We give a simple proof of $L(1,\chi )\sqrt{q}\gg 2^{\omega \left(q\right)}$ when $\chi$ is an odd primitiv quadratic Dirichlet character of conductor $q$. In particular we do not use the Dirichlet class-number formula.

Let ${\Phi }_n\left(q\right)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1(mod4)$, $$\sum _{k=0}^{n-1}\frac{{(q^{-1};q^2)}_k^2{(q^{-2};q^4)}_k}{{(q^2;q^2)}_k^2{(q^4;q^4)}_k}{q}^{6k}\equiv 0(mod{\Phi }_n{\left(q\right)}^2),$$ where ${(a;q)}_m=(1-a)(1-aq)\cdots (1-a{q}^{m-1})$. In this note, we give a generalization of the above $q$-congruence to the modulus ${\Phi }_n{\left(q\right)}^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo ${\Phi }_n{\left(q\right)}^2$ for $n\equiv 3(mod4)$. Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a ${}_4{\varphi }_3$ summation formula.

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight $k$ for the full modular group $\mathrm{SL}(2,\mathbb{Z})$, and denote its $n$th Fourier coefficient by ${\lambda }_f\left(n\right)$. We consider the hybrid problem of quadratic forms with prime variables and Hecke eigenvalues of normalized primitive holomorphic cusp forms, which generalizes the result of D. Y. Zhang, Y. N. Wang (2017).

Let K be an algebraic number field with non-trivial class group G and ${}_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_k\left(x\right)$ denote the number of non-zero principal ideals $a_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_k\left(x\right)$ behaves for x → ∞ asymptotically like $x{\left(logx\right)}^{1-1/\left|G\right|}{\left(loglogx\right)}^{{}_k\left(G\right)}$. We prove, among other results, that $₁\left(C_{n₁}\oplus C_{n₂}\right)=n₁+n₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

Let K be an algebraic number field with non-trivial class group G and ${}_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_k\left(x\right)$ denote the number of non-zero principal ideals $a_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_k\left(x\right)$ behaves, for x → ∞, asymptotically like $x{\left(logx\right)}^{1/\left|G\right|-1}{\left(loglogx\right)}^{{}_k\left(G\right)}$. In this article, it is proved that for every prime p, $₁\left(C_p\oplus C_p\right)=2p$, and it is also proved that $₁\left(C_{mp}\oplus C_{mp}\right)=2mp$ if $₁\left(C_m\oplus C_m\right)=2m$ and m is large enough. In particular, it is shown that for...