∞ ∑ n=1 1 -ζ(2n + 1) (2n + 1)2 2n , ( 13
) ∞ ∑ n=1 ρ(n) n , ρ(n) = -(2k -1) if n = k 2 , ρ(n) = 1 otherwise, ( 14
) ∞ ∑ n=2 Λ(n) -1 n , Λ(n) = log(p) if n = p k , Λ(n) = 0 otherwise. (15)
Some of these expressions are merely variations of one another, but it is dicult to distinguish a common pattern because of the diversity of elementary functions involved: oor, square root, exponential, cosine, logarithm, zeta, digital, arithmetic functions, binomial coecients, etc. Presumably, π is the only other classical constant with at least as many representations. However, contrary to π, the arithmetic nature of γ is still an open problem. Nobody knows whether it is a rational or an irrational number, an algebraic or a transcendental number, though γ is widely expected to be transcendental. Moreover, π = ∫ ∞ -∞ dx 1+x 2 is a period; none of the above representations proves that γ is a period and in fact a folklore conjecture states that γ is not a period, though it is one of the simplest examples of an exponential period in the sense of Kontsevich-Zagier [START_REF] Kontsevich | Periods, Mathematics unlimited 2001 and beyond[END_REF].

Despite very general transcendence results for various classes of arithmetic special functions, none of them is known to apply to γ. In the sequel, we explain that even though γ is conjecturally not a value at an algebraic point of these arithmetic special functions, certain representations hidden in (1) [START_REF] Pilehrood | Vacca-type series for values of the generalized Euler constant function and its derivative[END_REF] show that it is not very far from being one.

Arithmetic special functions.

It is not easy to dene the meaning of special functions though there exist many books devoted to them, for instance [START_REF] Andrews | Special functions, Encyclopedia of Mathematics and Its Applications %[END_REF]; a voluntarily imprecise denition might be a mathematical function used for ages in some part of science and satisfying some sort of functional equation. In our Diophantine context, the arithmetic special functions we have in mind are of three types. Siegel dened E-functions and G-functions in [START_REF] Siegel | Über einige Anwendungen diophantischer Approximationen[END_REF]: these are series of ! the form

∑ ∞ n=0 an n! z n , respectively ∑ ∞ n=0 a n z n
, where the height of a n ∈ Q satises a certain growth condition (see [START_REF] Fischler | On the values of G-functions[END_REF]), and solutions of a dierential equation

∑ d j=0 p j (z)y (j) (z) = 0 with coecients p j (z) ∈ Q(z).
Prototypical examples are the exponential and logarithmic functions respectively, and the intersection of both classes is reduced to polynomials functions. M -functions are power series ∑ ∞ n=0 a n z n with a n ∈ Q, and solutions of a functional equation ∑ d j=0 p j (z)y(z ℓ j ) = 0 for some integers ℓ ≥ 2 and d ≥ 1, and p j (z) ∈ Q(z). A simple example is the series

∑ ∞ n=0 z 2 n
. They were rst studied by Mahler [START_REF] Mahler | Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen[END_REF] in a broader context. E-functions are entire and G-functions can be analytically continued to C minus a nite number of cuts; an M -function is either in Q(z) or has the unit circle as natural boundary ( [START_REF] Randé | Équations fonctionnelles de Mahler et applications aux suites p-régulières[END_REF][START_REF] Bell | The rational-transcendental dichotomy of Mahler functions[END_REF]) and moreover if it satises a linear dierential equation, then it must be a in Q(z) by [START_REF] Bézivin | Sur une classe d'equations fonctionnelles non linéaires[END_REF].

The arithmetic nature of the values of E-functions at algebraic points have been studied

for a long time: the Siegel-Shidlovsky Theorem [START_REF] Shidlovsky | Transcendental Numbers[END_REF] is the landmark result, with renements in [START_REF] Adamczewski | Exceptional values of E-functions at algebraic points[END_REF][START_REF] André | Séries Gevrey de type arithmétique, I and II[END_REF][START_REF] André | Arithmetic Gevrey series and transcendence. A survey[END_REF][START_REF] Beukers | A rened version of the Siegel-Shidlovskii theorem[END_REF][START_REF] Yu | On the linear independence of values of E-functions[END_REF]. As a result, there exists now an algorithm which, for any given Efunction f (z), provides explicitly the nite list of algebraic numbers α such that f (α) ∈ Q. Essentially the same things can be said of M -functions. An analogue of the Siegel-Shidlovsky Theorem was proved by Nishioka [START_REF] Ku | Algebraic independence measures of the values of Mahler functions[END_REF] and rened in [START_REF] Adamczewski | Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques[END_REF][START_REF] Adamczewski | Méthode de Mahler, transcendance et relations linéaires : aspects eectifs[END_REF][START_REF] Philippon | Groupes de Galois et nombres automatiques[END_REF]. Again, there exists now an algorithm which, for any given M -function f (z), provides explicitly the list of algebraic numbers α such that f (α) ∈ Q. On the other hand, the situation is not completely understood for G-functions. The best general Diophantine result is due to Chudnosvky [START_REF] Chudnovsky | On applications of diophantine approximations[END_REF], but we are far from having an algorithm to perfom the same task as for E-functions and M -functions. Let us denote by E, G and M the subsets of C of all the values taken by E-functions and (analytic continuation of ) G-functions and M -functions at algebraic points, respectively.

The rst two sets have been introduced and studied in [START_REF] Fischler | On the values of G-functions[END_REF] and [START_REF] Fischler | Arithmetic theory of E-operators[END_REF] respectively, see also [START_REF] Fischler | Microsolutions of dierential operators and values of arithmetic Gevrey series[END_REF]; in particular it was proved there that E and G are in fact subrings of C. The set M does not seem to have been considered so far but it is easy to prove that it is a ring as well. As said above, none of the above Diophantine results has been applied so far to γ because this number is not known to be in either E or G or M.

In fact, given the long list of series and integral expressions for γ, one can even argue that if it were such a value, this would be known for a long time. Hence it is plausible that γ is not in E ∪ G ∪ M and in this sense, the answer to the title of this note is negative. Moreover, for similar reasons it is believed that

E ∩ G = Q, E ∩ M = Q and G ∩ M = Q.
Let us also say that it is also believed no Liouville number is in E ∪ G ∪ M, but it would certainly be surprising if γ were itself a Liouville number.

Extension of arithmetic sets.

We claim that γ is in some sense close to be in E ∩ G ∩ M. Of course, this statement is paradoxical because this intersection is expected to contain only algebraic numbers, while γ is expected to be transcendental. We mean the following: we can slightly extend the sets E and M to certain sets E and M respectively so that γ ∈ E ∩ M. The extension process is the same for both classes of functions, which adds value to this observation. The process can be applied to G but we get G = G. However, we shall prove the non-trivial fact that G ⊂ E, so that E can be viewed as a common extension of both E and G.

