International Journal of Number Theory

We establish the triple integral evaluation

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of $n$. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here.

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt’s volumes on Ramanujan’s notebooks. We give two alternate approaches to proving Ramanujan’s Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by the theory of orthogonal polynomials.

We describe an extension of Fine’s method of deriving basic hypergeometric series transformations and derive new transformations from the method. Combinatorial proofs of two of the examples are provided.

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function $\omega (q)$ and Ramanujan’s third-order mock theta function $f(q)$.

In this paper, we revisit an identity which was proven by Ramanujan and from which he deduced the famous identities that are named after him and L. J. Rogers. Unlike Ramanujan’s proof (which uses the method of $q$-difference equations), we examine directly the $q$-coefficients involved. We isolate and identify terms that cancel each other. Once these terms are paired up and canceled, we only need the geometric series to complete the proof.

We prove a new mock theta function identity related to the partition rank modulo 3 and 9. As a consequence, we obtain the $3$-dissection of the rank generating function modulo $9$. We also evaluate all of the components of the rank–crank differences modulo $9$. These are analogous to conjectures of Lewis [The generating functions of the rank and crank modulo 8, Ramanujan J. 18 (2009) 121–146] on rank–crank differences modulo $8$, first proved by Mortenson [On ranks and cranks of partitions modulo 4 and 8, J. Combin. Theory Ser. A 161 (2019) 51–80].

In this paper, we consider the congruence equation $\overline{a}+\overline{b}\equiv \overline{c}(modp)$ with $1\le a,b,c\le H$ where $\overline{x}$ denotes the multiplicative inverse of $x(modp)$. We prove that its number of solutions is asymptotic to $H^3/p$ when $H>p^{2/3+o(1)}$ by estimating a certain average of Kloosterman sums via Gauss sums. On the other hand, when $H<p^{1/2}\sqrt{logp}$, the number of solutions has order of magnitude at least $HlogH$. It would be interesting to better understand its transition of behavior. By transforming the question slightly, one can relate the problem to a certain first moment of Dirichlet $L$-functions at $s=1$.

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

Given an integer base $b\ge 2$, a number $\rho \ge 1$ of colors, and a finite sequence $\mathrm{\Lambda }=({\lambda }_1,\dots ,{\lambda }_{\rho })$ of positive integers, we introduce the concept of a $\mathrm{\Lambda }$-restricted $\rho$-colored $b$-ary partition of an integer $n\ge 1$. We also define a sequence of polynomials in ${\lambda }_1+\cdots +{\lambda }_{\rho }$ variables, and prove that the $n$th polynomial characterizes all $\mathrm{\Lambda }$-restricted $\rho$-colored $b$-ary partitions of $n$. In the process, we define a recurrence relation for the polynomials in question, obtain explicit formulas, and identify a factorization theorem.

We obtain a new proof of Hurwitz’s formula for the Hurwitz zeta function $\zeta (s,a)$ beginning with Hermite’s formula. The aim is to reveal a nice connection between $\zeta (s,a)$ and a special case of the Lommel function $S_{\mu ,\nu }(z)$. This connection is used to rephrase a modular-type transformation involving infinite series of Hurwitz zeta function in terms of those involving Lommel functions.

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews’ famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank moments, analogous to Garvan’s conjecture. In this paper, we give a proof of this conjecture.

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as $q$ tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function $f(q)$ were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as $q$ tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity.

In this paper, we prove modular identities conjectured by Koike and Somos for the Rogers–Ramanujan functions. Our methods focus on approaches that Ramanujan could have employed, including the theory of modular equations, dissections of identities, and methods derived from the approaches of Watson and Rogers.

Using a modular equation of level $3$ and degree $23$ due to Chan and Liaw, we prove the fastest known (conjectured to be the fastest one) convergent rational Ramanujan-type series for $1/\pi$ of level $3$.

In this paper, we study the order of the pole of the triple tensor product $L$-functions $L(s,{\pi }_1\times {\pi }_2\times {\pi }_3,{\otimes }^3)$ for cuspidal automorphic representations ${\pi }_i$ of ${\mathrm{\text{GL}}}_{n_i}({𝔸}_F)$ in the setting where one of the ${\pi }_i$ is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands’ beyond endoscopy proposal.

Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series $E_k$, especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized Petersson’s approach via Poincaré series, to handle the general case. We introduce a third approach. We attach recursively defined polynomials to reciprocals of Eisenstein series. This provides easy access to the signs of the Fourier coefficients of reciprocals of Eisenstein series, sheds some light on reciprocals of $E_k$ of general weight, and provides some upper and lower bounds for their growth.

Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level seven. The work determines a general strategy for identifying and proving identities and associated congruences for modular forms on the principal congruence subgroup of level $7$. Ramanujan’s partition congruence modulo $7$ serves as a prototype for the process used to prove new congruences for modular forms of level $7$.

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.