Paul Erdös, Janos Galambos and others have studied the relative size of the consecutive prime divisors of an integer. Here, we further extend this study by examining the distribution of the consecutive neighbour spacings between the prime divisors $p_1(n)<p_2(n)<\cdots <p_r(n)$ of a typical integer $n\ge 2$. In particular, setting $\gamma_j(n):=\log p_j(n)/\log p_{j+1}(n)$ for $j=1,2,\ldots,r-1$ and, for any $\lambda\in(0,1]$, introducing $U_\lambda(n):= \#\{j\in\{1,2,\ldots,r-1\}: \gamma_j(n)<\lambda\}$, we establish the mean value of $U_\lambda(n)$ and prove that $U_{\lambda}(n)/r \sim \lambda$ for almost all integers $n\ge 2$. We also examine the shifted prime version of these two results and study other related functions.

Fix a positive integer $N \geq 2$. Following Chowla, we associate the $L$-series$L(s,f):=\sum_{n\geq 1}f(n)/n^s$ to each function $f:\mathbb{Z}\to\mathbb{C}$ with period $N$. Using a characterization derived by Okada for the vanishing of $L(1,f)$, we construct an explicit basis for the $\mathbb{Q}$-vector space, $$\mathscr{O}(N) = \{f \bmod N : f(n) \in \mathbb{Q}, L(1,f) = 0 \}.$$ We analyze the structure of this space and use the explicit basis to extend earlier works of Baker-Birch-Wirsing and Murty-Saradha.The arithmetical nature of Euler's constant $\gamma$ emerges as a central question in these extensions.

We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, Bäumer, Lumer and Neubrander, and Langenbruch.

Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field $\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound $\sqrt{q}$ on the clique number of $GP(q,d)$. In this paper, we investigate the connection between Gauss sums over a finite field and the maximum cliques of their corresponding generalized Paley graphs. We show that the trivial upper bound on the clique number of $GP(q,d)$ is tight if and only if $d \mid (\sqrt{q}+1)$, which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickelberger's theorem on evaluating semi-primitive Gauss sums.