We dene E, respectively M, as the set of all convergent evaluations at z ∈ Q ∪ {∞} of the twisted primitives

∫ z 0 Q(x)F (x)dx (16)
where Q(z) ∈ Q(z) and F (z) is an E-function, respectively an M -function, such that QF is holomorphic at z = 0. Changing the variable z to αz for any algebraic number, we can in fact assume that z ∈ {1, ∞} in [START_REF] Finch | Mathematical constants, Encyclopedia of Mathematics and Its Applications[END_REF]. The drawback of this voluntarily simple denition is that we do not know if E and M carry an algebraic structure. Of course, we get two Q-vector spaces by considering the a priori larger sets of numbers

∫ β α Q(x)F (x)dx where α, β ∈ Q ∪ {∞} and F is an E-function, respectively an M -function.
We have E ⊂ E and M ⊂ M because if F (z) is an E-function, respectively an Mfunction, then F ′ (z) is an E-function, respectively an M -function, and F (z) = ∫ z 0 F ′ (x)dx. It is likely however that E ̸ = E and M ̸ = M, because when F (z) is an E-function or an M -function, then an integral of the form ( 16) is in general neither an E-function nor an M -function. Taking F (z) = 1 and Q

(z) = 1 1+z , we see that log(Q * ) ⊂ E ∩ M.
Values at algebraic points of iterated primitives vanishing at 0 of E-functions and M -functions are in E and M respectively because for any integer p ≥ 1

( ∫ z 0 ) p F (x)dx = (-1) p-1 (p -1)! ∫ z 0 (x -z) p-1 F (x)dx. (17)
But since the class of E-functions is stable by d dz and

∫ z 0 , we get nothing more than E if F is an E-function in [START_REF] Glaisher | On Dr. Vacca's series for γ[END_REF]. On the other hand, F is an M -function if and only if QF is an M -function, so that M is exactly the set of values at algebraic points of primitives of M -functions vanishing at 0.

Much more interesting is the property that γ ∈ E ∩ M. Indeed, by (6), we have

γ = ∫ ∞ 0 1 -(1 + x)e -x x(1 + x) dx ∈ E (18) = ∫ 1 0 1 x(1 + x) ∞ ∑ n=1 x 2 n dx ∈ M (19) = ∫ 1 0 2 + x x(1 + x + x 2 ) ∞ ∑ n=1 x 3 n dx ∈ M (20) because 1 -(1 + z)e -z is an E-function and ∑ ∞ n=1 z 2 n , ∑ ∞ n=1 z 3 n are M -functions. Eq. ( 18
)
is apparently due to Dirichlet, while Eqs. ( 19) and ( 20) are due to Catalan and Ramanujan, respectively. All the series in ( 8) and ( 9) can be transformed in a straighforward manner as integrals

∫ 1 0 Q(x)F (x)dx for some Q(z) ∈ Q(z)
and some M -function F (z). Moreover, the series ( 11) is just a rewritting of the rst series in [START_REF] Appell | Sur les séries numériques convergentes à termes positifs dont la somme est rationnelle[END_REF].

#

With the algebraic numbers, the numbers π and ζ(n) (n any integer ≥ 2) are amongst the simplest elements of G. So far, no simple relation between γ and G-functions is known, though γ is often considered to be the regularized value of the divergent ζ [START_REF] Adamczewski | Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques[END_REF]. The tilde extension presented here can also be applied to G-

functions but if F (z) is a G-function then ∫ z 0 Q(x)F (x)dx is still a G-function and a convergent integral ∫ α 0 Q(x)F (x)dx with α ∈ Q ∪ {∞} is in G.
Hence G = G. However, we have the following property. Proposition 1. We have G ⊂ E.

More precisely, G coincides with the set of all convergent integrals

∫ ∞ 0 F (x)dx of E- functions F .
Equivalently, G coincides with the set of all nite limits of E-functions at innity (along some direction). ∫ ∞ 0 F (x)dx of E-functions and the set of nite limits of E-functions along some direction as z → ∞ are the same.

Theorem 2(iii) in [START_REF] Fischler | Arithmetic theory of E-operators[END_REF] implies that if an E-function has a a nite limit as z → ∞ along some direction, then this limit must be in G.

Conversely, let β ∈ G. By Theorem 1 in [START_REF] Fischler | On the values of G-functions[END_REF], there exists a G-function

G(z) = ∑ ∞ n=0 a n z n of radius of convergence ≥ 2 (say) such that G(1) = β. Let F (z) = ∑ ∞ n=0
an n! z n be the associated E-function. Then for any z such that Re(z) > 1

2

, we have

1 z G ( 1
z ) = ∫ +∞ 0 e -xz F (x)dx. Hence, β = ∫ +∞ 0 e -x F (x)dx. Now, e -z F (z) is an E-function, so that β ∈ E. □
This proposition shows how dicult it would be to obtain a Siegel-Shidlovsky like result for the elements of E. A generalization of Nishioka's Theorem to elements of M would probably be dicult to obtain as well. On the other hand, it is not formally necessary to prove such general theorems if one is only interested in γ, as ad hoc Diophantine arguments might be simpler to obtain for the integrals ( 18), [START_REF] Havil | Gamma: Exploring Euler's constant[END_REF] or [START_REF] Kontsevich | Periods, Mathematics unlimited 2001 and beyond[END_REF].

The number log(4/π) is sometimes named the alternating Euler's constant because [START_REF] Sondow | New Vacca-type rational series for Euler's constant γ and its alternating analog ln(4/π)[END_REF] log

(4/π) = ∞ ∑ n=1 (-1) n-1 ( 1
n -log ( 1 + 1 n
) ) .

(Compare with [START_REF] Adamczewski | Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques[END_REF].) The identity [START_REF] Pilehrood | Vacca-type series for values of the generalized Euler constant function and its derivative[END_REF] log(4/π) =

∫ 1 0 x 2 -x -1 1 -x 2 ∞ ∑ n=0 (1 -x 2 n )x 2 n -1 1 + x 2 n dx
proves that log(4/π) ∈ M. There exist other representations similar to certain for γ, for instance [START_REF]Polynômes de type Legendre et approximations de la constante d'Euler[END_REF][START_REF] Sondow | New Vacca-type rational series for Euler's constant γ and its alternating analog ln(4/π)[END_REF][START_REF] Allouche | Summation of series dened by counting blocks of digits[END_REF]:

log(4/π) = ∫ 1 0 1 -x 1 + x ( 1 log(x) + 1 1 -x ) dx = ∫ 1 0 ∫ 1 0 x -1 (1 + xy) log(xy) dxdy = ∞ ∑ n=1 N 0 (n) -N 1 (n) 2n(2n + 1
) , but it does not seem to be known whether log(4/π) ∈ E or not.

Finally, let us mention that arithmetic special functions could also include q-analogues.

It is proved in [START_REF] Kurokawa | On q-analogues of the Euler constant and Lerch's limit formula[END_REF] that

γ = lim q→1 + ( ∞ ∑ n=1 ( 1 -q 1 -q n ) + (q -1) log(q -1) log(q) + 1 -q 2 )
which is a q-analogue of the denition of γ stated in [START_REF] Adamczewski | Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques[END_REF]. But no evaluation of a q-series or a q-integral in terms of γ seems to be known.

