In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

Let $m$ be a positive integer and $p$ a prime number. We prove the orthogonality of some character sums over the finite field $\mathbb{F}_{p^{m}}$ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension $p^{m}$ over the complex number field $\mathbb{C}$, especially with $p=2$.

For three points , and in the n-dimensional space 𝔽nq over the finite field 𝔽q of q elements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points define acute angle triangles. A similar question in the real space ℛn dates back to P. Erdős and has been studied by several authors.

Let q≥2 and N≥1 be integers. W. Zhang recently proved that for any fixed ε>0 and qε≤N≤q1/2−ε, where the sum is taken over all nonprincipal characters χ modulo q, L(1,χ) denotes the L-functions corresponding to χ, and αq=qo(1) is some explicit function of q. Here we improve this result and show that the same asymptotic formula holds in the essentially full range qε≤N≤q1−ε.