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 Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary Number of partitions and formula of Hardy-Ramanujan A partition of a positive integer n , also called an integer partition, is a way of 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. n ≥ 0 p ( n ) q n = 1 Since Euler, one knows that � ( q ; q ) ∞ , where and in the following : � (1 − aq n ) , ( a ; q ) ∞ = | q | < 1 . n ≥ 0 An asymptotic expression for p ( n ) is given by Hardy and Ramanujuan : � � � 1 2 n p ( n ) ∼ √ exp π as n → ∞ . 3 4 n 3 Rademacher has completed this asymptotic formula into an exact formula. A key point consists of the fact that the Dedekind η -function η ( τ ) := q 1 / 24 ( q ; q ) ∞ is modular, where q = e 2 π i τ , τ ∈ H . Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan
Introduction Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary 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 q n 2 a rank ( λ ) q n = � � � R ( a ; q ) := m =1 (1 − aq m )(1 − q m / a ) . � n n ≥ 0 λ ∈P n n ≥ 0 Given an integer m , knowing the number of partitions of n with rank congruent to r modulo m for all r ∈ Z / m Z and all n is equivalent to knowing the specialization of R ( a ; q ) to all m -th roots of unity a = e 2 π ik / m . Gordon and McIntosh give that ∞ ( − 1) n q (3 n 2 + n ) / 2 1 − a � R ( a ; q ) = , ( q ; q ) ∞ 1 − q n a n = −∞ 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 Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary Specialization at a = − 1 for R ( a ; q ) Putting a = − 1 into the expression q n 2 � R ( a ; q ) = m =1 (1 − aq m )(1 − q m / a ) . � n n ≥ 0 yields q n 2 � R ( − 1; q ) = 1 + m =1 (1 + q m ) 2 . � n n ≥ 1 Ramanujan claimed that α n ∼ ( − 1) n +1 � � � n √ n exp π as n → ∞ 6 where α n denotes the coefficient of q n in the power series expansion R ( − 1; q ) = 1 + � n ≥ 1 α n q n . 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 Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary 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 : ∞ q n 2 � f ( q ) = (1 + q ) 2 (1 + q 2 ) 2 ... (1 + q n ) 2 , n =0 q n 2 ∞ � φ ( q ) = (1 + q 2 )(1 + q 4 ) ... (1 + q 2 n ) , n =0 ... 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 Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary 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 : infinitely many roots of unity are exponential singularities, 1 for every root ζ of unity, there is a theta function T ζ ( q ) such that the 2 difference f ( q ) − T ζ ( q ) is bounded as q → ζ radially, f is not the sum of two functions, one of which is a theta function and the 3 other 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 Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary What might mean theta for Ramanuajan ? One finds that four functions play a particular role in his theory : ( q ; q 2 ) ∞ , ( − q ; q 2 ) ∞ . ( q ; q ) ∞ , ( − q ; q ) ∞ , Since ( a 2 ; q 2 ) ∞ = ( a , − a ; q ) ∞ and ( a ; q ) ∞ = ( a , aq ; q 2 ) ∞ , one finds that ( − q ; q ) ∞ = ( q 2 ; q 2 ) ∞ ( q ; q ) ∞ ( q ; q 2 ) ∞ = ( q ; q ) ∞ , ( q 2 ; q 2 ) ∞ and that ( − q ; q 2 ) ∞ = ( q 2 ; q 4 ) ∞ ( q 2 ; q 2 ) 2 ∞ ( q ; q 2 ) ∞ = ( q ; q ) ∞ ( q 4 ; q 4 ) ∞ . Thus, all the four above functions can be expressed in terms of (quotients of) ( q k ; q k ) ∞ , where k may be 1, 2 or 4. Remember that q 1 / 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 Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary A brief revisit of Jacobi’s Theta functions (I) Let z ∈ C ∗ , x = e ( z ) = e 2 π iz , τ ∈ H , q = e ( τ ) = e 2 π i τ , and let θ ( x ; q ) = θ ( z | τ ) with q n ( n − 1) / 2 x n = ( q , − x , − q � θ ( z | τ ) = x ; q ) ∞ . n ∈ Z The classical θ -modular formula says that � θ ( z + 1 A τ e ( − 1 i τ ( z + 1 2 m − ˆ τ 2 ) 2 ) θ ( z τ + 1 2 | τ ) = √ m 2 | τ m ) , ˆ 2ˆ m ˆ where A = A ( m , p , α, β ) denotes some unity root, � � α β ∈ SL (2 , Z ) , m ∈ Z > 0 − m p and where τ = τ − p τ m = − α 1 ˆ m , m − τ . m 2 ˆ Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan
Introduction Motivation Appell-Lerch series and q -difference equations Letter of Ramanuajan and possible definitions Continued fractions and modularity Asymptotics deduced from Modularity Algorithms (Mock) Theta-type functions by asymptotics Summary A brief revisit of Jacobi’s Theta functions (II) As τ → p m , it follows that ˆ τ → 0 and τ m → i ∞ in H . This implies that if q m = e ( τ m ), then q m → 0 exponentially. Lemma Given z 0 , z 1 ∈ R , if z = z 0 + z 1 τ and τ → p m , then � 2 | τ ) = A ′ θ ( z + 1 τ e ( λ − i τ ) θ ( z τ + 1 √ m + λ + ˆ 2 | τ m ) , ˆ τ ˆ m ˆ τ = τ − p with ˆ m → 0 , | A ′ | = 1 , λ − = − 1 2 ( z 0 + z 1 p 2 m ) 2 , λ + = − 1 1 2 ( z 1 − 1 2 ) 2 , m + 1 τ + 1 2 | τ m ) = C q δ m (1 + O ( q κ θ ( z m )) , where δ ∈ R and κ > 0 . 2 m ˆ Changgui ZHANG An analytic pointview on the Mock Theta functions of Ramanujan
Recommend
More recommend