Other extensions.

The form of the integral in ( 16) also suggests to dene a subset EGM of C as follows:

it is the set of all convergent integrals [START_REF] André | Arithmetic Gevrey series and transcendence. A survey[END_REF] shows in many dierent ways that γ ∈ EGM. However, this set is quite big because it contains periods, exponential periods, E, G, M, E and M. This likely makes EGM very dicult to study from a Diophantine point of view.

∑ ∞ n=1 z 2 n -1 is an M -function,
To get other constants than γ without extending too much E, G or M, we consider the twisted primitives

∫ z 0 Q(x)F (x)dx
where Q(z) is now an algebraic function over Q(z) and F (z) is either an E-function or an M -function such that QF is integrable at z = 0. When F (z) is an E-function, respectively an M -function, any convergent integral ∫ α 0 Q(x)F (x)dx, with α ∈ {1, ∞}, denes an element of E and M. Again, the double tilde extension is not interesting for G-functions and G = G, but obviously we still have G ⊂ E. % For instance, for any rational number s > 0, we have

Γ(s) = ∫ ∞ 0 x s-1 e -x dx ∈ E and [10] Ψ(s) := Γ ′ (s) Γ(s) = ∫ 1 0 ( x s-1 1 -x - 2x 2s-2 1 -x 2 ) ∞ ∑ n=0 x 2 n dx ∈ M.
However, it does not seem to be known whether Γ(s) ∈ M and Ψ(s) ∈ E.

We don't know if EGM, E and M carry any interesting algebraic structure. We get three Q-vector spaces by considering the set of convergent integrals ∫ β α e(z)g(z)m(z)dz, where α, β ∈ Q ∪ {∞}, e, g, m are an E-fonction, a G-function and an M -function respectively, as well as the two sets of convergent integrals

∫ β α Q(x)F (x)dx where α, β ∈ Q ∪ {∞}, Q is algebraic over Q(z)
and F an E-function, respectively an M -function. [START_REF] André | Arithmetic Gevrey series and transcendence. A survey[END_REF]. A generalization of Vacca's second identity.

Vacca [START_REF] Vacca | A new series for the Eulerian constant γ = 0 • 577[END_REF] proved in a geometric way the identity

γ = ∞ ∑ n=1 n ( 2 n+1 -1 ∑ k=2 n (-1) k k ) ,
which is a variant of the rst identity displayed in [START_REF] Appell | Sur les séries numériques convergentes à termes positifs dont la somme est rationnelle[END_REF]. Analytic proofs were quickly found by Glaisher [START_REF] Glaisher | On Dr. Vacca's series for γ[END_REF] and Hardy [START_REF] Hardy | Note on Dr. Vacca's series for γ[END_REF], the latter showing that Vacca's identity is also a variant of Catalan's integral [START_REF] Catalan | Sur la constante d'Euler et la fonction de Binet[END_REF]

γ = ∫ 1 0 1 x(1+x) ∑ ∞ n=1 x 2 n dx, displayed in (6).
Later on, Vacca proved in [START_REF] Vacca | Nuova serie per la costante di Eulero C = 0, 577[END_REF] a second identity, stated in [START_REF] Bézivin | Sur une classe d'equations fonctionnelles non linéaires[END_REF], namely

γ + ζ(2) = ∞ ∑ n=1 ( 1
⌊ √ n⌋ 2 - 1 n ) . ( 21
)
Because the paper [START_REF] Vacca | Nuova serie per la costante di Eulero C = 0, 577[END_REF] might be dicult to nd, for the reader's convenience, we present a proof of the following generalization of (21).

Proposition 2. For any integer s ≥ 2, we have

γ = 1 s -1 ∞ ∑ n=1 ( 1
⌊n 1/s ⌋ s - 1 n - s ∑ ℓ=2 ( s ℓ ) n ℓ
) . The case

s = t = 2 reads ∞ ∑ n=1 n ( 1
⌊ √ n⌋ 4 - 1 n 2 ) = γ + 3ζ(2) + ζ(3) or γ = ∞ ∑ n=1 n ( 1
⌊ √ n⌋ 4 - 1 n 2 - 3 n 3 - 1 n 4
) .

  Proof. From the stability of the class of E-function by d dz and ∫ z 0 , we deduce that the set of convergent integrals

∫ α 0 e

 0 (z)g(z)m(z)dz where α ∈ {1, ∞}, and e(z) is E-fonction, g(z) is a G-function g(z) and m(z) is an Mfunction. Since 1-(1+z)e -z z is an E-function, 1 1+z is a G-function (and an M -function as well), and

1 N 1 1 j n=1 ( n t- 1 ⌊n 1 n=1 ( n t- 1 ⌊n 1

 111n=111n=111 -1)γ + s log(⌊N 1/s ⌋) -log(N ) + O ( ) + o(1), N → ∞ where we used∑ 1≤j≤x = γ + log(x) + O( 1x ) as x → +∞. Eq. (22) follows. □The convergence is slow because 1⌊n 1/s ⌋ s -1 n = O( 1 n 1+1/s) and the exponent can not be improved.A similar argument enables one to sum the series∞ ∑ /s ⌋ st -1 n) for any integers s ≥ 2 and t ≥ 1. The formula is slightly more complicated to write down explicitly. We use the summation formula (j+1) s -1 s,t j m where P k (X) is the k-th Bernoulli polynomial, and the α m,s,t ∈ Q could be explicited in terms of the Bernoulli numbers and binomial coecients. We then get ∞ ∑ s,t ζ(st -m).


Thierry Chanier

Marie-Noëlle Lamy

Researching Technology-mediated Multimodal Interaction

, and for a summary about technology-mediated language learning see Hampel (2012). These sources provide a starting point for the investigation of multimodality in language learning, which is relatively recent computer-assisted language learning (CALL) research. In view of the importance of such meaning-making resources for learning, this chapter introduces the meaning of "mode" and "multimodality" before introducing the research and theoretical issues in the study of multimodality in language learning. It then describes a corpus-based approach for studying multimodality in CALL, drawing attention to important methodological factors such as transcription that come into play in conducting such research.

Multimodality in online learning

During online communication in a second language (L2), learners orchestrate various resources including language, in its written and spoken forms, as well as images, colors, movements, and sounds. Responding to even a simple written post requires at minimum two such resources: the linguistic mode, that is, the written language, and the visual mode, which involves the choice of fonts and the organization of spaces on the screen. In addition to language and image, online tools such as a floating magnifier facilitates the understanding not only of written communication but also of pictures.

In online shops such tools can be used to reveal enlargeable images of the product to allow the shopper to see products in greater detail. The floating magnifier tool in itself has no meaning but the enlarged image or written text probably does. For language learners, the floating magnifier may be used to support a vocabulary or grammatical gloss, and an image may carry a cultural reference. The physical material also includes keys to be pressed for text creation, screens to be tapped, pads to be stroked or hotspots to be clicked for opening up video or audio channels. In computer-mediated interactive language learning (henceforth CMILL), learning is affected by the resources that are available to learners and their use. Therefore, the design of learning activities and research on their use needs to take into account of the materiality of the modes available to learners and how they are used to create meaning multimodally (Lamy 2012a).

