An analytic pointview on the Mock Theta functions of Ramanujan - - PowerPoint PPT Presentation

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary

FELIM 2016 Limoges, France

An analytic pointview on the Mock Theta functions of Ramanujan

Changgui ZHANG

University of Lille, France

March 29-31, 2016

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary

Plan and key-words Motivation : Dedekind eta-function, integer partitions, Ramanujan-Hardy formula, rank and co-rank, Appell-Lerch of the third order, mock theta functions of Ramanujan. Asymptotics found from modular formula, definition of theta-type function, Eulerian series and Ramanujan theta-functions. Singular q-difference equation, heat kernel (Gaussian) and Jacobi theta function, Stokes phenomenon, modular-like relation. Continued fractions, asymototic behavior of Appell-Lerch series, some technical remarks Conclusion : Stokes phenomenon implies modularity.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

Number of partitions and formula of Hardy-Ramanujan A partition of a positive integer n, also called an integer partition, is a way

• f writing n as a sum of positive integers. The number of partitions of n is

denoted by p(n). Example : 4 = 4 × 1(= 1 + 1 + 1 + 1) = 2 × 1 + 1 × 2 = 1 × 1 + 1 × 3 = 2 × 2 = 1 × 4, so p(4) = 5. Since Euler, one knows that

n≥0 p(n)qn = 1 (q;q)∞ , where and in the

following : (a; q)∞ =

• n≥0

(1 − aqn) , |q| < 1 . An asymptotic expression for p(n) is given by Hardy and Ramanujuan : p(n) ∼ 1 4n √ 3 exp

• π
• 2n

3

• as n → ∞ .

Rademacher has completed this asymptotic formula into an exact formula. A key point consists of the fact that the Dedekind η-function η(τ) := q1/24(q; q)∞ is modular, where q = e2πiτ, τ ∈ H.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

Rank of a partition and Appell-Lerch series of order 3 Following Dyson, the rank of a partition is its largest part minus the number of its parts. Thanks to Garvan and Andrews, one knows that R(a; q) :=

• n≥0
• λ∈Pn

arank(λ)qn =

• n≥0

qn2 n

m=1(1 − aqm)(1 − qm/a) .

Given an integer m, knowing the number of partitions of n with rank congruent to r modulo m for all r ∈ Z/mZ and all n is equivalent to knowing the specialization of R(a; q) to all m-th roots of unity a = e2πik/m. Gordon and McIntosh give that R(a; q) = 1 − a (q; q)∞

∞

• n=−∞

(−1)nq(3n2+n)/2 1 − qna , where the last series is an Appell-Lerch series of order 3.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

Specialization at a = −1 for R(a; q) Putting a = −1 into the expression R(a; q) =

• n≥0

qn2 n

m=1(1 − aqm)(1 − qm/a) .

yields R(−1; q) = 1 +

• n≥1

qn2 n

m=1(1 + qm)2 .

Ramanujan claimed that αn ∼ (−1)n+1 √n exp

• π
• n

6

• as n → ∞

where αn denotes the coefficient of qn in the power series expansion R(−1; q) = 1 +

n≥1 αnqn.

This asymptotic relation has first been completed into an exact formula as conjecture by Andrews-Dragonette and proved later by K. Ono and K. Bringmann : by using Cauchy + Unit circle method.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

Ramanujan mock-theta functions of order 3 In The final problem (1935), G. N. Watson discussed the last letter of Ramanujuan to Hardy in which a list of q-series had been introduced : f (q) =

∞

• n=0

qn2 (1 + q)2(1 + q2)2...(1 + qn)2 , φ(q) =

∞

• n=0

qn2 (1 + q2)(1 + q4)...(1 + q2n) , ... These series are named by Ramanujuan mock theta functions. Some of these functions (NOT ALL) can be directly expressed in terms of R(a; q) : f (q) = R(−1; q), φ(q) = R(i; q), ...

