The Ramanujan Journal最新文献

A formula is established for the evaluation of double series of Hecke–Rogers type in terms of theta functions and Appell–Lerch functions. This formula is similar to others obtained previously by Hickerson and Mortenson, and by Mortenson and Zwegers. It is applied to the rank function, leading to an expansion closely analogous to two of Garvan’s double series identities. Several identities involving third-order mock theta functions are obtained as special cases.

Let (K_n) be a tamely ramified cyclic quintic field generated by a root of Emma Lehmer’s parametric polynomial. We give all normal integral bases for (K_n) only by the roots of the polynomial, which is a generalization of the work of Lehmer in the case that (n^4+5n^3+15n^2+25n+25) is prime number, and Spearman–Willliams in the case that (n^4+5n^3+15n^2+25n+25) is square free.

In the article, we study the Oberdieck derivative defined on the space of weak Jacobi forms. We prove that the Oberdieck derivative maps a Jacobi form to a Jacobi form. Moreover, we study the adjoint of the Oberdieck derivative of a Jacobi cusp form with respect to the Petersson scalar product defined on the space of Jacobi forms. As a consequence, we also obtain the adjoint of the Jacobi–Serre derivative (defined in an unpublished work of Oberdieck). As an application, we obtain certain relations among the Fourier coefficients of Jacobi forms.

We reformulate the Kanade–Russell conjecture modulo 9 via the vertex operators for the level 3 standard modules of type (D^{(3)}_{4}). Along the same lines, we arrive at three partition theorems which may be regarded as an (A^{(2)}_{4}) analog of the conjecture. Andrews–van Ekeren–Heluani have proven one of them, and we point out that the others are easily proven from their results.

Let (pi ) be a cuspidal automorphic representation of (textrm{GL}_2(mathbb {A}_mathbb {Q})) associated to holomorphic forms with Fourier coefficients (a_{ pi }(n)). Consider an automorphic representation (Pi ) which is equivalent to (textrm{sym}^m pi ) or (pi times textrm{sym}^m pi ). We establish uniform upper bounds for (sum _{nleqslant X} |a_{Pi } (|f(n)|)|), where (f(x)in mathbb {Z}[x]) is a polynomial of arbitrary degree. This builds on the work of Chiriac and Yang, and refines one of their results.

A Diophantine m-tuple over a finite field ({mathbb F}_q) is a set ({a_1,ldots , a_m}) of m distinct elements in (mathbb {F}_{q}^{*}) such that (a_{i}a_{j}+1) is a square in ({mathbb F}_q) whenever (ine j). In this paper, we study M(q), the maximum size of a Diophantine tuple over ({mathbb F}_q), assuming the characteristic of ({mathbb F}_q) is fixed and (q rightarrow infty ). By explicit constructions, we improve the lower bound on M(q). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor.

In 1980, Bressoud conjectured a combinatorial identity (A_j=B_j) for (j=0) or 1. In this paper, we introduce a new partition function (widetilde{B}_0) which can be viewed as an overpartition analogue of the partition function (B_0). An overpartition is a partition such that the last occurrence of a part can be overlined. We build a bijection to get a relationship between (widetilde{B}_0) and (B_1), based on which an overpartition analogue of Bressoud’s conjecture for (j=0) is obtained.

Let (n, q) denote the greatest common divisor of positive integers n and q, and let (f_{r}) denote the characteristic function of r-full numbers. We consider several asymptotic formulas for sums of the modified square-full ((r=2)) and cube-full numbers ((r=3)), which is (sum _{nle y}sum _{qle x}sum _{d|(n,q)}df_{r}left( frac{q}{d}right) log frac{x}{q}) with any positive real numbers x and y. Moreover, we derive the asymptotic formula of the above with (r=2) under the Riemann Hypothesis.

Various vanishing coefficient results on q-series expansions have been widely studied by many authors in recent years. Motivated by these works, we establish a general vanishing coefficient result on odd powers of Ramanujan’s theta functions.

Over any fixed totally real number field with narrow class number one, we prove that the Rankin–Cohen bracket of two Hecke eigenforms for the Hilbert modular group can only be a Hecke eigenform for dimension reasons, except for a couple of cases where the Rankin–Selberg method does not apply. We shall also prove a conjecture of Freitag on the volume of Hilbert modular groups, and assuming a conjecture of Freitag on the dimension of the cuspform space, we obtain a finiteness result on eigenform product identities.