What are modes?

Mode in linguistics refers to the resources used to express meaning. For example, [START_REF] Kress | Multimodal Discourse: The Modes and Media of Contemporary Communication[END_REF] showed that readers make sense out of a page of a newspaper by combining their understanding of the linguistic content (linguistic mode), with their interpretation of the layout and photos or cartoons (visual mode). In other words, language users combine semiotic resources to convey messages through simultaneous realization of linguistic and non-linguistic modes in printed media. In CMILL, the resources to be co-orchestrated by participants are made up of a greater variety of modes than those in print media or audio-video alone. CMILL is carried out through the use of modes, which are accessed and manipulated with tools to carry out certain learning objectives. The integration of these three aspects of communication makes up modality. The relationships between the three are illustrated in Table 28.1. The objectives refer to the types of communication and learning activities that the students engage in such as information seeking as part of a communication task, reflection on their past work, and peer collaboration to produce a written product. The tools are the software and hardware configured in a way that allows learners to use them to accomplish learning objectives. These tools are socially-shaped and culturally-constructed to provide access to the modes of communication. [START_REF] Kress | Multimodality A Social Semiotic Approach to Contemporary Communication[END_REF] explains the connection between tools and modes, by noting, for example, that an image shows the world and a written text tells the world. Therefore modes offer "distinctive ways of representing the world" (96). So while a tool may borrow a representation from the world (a cursor may be materialized on screen as a hand or a pair of scissors) the tool merely indicates by this that its function is to grab or to cut. Its function is not to represent hands or scissors to the user. Materialized through the use of tools, modes combine together to facilitate meaning-making in learning. Thus we can define modality as the relationship between tools and modes.

What is multimodality?

Multimodality is the complex relationship that develops between multiple tools and modes when they are co-deployed in different combinations, in learning situations to work toward particular objectives. In online audio-visual environments the complex meaning-making (or semiotic) possibilities that open up to language learners are materialized through hardware and software. New meanings emerge in learning situations through learners' physical relati onship to tools (sometimes called embodiment), through participants' body language on screen (another form of embodiment), through learners' engagement with still and moving images, with sounds, and with each other's language outputs. All of this is experienced in simultaneous integration or, as some multimodality researchers put it, co-orchestration.

Another way of expressing this integration is by examining the notion of mediation.

Mediation is always present in any kind of human interaction, including multimodal ones. The separate meaning-making resources of mediation in online learning are illustrated in Figure 28.1 as three circles, labeled A, B and C. Circle A represents the resources available through the participants' physical bodies. Circle B represents the resources that are available through the technology. Circle C symbolizes the meaning-making resources available through language. In any situation of meaning-making, these separate components of mediation form intersecting areas (shown in the figure with Roman numerals). In offline meaning-making, only two types of resources are present: for instance when we hail friends across a coffee-bar, we are making meaning with our smiling face, our waving arm, and our cheerful call of their name (represented by Area II, where Circles C and A

Circle B Physical material and technologies III Circle C Language IV I II

Circle A Body language, gestures, resources intersect, that is, where language and body language meet). Language II often part of meaning-making, but Area I, at the intersection of Circles A and B, represents instances when language is not involved. For example, a piano tuner using auditory senses and hands (Circle A) to tune a piano (Circle B) would create meaning without language. Finally, an example of Area III might be two computers programmed to communicate with each other.

In online multimodality, the three components intersect: for example when we edit our photos and upload them to a social network site, we are working mainly with our hands and eyes (resources in Circle A), with our tablet/phone/laptop (resources in Circle B), with the language of our editing suite's instructions, and with our own language (resources in Circle C) to create captions and label albums. In addition if we interact with others (for example by engaging with them in "commenting" on the photographic network) we again bring in resources from Circle C. Our online interaction is thus mediated through technology, through our own body and through language (a triple overlap represented by Area IV in the middle of the figure). Area IV is therefore the space where phenomena of interest to researchers in multimodality within CMILL reside.

Research on multimodality in online learning

Research on multimodality examines two questions: "What aspect of the learning is mediated through the technology?" and "How is the learning situation experienced, especially in terms of affect?" [START_REF] Erben | Constructing Learning in a Virtual Immersion Bath: LOTE Teacher Education through Audiographics[END_REF] showed participants creating new discursive practices by altering the content of their exchanges (through "reducing" and "amplifying" meanings) as they strove to find workaround responses to what they experienced as an unwieldy communication platform. His findings made it clear that the "what" of communication was affected, as participants adapted to the crudeness of the mediation tools at their disposal at the time. On the other hand, [START_REF] Goodfellow | Face-to-Face Language Learning at a Distance? A Study of a Videoconference Try-Out[END_REF] as well as [START_REF] Mcandrew | Videoconferencing in a Language Learning Application[END_REF] reported on the "how," showing that video-based learning inter-actions were experienced as stressful, although at the same time video was seen by learners as motivational.

These two lines of investigation have continued to structure much CMILL research in recent years. First, the issue of the "what" needs to be thoroughly understood if communication distortion/breakdown is to be avoided and also, more positively, if the communicative affordances of the multimodal environment are to be maximized. In the decades that followed the pioneering studies above, scholars continued to explore many aspects of mediation as reflected in several of the articles on multimodal learning in ReCALL 25 (2013) and elsewhere. For example, [START_REF] Codreanu | Effects of Webcams on Multimodal Interactive Learning[END_REF] found that tool management influenced the content of the learning while [START_REF] Satar | Multimodal Language Learner Interactions via Desktop Conferencing within a Framework of Social Presence: Gaze[END_REF] analyzed the mediating effects of gaze, and Wigham andChanier (2013a, 2013b) studied the mediation of the learning through digital gestures in Second Life. [START_REF] Dooly | Filling in the Gaps: Linking Theory and Practice through Telecollaboration in Teacher Education[END_REF] looked at how Second Life mediated the transfer of knowledge from theory to practice, and [START_REF] Monteiro | An Experimental Study of Corrective Feedback during Video-Conferencing[END_REF] studied video-mediated corrective feedback. As Wang and Chen pointed out in 2012, "in-depth research is needed to establish the extent to which visual cues mediated through videoconferencing tools are important to collaborative language learning. Such research is more urgently needed now than it was 5 years ago as broadband technology has already made good quality of video transmission a reality" (325). A wider discussion of what tools and affordances best mediate learning in CMILL can be found in [START_REF] Hampel | Making Meaning Online: Computer-mediated Communication for Language Learning[END_REF] and in the domain of synthetic worlds, a review of mediational tools appears in [START_REF] Schwienhorst | The State of VR: A meta-analysis of Virtual Reality Tools in Second Language Acquisition[END_REF]. Finally, Reinhardt and Sykes' (2014) edited issue on game-informed L2 teaching and learning.

The second line of research, centering on "how" the learning situation is experienced, that is, on affect, has produced literature on psychological variables. These range across anxiety, anger, regret, desire and poor self-esteem. An early study of Web use [START_REF] Yang | Language Learning on the World Wide Web: An Investigation of EFL Learners' Attitudes and Perceptions[END_REF]) identified anxiety as a factor connected to cognitive overload. However, other anxiogenic factors in language interaction on platforms did not come to the fore until the late 2000s, with work by [START_REF] Hurd | Anxiety and Non-anxiety in a Distance Language Learning Environment: The Distance Factor as a Modifying Influence[END_REF], de los Arcos (2009), de los Arcos et al. ( 2009), [START_REF] Coryell | One Right Way, Intercultural Participation, and Language Learning Anxiety: A Qualitative Analysis of Adult Online Heritage and Non heritage Language Learners[END_REF], [START_REF] Develotte | The Use of the Webcam for Teaching a Foreign Language in a Desktop Videoconferencing Environment[END_REF] analysis of gaze in desktop conferencing from a socio-affective viewpoint and Tayebinik's and Puteh's (2012) literature review article on self-esteem.

Scholars whose work has included both what and how of learning during the last decade are Wang and to a lesser extent her co-authors Chen and Levy. They studied desktop videoconferencing, which they later called "synchronous cyber face-to-face" from several angles: participants' perceptions of the benefits of tools [START_REF] Wang | Supporting Synchronous Distance Language Learning with Desktop Videoconferencing[END_REF]), focus on form in task completion [START_REF] Wang | Negotiation of Meaning in Desktop Videoconferencing-Supported Distance Language Learning[END_REF], task design [START_REF] Wang | Task Design in Videoconferencing-Supported Distance Language Learning[END_REF], teacher training [START_REF] Wang | Teacher Training in a Synchronous Cyber face-to-face Classroom: Characterizing and Supporting the Online Teachers' Learning Process[END_REF], collaboration [START_REF] Wang | The Collaborative Language Learning Attributes of Cyber Face-to-face Interaction: The Perspectives of the Learner[END_REF] and the question of how principles of instructed language learning can be applied in these environments [START_REF] Chen | Testing Principles of Language Learning in a Cyber Face-to-Face Environment[END_REF]). They found that potential learning benefits could arise in multimodal environments provided tools were used in a way that was balanced and appropriate to pedagogies, to technologies, and to audiences. They also found potential learning losses due to synchrony nerves and anxiety (including among teachers). There are indications that the affect-oriented strand of multimodality research in CMILL is a continuing concern of scholars.

Theoretical underpinnings

Few scholars of multimodality in language learning ground their work theoretically except to claim an affiliation to "sociocultural approaches." Among those who identify a theoretical basis for their work, the literature is broadly divided into those using technoliteracy frame-works and those relying on semiotic theories. In the former category, the aim is to investigate obstacles and facilitators in the development of learner competence with the platforms and tools, while the latter are more interested in understanding how learners orchestrate meanings mediated to them through a variety of modes in the online situation.

On the one hand the technoliteracy-oriented CMILL community has long insisted that task design is key and should closely match the communicative affordances of the environments. The practical consequences of this are outlined by [START_REF] Hampel | Making Meaning Online: Computer-mediated Communication for Language Learning[END_REF] "Tasks need to be appropriate to the environment, and it is crucial that activities that make use of digital environments take account of their functionalities and affordances" (18). Others have also stressed the centrality of pedagogy and task design, with studies focusing on audiographic conferencing; webcam-assisted conferencing; Second Life; and processing overload. For an overview on multimodality research and literacy in CMILL, see [START_REF] Ho | Design and Implementation of a Student-generated Virtual Museum in a Language Curriculum to Enhance Collaborative Multimodal Meaning-making[END_REF].

On the other hand, CMILL scholars using social-semiotic theories have responded to a different priority: the need to understand a learner's meaning-making in multimodal situations. One seminal text explaining how social semiotics can account for multimodal communication is [START_REF] Kress | Multimodality A Social Semiotic Approach to Contemporary Communication[END_REF]. However critics have complained about what they see as the over-prominent role played by language in Systemic Functional Linguistics, a theory that Kress and others (O'Halloran and Smith 2011) consider to be core to the shaping of social-semiotic theories. These critics have argued that for online language-learning set-tings, it is necessary to further root social-semiotic analysis in notions of place and embodiment. Among these scholars some concentrate on the relationship of the body with the physical environment of the virtual experience [START_REF] Jones | The Problem of Context in Computer-mediated Communication[END_REF]Lamy 2012a), others on silence and gaze [START_REF] Stickler | Distant Classmates: Speech and Silence in Online and Telephone Language Tutorials[END_REF][START_REF] Develotte | The Use of the Webcam for Teaching a Foreign Language in a Desktop Videoconferencing Environment[END_REF], and others yet on social presence [START_REF] Dahlberg | Communication in the Virtual Classroom in Higher Education: Languaging Beyond the Boundaries of Time and Space[END_REF]Bagga-Gupta 2011, 2013;[START_REF] Lamy | Describing Online Conversations: Insights from a Multimodal Approach[END_REF]. However an unresolved issue in the social-semiotic analysis of CMILL exchanges remains that of their linguistic component, and of how to transcend language-based methods such as discourse analysis [START_REF] Sindoni | Spoken and Written Discourse in Online Interactions: A Multimodal Approach[END_REF] or conversational analysis, so as to fully recognize their multimodal dynamics.

Given that technology has now opened up possibilities for fully-documented, accurate, and exhaustive capture of multimodal exchanges, both the technoliteracy and the social semiotics research communities should be able to establish a synergistic relationship bet-ween their theories and the empirical data that they collect. However the complexity of multimodal environments, and the sophistication of the tools that can capture it, combine to create massive datasets and not all researchers need all data. So the choice of an appropriate window of observation for each particular project and the selection of relevant categories of data for collection, storage, and analysis are key.

The need to analyze multimodal data in education

The difficulties of collecting and handling online multimodal data have been problematized in non-educational fields. See for example [START_REF] Smith | Analyzing Multimodality in an Interactive Digital Environment: Software as Metasemiotic Tool[END_REF]. Also, for an overview of almost two decades of semiotics-based investigation of multimodal data, see O'Halloran and [START_REF] Smith | Analyzing Multimodality in an Interactive Digital Environment: Software as Metasemiotic Tool[END_REF][START_REF] O'halloran | Interactive Software for Multimodal Analysis[END_REF]. In contrast, the CMILL literature provides few methodological publications focusing on the treatment of multimodal data in language-learning con-texts online. An article entitled "What are multimodal data and transcription?" applied to education more generally [START_REF] Flewitt | What Are Multimodal Data and Transcription?[END_REF] pointed to this gap in the field, and will serve as our introduction the problems of working with multimodal data. "Drawing on findings from across a number of disciplines and research fields including applied linguistics, visual ethnography, symbolic representation and computer-mediated communication" (53), the authors outline issues such as the transcription, description, and analysis of multimodal texts and, before all else, the definition of a unit of analysis for research. They state that in dynamic texts (i.e., conversations) "units of transcription are usually measured as turns of speech, but it is questionable whether this convention is useful for multimodal analysis" (47). Because other modes come into play, Flewitt et al. suggest either linking measurement to the visual mode and using visual frames as units or timeframes can provide definitions for the unit for analysis.