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

What might mean mock-theta ? Following the lastest letter of Ramanujan to Hardy, one may guess that a mock theta function means a function f of the complex variable q , defined by a q-series of a particular type (Ramanujan calls this the Eulerian form), which converges for |q| < 1 and satisfies the following conditions :

1

infinitely many roots of unity are exponential singularities,

2

for every root ζ of unity, there is a theta function Tζ(q) such that the difference f (q) − Tζ(q) is bounded as q → ζ radially,

3

f is not the sum of two functions, one of which is a theta function and the

• ther a function which is bounded radially toward all roots of unity.

Question – theta functions = ? Following Andrews and Hickerson, one can define θ-products by using Jacobi’s theta functions and one can guess that a theta is the sum of a finitely many theta-products.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

What might mean theta for Ramanuajan ? One finds that four functions play a particular role in his theory : (q; q)∞, (−q; q)∞, (q; q2)∞, (−q; q2)∞ . Since (a2; q2)∞ = (a, −a; q)∞ and (a; q)∞ = (a, aq; q2)∞, one finds that (−q; q)∞ = (q2; q2)∞ (q; q)∞ , (q; q2)∞ = (q; q)∞ (q2; q2)∞ and that (−q; q2)∞ = (q2; q4)∞ (q; q2)∞ = (q2; q2)2

∞

(q; q)∞(q4; q4)∞ . Thus, all the four above functions can be expressed in terms of (quotients

• f) (qk; qk)∞, where k may be 1, 2 or 4.

Remember that q1/24(q; q)∞ is the Dedekind eta-modular function, so Ramanuajan’s theta functions might merely be related to modular functions.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

A brief revisit of Jacobi’s Theta functions (I) Let z ∈ C∗, x = e(z) = e2πiz, τ ∈ H, q = e(τ) = e2πiτ, and let θ(x; q) = θ(z | τ) with θ(z | τ) =

• n∈Z

qn(n−1)/2 xn = (q, −x, −q x ; q)∞. The classical θ-modular formula says that θ(z + 1 2 | τ) = A √m

• i

ˆ τ e(− 1 2ˆ τ (z + 1 2m − ˆ τ 2 )2) θ( z mˆ τ + 1 2 | τm) , where A = A(m, p, α, β) denotes some unity root,

• α

β −m p

• ∈ SL(2, Z),

m ∈ Z>0 and where ˆ τ = τ − p m , τm = − α m − 1 m2ˆ τ .

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

A brief revisit of Jacobi’s Theta functions (II) As τ → p

m, it follows that ˆ

τ → 0 and τm → i∞ in H. This implies that if qm = e(τm), then qm → 0 exponentially. Lemma Given z0, z1 ∈ R, if z = z0 + z1τ and τ → p

m, then

θ(z + 1 2 | τ) = A′ √m

• i

ˆ τ e(λ− ˆ τ + λ+ˆ τ) θ( z mˆ τ + 1 2 | τm) , with ˆ τ = τ − p

m → 0,

1

|A′| = 1, λ− = − 1

2(z0 + z1p m + 1 2m)2, λ+ = − 1 2(z1 − 1 2)2,

2

θ( z

mˆ τ + 1 2 | τm) = C qδ m (1 + O(qκ m)), where δ ∈ R and κ > 0.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

This revisit about Jacobi’s theta functions shows that : All modular function f (τ) has a similar asymptotic behavior at every root

• f unity e(p/m) = e2πip/m :

f (τ) = A′ √m

• i

ˆ τ e(λ− ˆ τ + λ+ˆ τ) (1 + O(e−κ/|ˆ

τ|)) ,

with ˆ τ = τ − p

m → 0.

