The Ramanujan Journal

Let $$\alpha >1$$ be an irrational number. We establish asymptotic formulas for the number of partitions of n into summands and distinct summands, chosen from the Beatty sequence $$(\lfloor \alpha m\rfloor )$$ . This improves some results of Erdös and Richmond established in 1977.

The Hilbert genus field of the real biquadratic field $$K=\mathbb {Q}(\sqrt{\delta },\sqrt{d})$$ is described by Yue (Ramanujan J 21:17–25, 2010) and by Bae and Yue (Ramanujan J 24:161–181, 2011) explicitly in the case $$\delta =2$$ or $$p$$ with $$p\equiv 1 \, \mathrm{mod}\, 4$$ a prime and $$d$$ a squarefree positive integer. In this article, we describe explicitly the case that $$\delta =p, 2p$$ or $$p_1p_2$$ where $$p$$ , $$p_1$$ , and $$p_2$$ are primes congruent to $$3$$ modulo $$4$$ , and $$d$$ is any squarefree positive integer, thus complete the construction of the Hilbert genus field of real biquadratic field $$K=K_0(\sqrt{d})$$ such that $$K_0=\mathbb {Q}(\sqrt{\delta })$$ has an odd class number.

The 15 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 are called the supersingular primes: they occur in several contexts in number theory and also, strikingly, they are the primes that divide the order of the Monster. It is also known that the moduli space of (1, p)-polarised abelian surfaces is of general type for these primes. In this note, we explain that apparently coincidental fact by relating it to other number-theoretic occurences of the supersingular primes.

In this note, we describe the parity of the coefficients of the McKay–Thompson series of Mathieu moonshine. As an application, we prove a conjecture of Cheng, Duncan, and Harvey stated in connection with umbral moonshine for the case of Mathieu moonshine.

Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $${\mathbb {Q}}$$ and by the subsequent generalization of Cojocaru–Davis–Silverberg–Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over $${\mathbb {Q}}$$ that are not $${\overline{{\mathbb {Q}}}}$$ -isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.

Kaneko and Koike gave the “extremal” quasimodular forms of depth 1 for PSL2(ℤ) and modular differential equations they satisfy. In this paper, we study modular solutions of their modular differential equations.

For every positive integer d, we define a meromorphic function F d (n;z), where n,z∈ℂ d , which is a natural extension of the multivariable Askey–Wilson polynomials of Gasper and Rahman (Theory and Applications of Special Functions, Dev. Math., vol. 13, pp. 209–219, Springer, New York, 2005). It is defined as a product of very-well-poised 8 φ 7 series and we show that it is a common eigenfunction of two commutative algebras ${\mathcal{A}}_{z}$ and ${\mathcal{A}}_{n}$ of difference operators acting on z and n, with eigenvalues depending on n and z, respectively. In particular, this leads to certain identities connecting products of very-well-poised 8 φ 7 series.

Let NT(m, k, n) denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck’s conjecture on congruences for NT(m, k, n) modulo 5 and 7. Generalizing Andrews’ results, Chern obtained congruences for NT(m, k, n) modulo 11 and 13. More recently, the second author used the theory of Hecke operators to establish congruences for such partition statistics modulo powers of primes $$\ell \ge 7$$ . In this paper, we obtain Andrews-Beck type congruences modulo powers of 5.