The topics include

theta functions and holomorphic Jacobi forms, classical Maass forms, differential operators and mock modular forms, examples of harmonic Maass forms, Ramanujan's mock

theta functions, the mock modular Eichler-Shimua theory, asymptotics for coefficients of modular-type functions, harmonic Mass forms as arithmetic and geometric generating functions, generalized Borcherds products, and representation theory and mock modular forms.

The study of

theta functions and theta constants has a long history, and they are very important objects in arithmetic and geometry.

Zhang, "Riemann

theta functions periodic wave solutions and rational characteristics for the nonlinear equations," Journal of Mathematical Analysis and Applications, vol.

Ojah, "Analogues of Ramanujan's partition identities and congruences arising from his

theta functions and modular equations," Ramanujan Journal, vol.

A Brief Introduction to

Theta Functions (reprint, 1961)

Tian and Zhang gave the exact periodic solutions for some evolution equations with the aid of the Hirota bilinear method and

theta functions identities [18,19].

As usual, the classical Jacobi

theta functions are defined as follows,

Undoubtedly the most famous are mock

theta functions. In 1919, Ramanujan returned to India, after about five years in England.

Ramanujan's letter to Hardy described several new functions that behaved differently from known

theta functions, or modular forms, and yet closely mimicked them.

In four cases it is already known that the product of two distinct Jacobian

theta functions having the same variable z and the same nome q is a multiple of a single Jacobian

theta function, with the multiple independent of z.

A selection of 10 papers from it consider such topics as self-dual codes and invariant theory; vector bundles in error-correcting for geometric Goppa codes; combinatorial designs and code synchronization; real and imaginary hyper-elliptic curve cryptography; divisibility, smoothness, and cryptographic applications; a variant of the Reidemeister-Schreier algorithm for the fundamental groups of Riemann surfaces;

theta functions and algebraic curves with automorphisms; enumerative geometry and string theory; and the cryptographical properties of extremal algebraic graphs.

The close connection between q-calculus on the one hand, and elliptic functions and

theta functions on the other hand will be shown.