Moreover, the factor 1 + O(e−κ/|ˆ

τ|) can be expressed as a germ of

analytic function of the NEW (modular) variable e(−1/ˆ τ) at 0 ∈ C. Modularity = ⇒ Exponential scale for the asymptotics : transseries !.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

Radial limit behavior at each root of unity For simplify, a function f will be called theta-type at a given root ζ = e2πi p

m of

unity, and one writes f ∈ Tζ, if there exists a quadruplet (υ, λ, I, γ), composed

• f a couple (υ, λ) ∈ Q × R, a strictly increasing and unbounded sequence I,

and a C∗-valued map γ defined on I, such that the following relation holds for any N ∈ Z≥0 as τ

a.v

− → r : f (q) = i ˆ τ υ qλ

• k∈I∩(−∞,N]

γ(k) qk

m + o(qN m)

• .

In the above, ˆ τ = τ − p

m and qm denotes the modular variable related to ζ :

qm = e−2πi

1 m2 ˆ τ e2πi p′ m

with pp′ = −1 (mod m).

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

When Euler products is theta-type ? The infinite product (x; q)∞ = (1 − x)(1 − xq)(1 − xq2)... is called Euler product. Theorem The following conditions are equivalent for any Eulerian product (x; q)∞ while x = e2πiaqb with a, b ∈ R, q = e2πiτ, τ ∈ H.

1

(x; q)∞ ∈ T1,

2

(x; q)∞ ∈ Tζ for some root of unity,

3

(x; q)∞ ∈ Tζ for every root of unity,

4

x ∈ {q, −q, √q, −√q}.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

Strategy for the proof Let x1 = e2πi a+bτ

τ

and q1 = e−2πi 1

τ . Comparing (x; q)∞ with (x1; q1)∞

yields that (x; q)∞ = eT(x;q) (x1; q1)∞ , where T can be expressed with dilogarithm plus a Laplace integral. (x; q)∞ ∈ T1 means that the above Laplace integral defines a polynomial so we have NO divergence as q → 1. Continued fractions = ⇒ other roots of unity from 1. Counterexample – Knowing (q; q)∞ ∈ T1, one can see that (q2; q)∞ does not belong to T1. Indeed, (q2; q)∞ = (q;q)∞

1−q

and

1 1−q /

∈ T1 ! For 1 1 − e2πiτ = C i τ ν e(λ− τ + λ+τ)

• 1 + O(e−κ/|τ|)
• .

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Motivation Letter of Ramanuajan and possible definitions Asymptotics deduced from Modularity (Mock) Theta-type functions by asymptotics

The above-introduced theta-type property can be extended as follows (0 = e(i∞)) : Definition Let q = e(τ), τ ∈ H, ζ ∈ U ∪ {0}, where U is the set of the roots of unity.

1

One says that f is of almost theta-type as q → ζ and one writes f ∈ ˜ Tζ, if there exists fζ ∈ Tζ such that f − fζ = O(1) as q → ζ.

2

One says that f is a theta-type function and one writes f ∈ T, if f ∈ Tζ for all ζ ∈ U ∪ {0}.

3

One says that f is a false theta-type function and one writes f ∈ F, if f / ∈ T and there exists T ∈ T such that f (q) − T(q) = O(1) for all ζ ∈ U ∪ {0}.

4

One says that f is a mock theta-type function and one writes f ∈ M, if f ∈ ˜ Tζ for all ζ = U ∪ {0} and, moreover, for each given T ∈ T, there exists ζ ∈ U ∪ {0} such that f − T is unbounded.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Linear q-difference equations Different sum-functions of a divergent q-series

A singular q-difference equation From the analytic classification viewpoint of complex linear differential and q-difference equations, the series

• n≥0

q−n(n−1)/2(−x)n is a q-analog of the famous Euler series

n≥0 n! (−x)n. We are led to the

q-difference equation y(qx) + qx y(x) = 1 , that admits x = 0 as non-Fuchsian singular point. As the moment problem of the sequence (q−n2/2)n is undetermined, the above q-Euler series can be the asymptotic expansion of several solutions of this functional equation. Thus, the q-analog of Borel-sum is not unique.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Linear q-difference equations Different sum-functions of a divergent q-series

Instead of n! = ∞ e−t tn+1 dt

t

Let n ∈ Z and a > 0. Since ∞

−∞

e−(t+na)2 dt = 2 ∞ e−t2dt = √π , it follows that en2a2 = 1 √π ∞

−∞

e−t2 e−2ant dt . Putting ea2 = q− 1

2 and ξ = e−2at yields that

q− 1

2 n2 =

1

• 2π ln 1/q

∞ e

1 2 ln q log2 ξ ξn dξ

ξ . Let µ / ∈ (−qZ), and let θ(x; q) =

k∈Z qk(k−1)/2xk = (q, −x, −q/x; q)∞.