However, kinesics data (how the on-screen moves of the artifacts are understood), proxemics (how distant participants "feel" they are to one another), or postural, gestural, and gaze elements should also be included. The authors conclude that "the representation of the complex simultaneity of different modes and their different structure and materiality-has not been resolved in transcription, nor have satisfactory ways as yet been found to combine the spatial, the visual and the temporal within one system" (47). Researchers can negotiate these difficulties by making pragmatic decisions about priorities for recording/ transcribing, depending on the object of the research, but in any case they need to be aware that fully represented multimodal transcriptions may be "illegible," which out-weighs the advantage of their descriptive accuracy. Finally, Flewitt et al. discuss examples from various researchers prioritizing different semiotic components but they again conclude that all of these representations "pose significant disadvantages for research dissemination, where the dominant media are still printed, written or visual formats" (52). This work has important implications for research-oriented corpus-building (see [START_REF] Dooly | Researching Multimodal Communicative Competence in Video and Audio Telecollaborative Encounters[END_REF].

Methodology for developing CMILL research: The need to investigate multimodal Interactions

In online language learning means that studies should examine situations where participants (learners, teachers, natives, etc.) are involved in activities which span over several weeks, including several hours a week. The coverage of the data collected for analysis is a key factor. This is one of the reasons for adopting a corpus-based approach for systematically gathering, transcribing and coding large amounts of longitudinal multimodal data. Our corpus-based approach is intended to overcome the limitations of the approach to research introduced by [START_REF] Flewitt | What Are Multimodal Data and Transcription?[END_REF], which introduces different, incompatible ways of collecting, organizing, and analyzing data. Our corpus-based methodological approach which navigates from transcription to coding and analysis suggests a compatible way to support multiple analyses in addition to sharing data among researchers.

Motivations for a corpus-based approach

This corpus-based approach seeks to address a range of scientific criteria that apply to research on second language acquisition (Mackey and Gass 2005) such as validity (Do the data allow for meaningful interpretations? Is the scope of relevance of the results clear not only to our sample but to other contexts?), and reliability (Is the manner of data collection, organization, and analysis sufficiently consistent to allow for comparable results to be obtained by other researchers?). We developed a corpus-based approach based on our experience initiated in 2005 with the concept of LEarning and TEaching Corpus (LETEC) in online multimodal learning situations. It combines work on speech corpora related to first language acquisition and computer-mediated communication (CMC) corpora and its model, which encompasses multimodal interactions. It addresses issues of meaningfulness and consistency by fitting with the expectations and conceptual frameworks of researchers in applied linguistics in addition to encompassing quality criteria for gathering and analyzing data.

The experience of the language acquisition community

Researchers investigating interaction through spoken language have needs for data that are similar to those of CMILL researchers. For example, Jenks (2011, 71), who specializes in the topic of transcribing speech and interaction, reminds us this overlap:

In face-to-face another types of multimodal interaction, non-verbal conduct (e.g. gaze, body, posture, pointing and nodding) is equally as important, prevalent, and multifunctional, as stress, intonation, and voice amplitude.

In language acquisition, researchers have to start by defining what kinds of observations and measures will best help capture the development process over time. Generally interactions need to be captured in authentic contexts rather than in laboratory conditions. For example, decisions have to be made, when studying discourse addressed by adults to children (i.e., Child Directed Speech), about the number of subjects to be studied in order to fulfill scientific criteria, about the type of situations in natural settings, the length of every window of observation (What time of the day? What child activity? Should the researcher be present or absent?). Choices need to be made concerning the repetition of the observations. Hoff (2012) provides a good introduction to these issues. Opportunities and pitfalls have been extensively studied over time by the language acquisition community. All these research protocols may illuminate the way we can develop multimodal CALL research in informal settings.

The longstanding tradition of building research on corpora illustrates the benefits CMILL may expect when following a similar route. Large repositories of corpora have been built over time by an international set of researchers following a unique methodology such as CHILDES on first language acquisition (MacWhinney 2000).

The research planning is presented as a three-step process (O'Donnell cited in [START_REF] Segouat | Corpus de Langue des Signes: Premières Réflexions sur leur Conception et Représentativité[END_REF]), which we synthesize as: (1) May I find appropriate data in existing corpora, or when extracting parts of different corpora? (2) What if I recombined and rearranged them with a different perspective in mind? (3) I will consider developing a new corpus if (and only if) answers to the two previous questions are negative.

This perception of research as a cumulative process, and of the analysis as a cyclical one, with researchers reconsidering previous data, mixing them with new data, measuring things differently in accordance with new theoretical frameworks, is a reality in language acquisition studies, and more generally in spoken corpora. A good illustration is presented in (Liégeois 2014) around the Phonologie du Français Contemporain (PFC) corpus which gathered together a community of international researchers over 15 years to pave the way for fundamental discussions about the nature of language acquisition.

Quality criteria for CMILL corpora

Since research questions in CMILL are always connected to learning situations, data collection will not only refer to multimodal interactions, listed in the previous section. It will also encompass: (1) the learning situation, (2) the research protocol, and (3) the permissions for access. The learning situation refers to the learning design if it is in a formal situation, and to other elements of the context if it is in an informal situation. It also refers to all the necessary information about the participants (e.g., learners and teachers) bio-graphical and sociological information, level of expertise not only with respect to language but also with the technological environment, and so on. The research protocol refers to questionnaires, participants' interviews, methodology for data collection and coding, and so on. The permissions for access specifies how data have been collected (how the question of ethics and rights have been taken care of? It provides the consent form used, and explains the anonymization process on raw data), and how the corpus contents can be freely used by other researchers.

In order to become a scientific object of investigation a corpus has to meet several quality criteria: systematic collection (Were the data collected systematically in view of the research phenomena at stake? Is the coverage representative?); coherent data organization: coherence for packaging the various parts of the corpus, interlinking them (e.g., video files with transcriptions), coherence when coding and transcribing; data longevity, including a short-term window in order to be able to use several types of analysis tools and collect data in nonproprietary formats; and a middle-term window in order to deposit the data, share them, and store them in an archive; human readability, including information allowing researchers who did not participate in the experiment to work with the corpus; machine readability for data storage and, beforehand, for analysis purposes; the so-called OpenData criteria, related not only to the aforementioned permissions, but also to the guarantee of continuing access to the internet and to efficient identification of the corpus on Web search, as described by [START_REF] Chanier | Standardizing Multimodal Teaching and Learning Corpora[END_REF].

