We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials $1+x^{r₁}+\cdots +x^{r₅}$, where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2r_j<r_{j+1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct value of...

The main result of this paper implies that for every positive integer $d⩾2$ there are at least ${(d-3)}^2/2$ nonconjugate algebraic numbers which have their Mahler measures lying in the interval $(1,2)$. These algebraic numbers are constructed as roots of certain nonreciprocal quadrinomials.

We prove that there is no primitive nonic number field ramified only at one small prime. So there is no nonic number field ramified only at one small prime and with a nonsolvable Galois group.

Let $p=1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{\phi }$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ=\mathbb{Q}\left({\zeta }_{(p-1)pⁿ}\right)$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\phi }=1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T{r}_{n/1}\left(\xi B_{1,\chi }\right)\ne 0$ for any pⁿth root ξ...

Let $K=\mathbb{Q}\left(\sqrt{-D}\right)$ be an imaginary quadratic field, and denote by $h$ its class number. It is shown that there is an absolute constant $c\&gt;0$ such that for sufficiently large $D$ at least $c·h{\prod }_{p\mid D}(1-p^{-1})$ of the $h$ distinct $L$-functions $L_K(s,\chi )$ do not vanish at the central point $s=1/2$.

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and let ${𝒪}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in ${𝒪}_K$ under the assumption of the $abc$-conjecture for number fields.