Using θ(qkx; q) = q−k(k−1)/2x−kθ(x; q) gives that q− 1

2 n(n−1) =

• ξ=µqk ,k∈Z

ξn θ(ξ; q) .

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Linear q-difference equations Different sum-functions of a divergent q-series

q-Borel-Laplace transforms q-Borel :

• n≥0

anxn = ⇒

• n≥0

anqn(n−1)/2ξn . q-Laplace with (Gaussian) heat kernel ω(u; q) =

1

√

2π ln(1/q) e

log2(u/√q) 2 ln q

: φ = ⇒ ∞ φ(ξ)ω( ξ x ; q)dξ ξ . q-Laplace with Jacobi theta function θ(u; q) =

k∈Z qk(k−1)/2uk :

φ = ⇒

• ξ∈µqZ

φ(ξ) θ( ξ

x ; q)

, where µ is a given complex number that indicates the integration path.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Linear q-difference equations Different sum-functions of a divergent q-series

Appell-Lerch series and Mordell’s integral as solution of q-difference equation By following the above q-analogs of the Borel-Laplace summation method, one can check that both expressions give solutions of y(qx) + qx y(x) = 1 : L(x, µ; q) =

∞

• n=−∞

1 1 − µqn 1 θ(− µ

x qn; q)

and G(x; q) = ∞ ω(ξ/x; q) 1 + ξ dξ ξ . Physics – G(x; q) represents a solution for the “heat equation” 2q∂qf + x2∂2

xf = 0,

f (x, q)|q=1 = 1 1 + x , which is formally satisfied by the power series

n≥0 q−n(n−1)/2(−x)n .

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Linear q-difference equations Different sum-functions of a divergent q-series

Stokes phenomenon The functions L(x, µ; q) and G(x; q) are both sum-functions of the same power series, so their difference satisfies the homogeneous q-difference equation y(qx) + qxy(x) = 0. This approach is similar to the analysis of Stokes in the theory of differential equations. Let L(z, w | τ) = L(x, µ; q) and G(z | τ) = G(x; q), where x = e(z) = e2πiz, µ = e(w) and q = e(τ). It follows that L(z, w | τ) = G(z | τ) + C(z | τ)

• L( z

τ , w τ | − 1 τ ) − 1

• ,

where C(z | τ) has the following alternative expressions : C(z | τ) = −i

• i

τ e((z + τ

2 − 1 2)2

2τ ) = θ(− z

τ + 1 2 | − 1 τ )

τ θ(−z + 1

2 | τ) .

The last expression of C(z | τ) in the above requires that z / ∈ Z ⊕ τZ.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Analytic continuation of Mordell’s integral Continued fractions Appell-Lerch series is of Mock-theta type

Key Lemma Lemma Let φ to be a given germ of analytic function at ǫ = 0 ∈ C such that φ(0) = 0, and let Φ(ǫ) = G(φ(ǫ); e−ǫ) for all enough small ǫ > 0. Then Φ can be continued to be an analytic function in some sector ∆R possessing the following properties.

1

If arg(φ(0)) ∈ (−π, π] and φ(0) = −1, then Φ(ǫ) admits an asymptotic expansion as ǫ → 0 in ∆R and Φ(ǫ) =

1 1+φ(0) + O(ǫ).