These criteria aim at the issues of validity and reliability by striving to make clear what the corpus represents and how usable it is for the purpose of analysis. Representativeness relates not only to the systematic way of collecting data, but also to the way its scope has been delineated [START_REF] Baude | Corpus Oraux, Guide des Bonnes Pratiques 2006[END_REF]. All these kinds of data build up what we called a LETEC corpus [START_REF] Chanier | Standardizing Multimodal Teaching and Learning Corpora[END_REF][START_REF] Reffay | Multimodal Learning and Teaching Corpora Exchange: Lessons learned in 5 years by the Mulce project[END_REF]; see the Mulce corpora repository, 2013, from which 50 corpora can be downloaded). The LETEC contains not only data, but also their detailed metadata which describe: conditions of data collection, aggregation, organization, coding, general information about the learning situation, about the technological environment, and so on. These elements of information provide a basis for a meaningful analysis.

Conceptualization of multimodal acts

A critical aspect of making research interpretable across different projects is to have a scheme for classifying and analyzing the many different types of multimodal data. The analytic scheme needs to capture certain types of relevant actions performed and/or perceived by CMILL participants within a given space. [START_REF] Beißwenger | Special Interest Group on CMC of the Text Encoding Initiative (TEI) Consortium[END_REF] call the site of interactions the "interaction space" (henceforth IS), which is an abstract concept temporally situated at the point when interactions occur online. The IS is defined by the properties of the set of environments used by the participants. Participants in the same IS can interact (but do not necessarily do so, cf. lurkers). They interact through input devices mainly producing visual or oral signals. Hence when participants cannot hear or see other participants' actions, they are not in the same IS. Within a variety of different ISs, multiple types of multimodal acts have been studied in various research projects, from data collected and organized in LETEC corpora. All such acts can be classified as either verbal or non-verbal acts.

Verbal acts, such as those studied by corpus linguistics, are based on textual and oral modes. In synthetic worlds such as Second Life, every participant can choose to use audio and textchat in order to publicly communicate with other participants located close to her/ his avatar. S/he can also decide to communicate with people not co-present (Wigham and Chanier 2013a, §2.1). For the sake of simplicity, we will assume that audio and textchat can be heard or read by participants in the same location. When studying interactions between various kinds of acts, it is useful to distinguish between a verbal act which is realized as an en bloc message and an adaptive one. Once a textchat message has been sent to a server, it appears to the other participants as a non-modifiable piece of language (it becomes a chat turn and has lost any indication of the way it had been planned by its author before being sent). On the contrary, a participant's utterance (e.g., in an audio chat act) can be planned, then modified in the heat of the interaction while taking into account other acts occurring in other modalities of communication (Wigham and Chanier 2013b).

As regards non-verbal acts, a great deal of attention is paid in social semiotics and in CALL research to acts related to the body, whether generated by actual human body (mediated through webcams) or by avatars in synthetic worlds. Wigham and Chanier (2013a) presented a classification of such acts (for another viewpoint see Shih 2014): proxemics, kinesics (which includes gaze, posture, and gestures), and appearance. Another type of non-verbal derives from actions in groupware. These share-edited tools, such as word processor, concept map, whiteboard can be integrated within wider environments for example, audiographic, or used besides video conference environments. They have been largely developed and studied by the computer-supported collaborative learning (CSCL) community [START_REF] Dyke | Technological Affordances for Productive Multivocality in Analysis[END_REF]. Within these environments knowledge is collaboratively built and negotiated with interaction switching between non-verbal acts and verbal acts [START_REF] Ciekanski | Developing Online Multimodal Verbal Communication to Enhance the Writing Process in an Audio-graphic Conferencing Environment[END_REF]).

Lastly we will consider non-verbal acts based on the use of icons. A first type of iconic tool is specifically oriented toward conversation management: such tools ease the turn-taking (icon raised hand), reduce overlap, restrain talkative participants, so as to encourage the more reserved ones to take the floor; they allow quick decisions and clarifications to be made (voting yes/no); they focus on technical aspects without interrupting the conversation flow (icons talk-louder, talk-more softly) or they mediate social support (clapping icon). The second type of iconic tool displays signs of participants' presence, for example icons show when a participant enters or leaves the interaction space, or is momentarily absent.

From transcription and coding to analysis

The scheme for classifying multimodal acts in CMILL situations provides a conceptual framework for categorizing multimodal data in CMILL, but to carry out research, the data also need to be transcribed so they can be analyzed. A transcription is a textual version of material originally available in a non-textual (or non-exclusively textual) medium. It is generally considered as a biased, and reduced version of reality [START_REF] Flewitt | What Are Multimodal Data and Transcription?[END_REF][START_REF] Rowe | Recording, Transcribing, and Coding Interaction[END_REF]) that the researcher wants to study, or, more precisely, not of reality but of the data which have been collected. One may wonder whether this weakness primarily arise from the data collection itself. For example, on the one hand, linguistic and paralinguistic features of a spoken interaction can be precisely transcribed. But, on the other, an audio file captures only a limited part of the interaction: who is the exact addressee of a speech turn? What is the surrounding context? A video recording will bring much more information. However, in face-to-face situations, if a single camera is recording, important alternative perspectives may be missed. The "reality" of online multimodal interaction may be easier to capture than the face-to-face one. Actually a video-screen capture with audio-recording will accurately record the online context and participants' actions. It will not render the context and the action of an individual participant, but the shared context of all the participants. What is recorded into the video-screen capture is the common ground on which participants interact.

The need for shared transcription conventions

If individual studies are to build upon a common knowledge base, researchers need to be able to examine data across studies. For example, [START_REF] Sindoni | Spoken and Written Discourse in Online Interactions: A Multimodal Approach[END_REF] studied participants' use of modalities in (non-educational) online environments that integrate audio, video, and text chat. She focused on what she termed "mode-switching," when a participant moves from speech to writing or the other way round. When analyzing transcriptions of video-screen online conversations, she observed that participants could be classified according to their preferred interaction mode (oral or written). She also observed that "[a]s anticipated, both speakers and writers, generally carry the interaction forward without mode-switching. This was observed in the whole video corpus" (Sindoni, 2013, §2.3.5). Hence she concluded that "those who talked did not write, and those who wrote did not talk. Turn-taking adheres to each mode."

In several CMILL situations that we studied, we had similar research questions. When analyzing data assembled in LETEC corpora such as Copéas or Archi21 (Mulce repository 2013), we observed that learners had a preferred mode of expression (oral or written), at least those at beginner level [START_REF] Vetter | Supporting Oral Production for Professional Purpose, in Synchronous Communication With heterogeneous Learners[END_REF]. In contrast with [START_REF] Sindoni | Spoken and Written Discourse in Online Interactions: A Multimodal Approach[END_REF], analyses of audiographic and 3D environments show that learners were mode-switchers (even modality-switchers). Choices of mode/modality depended on the nature of the task, and on the tutor's behavior (see Wigham and Chanier 2013b).

