The notion of depth two and higher mock modular forms have found important applications in mathematical physics and enumerative geometry since their inception through indefinite theta functions with general signature. These theta functions generalize Zwegers' work on Lorentzian signature lattices and the framework of mock modular forms that emanated from it. Mock modular forms can also be studied through Eisenstein and Poincaré series. The interaction of this second point of view with the indefinite theta function approach yields a wealth of tools to unearth the rich structure behind mock modular forms. For mock modular forms of higher depth, on the other hand, indefinite theta functions and their variants largely remained the only available approach. In this paper, we show that one can indeed get mock modular forms of depth two by "coupling" a pair of Eisenstein series that yield depth one mock modular forms, thereby providing a new and independent approach to higher depth mock modular forms. We exemplify this new perspective on a depth two object that appeared in the context of Vafa-Witten invariants.

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo-Pak, proved that the partition function p(n) is eventually log-concave. Inspired by this and other results, Chern-Fu-Tang first conjectured the log-concavity of k-coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for k >= 2 We shed further light on this phenomenon by utilizing Hardy Littlewood-P & oacute;lya's notion of majorizing. We prove that for partitions a, b of n is an element of N, if b majorizes a, then P-k(a) > p(k)(b). Numerical calculations indicate that our result is sharp.

We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.

In this paper, we investigate class numbers of shifted quadratic lattices $L+\frac{\boldsymbol{u}}{c}$ with $\boldsymbol{u}\in L$ and odd conductor $c\in \mathbb{N}$. For a lattice $L$ whose genus only contains one class, we determine a lower bound for the number of classes in the genus of $L+\frac{\boldsymbol{u}}{c}$ depending on $c$. As a result, we obtain an explicit bound $c_0$ such that any such shifted lattice with one class in its genus must have conductor smaller than $c_0$, restricting the possible choices of such $L+\frac{\boldsymbol{u}}{c}$ to a finite set.

Motivated by the fact that the classical Jacobi theta function ϑ is the exponential generating function of the Eisenstein series, we study the exponential Taylor coefficients (in the elliptic variable) of a related natural partial theta function, as well as a false theta function corresponding to the Dedekind eta function. We prove that the space spanned by these objects is closed under differentiation, analogous to the space of quasimodular forms, and that it contains the quasimodular forms themselves. We further provide their Fourier expansions, establish quasimodular completions, and derive a recursive formula for the Taylor coefficients of the logarithm of the unimodal rank generating function, expressed as partition traces of the false and partial objects.

In this paper we study restricted overpartitions and concave compositions. In several cases the resulting generating functions involve simultaneously modular forms, mock theta functions, mock Maass theta functions, and false theta functions, illustrating the appearance of mixed modular structures in restricted partition problems. Moreover, we obtain their asymptotic main terms. We also study related rank statistics.

Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle these conjectures by relating the corresponding generating function to modular forms and mock theta functions.

In this paper, we investigate congruences for meromorphic modular forms F which have a pole at a single point z in the fundamental domain of SL_2(ℤ). For a prime p with good supersingular reduction at the elliptic curve corresponding to z, we show that there exists a cusp form f such that F|U_p^m ≡ f|U_p^m p^κ_m, where κ_m=αm -β with α only depending on the weight of F and β depending on F and p but is independent of m. In particular, if the space of cusp forms is trivial, then F|U_p^m≡ 0 p^κ_m vanishes p-adically to a high order. In order to prove these results, we use the fact that p has supersingular reduction to realize F as an overconvergent modular form and then utilize the theory of overconvergent forms to show the congruences.

We introduce an extension of the standard cohomology which is characterised by maps that fail to be classical cocycles by products of simpler maps. The construction is motivated by the study of Manin's noncommutative modular symbols and of false theta functions. We use this construction to obtain a cohomological interpretation of important iterated integrals that arise in that study. In another direction, our approach gives modular counterparts to the long-studied relations among multiple zeta values.

We prove an asymptotic formula for the number of d-fold partition diamonds of n and their Schmidt-type counterparts. In order to do so, we study the asymptotic behavior of certain infinite products. We also remark on interesting potential connections with mathematical physics and Bloch groups.

We study certain algebras of theta-like functions on partitions, for which the corresponding generating functions give rise to theta functions, quasi-Jacobi forms, Appell-Lerch sums, and false theta functions.

We study an extension of Ramanujan's identities for odd zeta values by Lim and introduce Jacobi analogues of classical Eichler integrals of Eisenstein series. In negative weight we construct explicit completions and embed these objects into a modular framework by showing that they are (singular) harmonic Maass–Jacobi forms. We further describe their non-holomorphic parts in terms of Eichler integrals, establish Ramanujan-type inversion formulas, and study their behavior under the Maass raising and lowering operators and at torsion points.

We derive an asymptotic expansion with effective error bound for u(n), counting the number of unimodal sequences of size n. We prove that u(n) satisfies the higher order Turán inequalities for n≥33 and that certain second j-shifted difference of u(n) is positive.

In this paper, we give a direct conceptual proof of the main result of Mono, Rolen, and Stumpenhusen, using differential operators. More precisely, we realize their functions ω_k+1,D as images of the quadratic form Poincaré series f_k,D under the Maass raising operator. This perspective gives a natural explanation for the modularity and Laplace eigenvalue prop erties of ω_k+1,D. We further extend these results by investigating the images of more general local Maass forms under the Maass raising and lowering operators.

In this paper we strongly improve asymptotics for s_1(n) (respectively s_2(n)) which sums reciprocals (respectively squares of reciprocals) of parts throughout all the partitions of n into distinct parts. The methods required are much more involved than in the case of usual partitions since the generating functions are not modular and also do not posses product expansions.

A classical class number relation of Hurwitz expresses the Fourier coefficients of the product of a unary theta function and the class number generating function. Here, we establish an infinite family of analogous class number relations obtained by replacing the unary quadratic form m^2 by positive-definite binary quadratic forms. These identities involve Cohen's generalized class numbers and depend only on the genus of the underlying quadratic form. For this, we construct a genus-dependent level-lowering operator.

In this paper, we prove a conjecture of Andrews and Bachraoui relating a generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) to a Hecke-type double sum. Our proof is based on Zwegers' theory of indefinite theta functions together with modular transformation properties of mock theta functions.

Recently, Andrews and Bachraoui considered a generating function F_k,m(q) associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for m=1. They showed this property for 1 ≤ k ≤ 4. In this note, we prove that F_k,1(q) has non-negative coefficients for 5 ≤ k ≤ 10. Moreover, we show that, as k→∞, F_k,1(q) is related to Ramanujan's third order mock theta function ω(q) and to quotients of certain q-binomial coefficients.

In this paper, we prove a conjecture of Andrews and El Bachraoui concerning the parity of certain two-color partitions. Precisely, we show that if the Fourier coefficient t_o(n) of the corresponding q-series is odd, then 8n+9 is represented by the binary quadratic form x^2+2y^2.

In this paper, we construct Hecke eigenforms for two families of quotient spaces of meromorphic cusp forms on SL_2(ℤ) . We show that each quotient space in the first (resp. second family) is isomorphic as a Hecke module to the space S_2k (resp. M_2k ) of cusp forms (resp. holomorphic modular forms) of the same weight on SL_2(ℤ) .