2

If φ(0) = −1 = eiπ, φ(ǫ) = −eǫ(ψ(ǫ)+ 1

2 ) and

˜ Φ(ǫ) = Φ(ǫ) − i

• π

2ǫ e− ǫ

2 (ψ(ǫ))2 ,

then ˜ Φ(ǫ) admits an asymptotic expansion as ǫ → 0 in ∆R and ˜ Φ(ǫ) = 1 + φ′(0) + O(ǫ) .

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Analytic continuation of Mordell’s integral Continued fractions Appell-Lerch series is of Mock-theta type

Closed-forms and half-periods Resurgence – The asymptotic expansions of Φ or ˜ Φ are generally divergent. By a Stokes’ analysis, one can find the following Theorem

1

When φ(0) = −1, the function Φ becomes analytic at ǫ = 0 if, and only if, φ(ǫ) = e(n+ 1

2 )ǫ for some n ∈ Z. 2

The function ˜ Φ becomes analytic at ǫ = 0 if, and only if, φ(ǫ) = eπi+ n

2 ǫ

for some n ∈ Z. As for the Eulerian product (x; q)∞, the following principle is applied : Analytic ⇔ Stokes factor is null ⇔ Without finite singularities in Borel plane

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Analytic continuation of Mordell’s integral Continued fractions Appell-Lerch series is of Mock-theta type

Passing from a unity root to the unity let us come back to the modular-like relation : L(z, w | τ) = G(z | τ) + C(z | τ)

• L( z

τ , w τ | − 1 τ ) − 1

• ,

If τ → 0 vertically in H, then −1/τ → i∞ and one gets easily the asymptotic behavior of L( z

τ , w τ | − 1 τ ) in terms of q1 = e(− 1 τ ) (→ 0 exponentially). So, the

asymptotic behavior of L(z, w | τ) is completely deduced from that of G(z | τ) as stated by Key Lemma. If τ → r ∈ Q ∩ (0, 1), one takes the continued fractions and makes use of the fact that G(z | r) is analytic at z = r (see key Lemma). This implies that, finally, one will come back to the case of τ → 0.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Analytic continuation of Mordell’s integral Continued fractions Appell-Lerch series is of Mock-theta type

In what follows, we write Ω = (−1 2, 1 2] × [−1 2, 1 2) , Ωτ = {a + bτ : (a, b) ∈ Ω} . This is a fundamental domain of R2 for the usual action of Z2 : (a, b) → (a + ℓ, b + m) for all (ℓ, m) ∈ Z2. Theorem Let z, w to be given in Ωτ. Assume that w = 0 and w = z, and consider f (q) = L(z, w | τ) = L(x, µ; q) . Then f ∈ M except in the following cases :

1

z ∈ { 1

2, 1 2 − τ 2 , − τ 2 } and w ∈ { 1 2, 1 2 − τ 2 , − τ 2 }, in which case f is a

constant function.

2

z ∈ { 1

2, 1 2 − τ 2 , − τ 2 } and w /

∈ { 1

2, 1 2 − τ 2 , − τ 2 }, in which case f is a false

theta-type function.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Definitions Decomposition

How to find the asymptotic expansion ? Given a ∈ U, consider the behavior as q → ζ = e2πi p

m of the series

R(a; q) :=

• n≥0
• λ∈Pn

arank(λ)qn =

• n≥0

qn2 n

m=1(1 − aqm)(1 − qm/a) .

Two steps

• 1. Define an average sum related to the root ζ.
• 2. Make a modular-like transform.

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Definitions Decomposition

Average summation associated with a root ζ Let ζ be a root of unity, with ζ = e

2 m πip ,

p ∈ Z , m ∈ Z≥1 , p ∧ m = 1 . Consider h(x; ζ) =

m−1

• n=0

ζ

1 2 n(n−1) xn ,

H(x; ζ) =

• n≥0

ζ

1 2 n(n−1) xn .

Both H(x; ζ) and h(x; ζ) are related to Gauss’ sums. For any integer n, we define Cn(ζ) = 1 m h(−e

1 m πiζ−n; ζ−1) .

When n = 0, we shall write C(ζ) instead of C0(ζ).

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Definitions Decomposition

Compare q-and |q|-Borel-Laplace By definition, the formal q-Borel transform Bq is the automorphism of the C-vector space C[[x]] that transforms each monomial xn into q

1 2 n(n−1) xn.

When q = ρζ with ρ = |q|, Bq is the (formal) Hadamard product of Bρ with the power series H(x; ζ) Lemma One has Bqˆ f (ξ) =

m−1

• n=0

Cn(ζ−1) Bρˆ f (−e− 1

m πiζ−nξ)

and Bρˆ f (ξ) =

m−1

• n=0

Cn(ζ) Bqˆ f (−e− 1

m πiζnξ) . Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Definitions Decomposition

Definition of Gq-summation associated with a root Definition Given a power series ˆ f ∈ C[[x]] and a real d ∈ R, one says that ˆ f is Gq-summable in the direction [0, ∞eid) if its q-Borel transform Bq(ˆ f ) represents a germ of analytic function φ at 0 ∈ C that can be continued in a sector | arg ξ − d| < δ with φ(ξ) = O(θ(A|ξ|; ρ) for some suitable δ > 0 and A > 0. In this case, the Gq-sum Sd

q (ˆ

f ) of ˆ f is defined by the following expression : Sd

q (ˆ

f ) =

m−1

• n=0

Cn(ζ) ∞eid φ(ξ) ω( ξ e(1− 1

m )πiζnx

; ρ) dξ ξ . Important remark – If ˆ f is a convergent power series, then Sd

q (ˆ

f ) equals to the usual sum of ˆ f .

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Definitions Decomposition

Generalized Appell-Lerch series and Mordell integrals The power series

n≥0 q− 1

2 n(n−1) (−x)n is transformed into

1 1+ξ by q-Borel

• transform. So, its Gq-sum is

m−1

• n=0

Cn(ζ) ∞eid 1 1 + ξ ω( ξ e(1− 1

m )πiζnx

; ρ) dξ ξ . This is to say :

m−1

• n=0

Cn(ζ) G(xζn; ρ) . In the above, ζn = e(1− 1

m )πi ζn ,

arg(ζn) = 2π αn , αn = 1 2 − 1 2m + n p m . (1)

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Definitions Decomposition

For all x ∈ ˜ C∗, we define : G(x, α; q) = ∞ (−αξ; q)∞ 1 + ξ ω( ξ x ; ρ) dξ ξ provided that the integral in the right-hand side is convergent, where q = ρζ, ρ ∈ (0, 1), and where the integration-path is any half straight-line (0, eid∞) with |d| < π. Important fact – The Gq-sum of 2φ0(q, α; µ, q, x) equals to (αqx; q)∞ G(x, µ; q)

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary Definitions Decomposition

Theorem

1

If q ∈ (0, 1), then : R(a; q) = −a(a, 1 a; q)∞

• L(a2

q , a; q) − G(a2 q ; q)

• + (1 − a) (a; q)∞ G(a2

q , q a ; q) .

2

If q = ρζ with ρ ∈ (0, 1) and ζ = e2πi p

m , then :

R(a; q) = −a(a, 1 a; q)∞

• L(a2

q , a; q) −

m−1

• n=0

Cn(ζ) G(a2 q ζn; ρ)

• + (1 − a) (a; q)∞

m−1

• n=0

Cn(ζ) G(a2 q ζn, q a ; q) .

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary

Conclusion

1

Ramanujan’s third order Mock-theta functions can be expressed by means

• f Mordell integrals and Appell-Lerch series.

2

This comes from an analysis of Stokes phenomenon for a second order confluent basic hypergeometric equation.

3

The average sum associated to a root yields an Gevrey asymptotic expansion when q tends to this root : real parameter asymptotics.

4

Modular like transform gives exponential smallness or bigness : theta part !

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan

Introduction Appell-Lerch series and q-difference equations Continued fractions and modularity Algorithms Summary

Thank you for your attention !

Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan