Let \(\textit{K}\) be a cubic number field. In this paper, we study the Ramanujan sums \(c_{\mathcal {J}}(\mathcal {I})\), where \(\mathcal {I}\) and {J}\) are integral ideals in {O}_\textit{K}\). The asymptotic behaviour of {I})\) over both is investigated.

We show, in the large $q$ limit, that average size of $n$-Selmer groups elliptic curves bounded height over $\mathbb F_q(t)$ is sum divisors $n$. As a corollary, again we deduce $100\%$ have rank $0$ or $1$.

This paper presents the design of a high-performance 0.45–0.50 THz antenna on chip (AoC) for fabrication 100-micron GaAs substrate. The is based metasurface and substrate-integrated waveguide (SIW) technologies. It constituted from seven stacked layers consisting copper patch–silicon oxide–feedline–silicon oxide–aluminium–GaAs–copper ground. top layer consists 2 × 4 array rectangular metallic p...