The Mordell integral, quantum modular forms, and mock Jacobi forms

Abstract

It is explained how the Mordell integral R e πiτ x 2 −2πzx cosh(π x) dx unifies the mock theta functions, partial (or false) theta functions, and some of Zagier's quantum modular forms. As an application, we exploit the connections between q-hypergeometric series and mock and partial theta functions to obtain finite evaluations of the Mordell integral for rational choices of τ and z. Mathematics Subject Classification: 11P55, 05A17Keywords: Jacobi forms; mock Jacobi Form; partial Jacobi theta function; False theta function; Partial theta function; Mock theta function; Mordell integral; Ramanujan The Mordell integralThe Mordell integral isThe integral was studied by Kronecker [19,20] and in the late 1800s. A few years later, the integral underwent an intensive investigation by Mordell [28,29]. He established relationships between this integral and class number formulas. Later, Siegel [37] used properties of the integral to determine the approximate functional equation of the Riemman zeta function as well as asymptotics for its first moment. Very recently, the Mordell integral has been used to efficiently compute values of the Riemann zeta function [21]. The integral also appears in the solution of the 1-dimensional heat equation, see [32] and the references therein. Andrews [1] gave an early investigation of the Mordell integral in the study of the mock theta functions of Ramanujan.The Mordell integral plays a central role in Sander Zwegers Ph.D. thesis [42] on the mock theta functions. His work shows that the Mordell integral may be viewed as the obstruction to modularity of the mock theta functions. Moreover, he showed how to reinterpret the integral in terms of a period integral of a certain weight 3/2 modular form. His thesis has paved the way for many applications of mock theta functions to the study

On the perturbative expansion of exact bi-local correlators in JT gravity

“…In the particular case of λ ∈ N/2, we derived an all-order formula for the perturbative contributions, that becomes particularly handful in the limit of infinite β. We have also shown that the exact expression for our bi-local correlators is closely related to the Mordell integral, a basic constituent in the theory of Mock-modular forms [31].…”

“…Moreover, the alternate sign of the perturbative orders points towards a possible Borel summability for the full series. As a final observation, we point out that the exact expression for the bi-local correlator can also be written in terms of Mordell integrals [29], suggesting a link with the world of Mock-modular forms [30,31]. The paper's structure is the following: in section 2, we review the exact results [17,19] and discuss the perturbative computation of the bi-local correlator in JT gravity from the Schwarzian perspective.…”

Error functions, Mordell integrals and an integral analogue of a partial theta function

“…Mordell showed that the above integral can be reduced to two standard forms, namely, ϕ(z, τ ) := τ Kuznetsov [28] has recently used h(z; τ ) to simplify Hiary's algorithm [22] for computing the truncated theta function n k=0 exp(2πi(zk+τ k 2 )), which in turn is used to compute ζ 1 2 + it to within ±t −λ in O λ (t 1 3 (ln t) κ ) arithmetic operations [21]. We refer the reader to a more recent article [9] and the references therein for further applications of Mordell integrals.…”

Error functions, Mordell integrals and an integral analogue of partial theta function

“…Kuznetsov [28] has recently used h(z; τ ) to simplify Hiary's algorithm [22] for computing the truncated theta function n k=0 exp(2πi(zk+τ k 2 )), which in turn is used to compute ζ 1 2 + it to within ±t −λ in O λ (t 1 3 (ln t) κ ) arithmetic operations [21]. We refer the reader to a more recent article [9] and the references therein for further applications of Mordell integrals.…”