At this stage, one may expect that scientific discussions could take place between researchers studying online interactions, to debate contradictions, fine differentiations of situations, tasks, and so on. In order to allow this, data from the different approaches need to be accessible in standard formats, with publications clearly relating to data and data analyses, and explicit information given about the format of the transcriptions, their code, and transcription alignments with video. However Sindoni's data are not available. The inability to accumulate findings across CMILL studies or to contrast data from one study with other examples, available in open-access formats, is still holding back the scientific advancement of the CALL field. Overcoming these limitations will require researchers' development of common guidelines for transcription, coding, and analysis. Jenks (2011, 5) describes the main functions of transcription as the following: (1) represent; (2) assist; (3) disseminate and (4) verify. Transcription seeks to represent the interaction, that is, a multimodal discourse which it would have been impossible to analyze in its "live" state. Transcription assists the analysis which will be made of the data. Henceforth this research depends on the coding of interaction, whether this code will be compatible with analysis tools. Dissemination refers to the repeated process of analysis, either by the data compiler when s/he plans to publish several articles out of her/his corpus, or by the community. This function is essential to conform to the principle of rigor in scientific investigation (Smith, et al. 2011, 377):

Transcription, coding, and analysis

The fact of being able to store, retrieve, share, interrogate and represent in a variety of ways […] the results of one's analysis means that a semiotician can conduct a variety of analyses, and then explore the range of such analyses […]. Different analyses and perspectives upon analysis are encouraged, so that an analyst may produce multiple interpretations of a text.

The last function of transcription refers to the need of the academic community to know, as Jenks (2011, 5) put it, "whether any given claim or observation made is demonstrably relevant to the interactants and interaction represented in the transcript." Checking this claim requires procedures for estimating the level of agreement between transcribers.

The first function, representation requires deliberation because transcription represents only a partial view of interaction corresponding to phenomena the researcher wants to examine. Obviously, the richness of multimodal acts, as sketched in Figure 28.2, cannot be simultaneously transcribed to represent all of the multimodal perspectives captured through the video. 1 It is therefore important to adopt a methodological approach when transcribing and coding by taking into account the decision steps enumerated in Table 28.2.

The first step (first line of Table 28.2) is to choose appropriate software for integrating video-screen capture and transcription layers of verbal and non-verbal acts for every participant. The aim is to appreciate whether these products and events support L2 language development. Co-occurrence of these modes could be compared to a concert performance, and the transcription task to writing a music score, where all the instruments/modes can be read and interpreted after having been aligned (cf. time priority for General act features in Table 28.2). Once the layout of the music score is "printed," that is, once the different layers, corresponding to each participant and the various modes have been opened within the transcription software, with the video ready for alignment, the coding process can start. In synchronous multimodal CALL environment, the alignment is preferably made around the speech verbal acts.

The second step concerns speech transcription (cf. coding scheme in Table 28.2), that is by at least typing the words corresponding to the sound signal. Table 28.2, line 2, lists key points to take into consideration and related issues. A large part of them are shared by the community of speech corpora. Out of this transcription (including words in the textchat tools), many software packages are able to automatically calculate word tokens, work types, and mean length utterance. This computation can measure individual participant contribution: numbers of turns; floor occupied in sessions; vocabulary diversity, and so on.

A further step (cf. Output format in Table 28.2), specific to CMILL concerns, is to relate this methodology to a more abstract model, such as the interaction space one. It is as an extension of the Text Encoding Initiative (TEI), well known in corpus linguistics and other fields of Humanities. It encompasses text types such as manuscripts, theater, literature, poem, speech, film and video scripts. The TEI is specifically designed to accept different levels of annotations, each one corresponding to a specific type of analysis (e.g., morpho-syntactic, semantic, discursive). The IS model, under development, is designed for CMC in general, and includes multimodal discourse. It is product of a European research work-group, TEI-CMC [START_REF] Beißwenger | Special Interest Group on CMC of the Text Encoding Initiative (TEI) Consortium[END_REF]. The CoMeRe repository [START_REF] Beißwenger | Special Interest Group on CMC of the Text Encoding Initiative (TEI) Consortium[END_REF] provides access to corpora of various CMC genres (SMS, blogs, tweets, textchat, combinations of email-forum-textchat, and multimodal acts coming from 3D, audiographic environments).

Conclusion

The multimodal corpus-based approach to research on online language learning described in this chapter is not used widely in the study of CALL today. However, there is a need to expand the community of CALL researchers who adopt a corpus-based approach. Learning designs for CMILL need to take full account of the material and multimodal nature of the technical and learning resources involved in order to promote learning mediated through these resources. In order for research to improve CMILL, researchers need to investigate learners' performance in such interaction spaces. Sharing these data and documenting the processes through which they were created is a necessary step for deepening research on multimodal CALL. The LETEC corpus provides an example of how the field might move forward to investigate CMILL by collecting a corpus of CMILL data. It has served as a site for exploration of the various challenges facing the researcher with the transcription of these data, their coding in a standard way, and analysis. As researchers continue to work with this and other multimodal corpora, this area will continue to see advances that will help to improve CMILL.

Note

Figure 28 . 1

 281 Figure 28.1 Schematic diagram showing components Mediation in CMILL.

Figure 28 . 2

 282 Figure 28.2 Multimodal acts as collected and studied within Mulce corpora repository (2013).

Table 28 .1

 28 Modality as a set of relationships among objectives, tools, and modes in CMILL. Adapted fromLamy (2012b).

	Main CMILL objectives facilitated	Tools	Modes
	Information seeking (preliminary to engagement with tasks or people)	Screens	Dual modes: linguistic (written) and visual
	Accessing and interacting with materials and people	Screens (conventional, tactile), pads, microphones, speakers	Multiple modes: linguistic (written, sometimes spoken), visual, kinetic, aural if music is involved)
	Reflective activities (revisiting tutorials and conversations), sharing	Recording and screen capture tools	Multiple modes: linguistic (written and spoken)
	audio-visual material
	Communication and
	interaction (e.g., Like button); create tele-presence; help with turn-taking (e.g., raised-hand	Hot buttons	Multiple modes: linguistic (written and spoken), visual, kinetic
	button)
	Written exchanges; peer collaboration; feedback; commenting; group bonding	Asynchronous sending/ receiving channels; synchronous messaging channels	Mainly written linguistic mode, with elements of visual mode
	Oral communication and interaction; collaborative work; feedback; group bonding	Webconferencing	Multiple modes: linguistic (spoken and some written), visual, kinetic

Table 28 . 2

 282 Decision steps with their respective main features and comments for transcription and coding.

	Verbal act			Non-verbal act
					<prod>
	Textual mode enbloc	Oral mode adaptative	Body	Groupware	Icons
	<post>	<u>
						Conversation
			Proxemics Kinesics Appearance	management Presence
					Word processor Whiteboard	Concept map
			Gesture	Gaze	Posture

Collecting multimodal data and transcribing them are time-consuming processes. As regards transcriptions, estimated ratios vary from 15:1 for speech (i.e., 10 mn of audio requires

2.5 hours to transcribe it) to 23:1 for both speech and gesture[START_REF] Rowe | Recording, Transcribing, and Coding Interaction[END_REF]).