We prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ , where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.

This work investigates the solution and the stability of a generalized system of bi-quadratic–Drygas functional equations in non-Archimedean normed spaces with unknown coefficients. The presence of asymmetric coefficients and reflection terms induces nontrivial coupling effects and symmetry-breaking phenomena, while simultaneously capturing additive, quadratic, and mixed additive–quadratic behaviors. An examination of the coefficients shows that the trivial solution is the only one satisfying the system of equations in asymmetric parameter configurations. For symmetric configurations, by exploiting the ultrametric structure of non-Archimedean norms and applying an iterative method combined with symmetry-based decomposition into even and odd parts, we establish the existence and uniqueness of an exact solution approximating a given mapping. Several known stability results for bi-Drygas functional equations are recovered with improvement as special cases.

We investigate the Hyers–Ulam–Rassias stability of a generalized quadratic functional equation of the asymmetric four-function form $F(x+y)+G(x-y)=L\left(x\right)+M\left(y\right)$, where F, G, L, and M are unknown mappings. This study is conducted within the framework of non-Archimedean normed spaces over the p-adic numbers. Our approach employs a separation technique, analyzing the even and odd parts of the functions to establish stability results. We show that all four functions are approximated by a combination of a quadratic function and two additive functions.

We introduce a generalized adjacency matrix in order to distinguish cospectral graphs. Our reasoning is motivated by the work of Johnson and Newman and properties of p-adic numbers. Using a polynomial time algorithm, we comment on computer experiments with which we can distinguish cospectral (non-isomorphic) graphs.

We consider linear recursions of length two and related gap recursions where the indices may not be consecutive integers. Given a linear recursion of length two, we prove the existence of an explicit linear recursion of length $\mathcal{l}$ with arbitrary distance between indices. Conversely, it is shown under some mild assumption that a linear recursion of length $\mathcal{l}$ can be reduced to one of length two.

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t^13/84+ϵ as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t^ϵ (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros.