We study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference ?, an integer with the Vorono? function weight Vk. In the case of V1(x)=e?x, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The ?=0 case is the divisor function, while the ?=1 case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan.

The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers. They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields. At the end, they provide some numerical examples supporting our results

Let \(d\) be a square-free integer and \(\mathbb{Z}[\sqrt{d}]\) a quadratic ring of integers. For a given \(n\in\mathbb{Z}[\sqrt{d}]\), a set of \(m\) non-zero distinct elements in \(\mathbb{Z}[\sqrt{d}]\) is called a Diophantine \(D(n)\)-\(m\)-tuple (or simply \(D(n)\)-\(m\)-tuple) in \(\mathbb{Z}[\sqrt{d}]\) if product of any two of them plus \(n\) is a square in \(\mathbb{Z}[\sqrt{d}]\). Assume that \(d \equiv 2 \pmod 4\) is a positive integer such that \(x^2 - dy^2 = -1\) and \(x^2 - dy^2 = 6\) are solvable in integers. In this paper, we prove the existence of infinitely many \(D(n)\)-quadruples in \(\mathbb{Z}[\sqrt{d}]\) for \(n = 4m + 4k\sqrt{d}\) with \(m, k \in \mathbb{Z}\) satisfying \(m \not\equiv 5 \pmod{6}\) and \(k \not\equiv 3 \pmod{6}\). Moreover, we prove the same for \(n = (4m + 2) + 4k\sqrt{d}\) when either \(m \not\equiv 9 \pmod{12}\) and \(k \not\equiv 3 \pmod{6}\), or \(m \not\equiv 0 \pmod{12}\) and \(k \not\equiv 0 \pmod{6}\). At the end, some examples supporting the existence of quadruples in \(\mathbb{Z}[\sqrt{d}]\) with the property \(D(n)\) for the above exceptional \(n\)'s are provided for \(d = 10\)

We study the exponential Diophantine equation x2+ pmqn= 2ypin positive integers x, y, m, n, and odd primes p and q using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation. © 2024 New Zealand Mathematical Society and Department of Mathematics, University of Auckland. All rights reserved.