We give a simple proof that critical values of any Artin $L$-function attached to a representation $\rho$ with character ${\chi }_{\rho }$ are stable under twisting by a totally even character $\chi$, up to the $dim\rho$-th power of the Gauss sum related to $\chi$ and an element in the field generated by the values of ${\chi }_{\rho }$ and $\chi$ over $\mathbb{Q}$. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.

Let $\chi$ be a nonprincipal Dirichlet character modulo a prime number $p⩾3$ and let ${𝔞}_{\chi }:=\frac{1}{2}(1-\chi (-1))$. Define the mean value $${ℳ}_p(-s,\chi ):=\frac{2}{p-1}\sum \psi (modp)\psi (-1)=-1L(1,\psi )L(-s,\chi \overline{\psi })(\sigma :=\Re s>0).$$ We give an identity for ${ℳ}_p(-s,\chi )$ which, in particular, shows that $${ℳ}_p(-s,\chi )=L(1-s,\chi )+{𝔞}_{\chi }2p^{s}L(1,\chi )\zeta (-s)+o\left(1\right)(p\to \infty )$$ for fixed $0<\sigma <\frac{1}{2}$ and $|t:=\Im s|=o\left(p^{(1-2\sigma )/(3+2\sigma )}\right)$.

Grosswald’s conjecture is that g(p), the least primitive root modulo p, satisfies g(p) ≤ √p - 2 for all p > 409. We make progress towards this conjecture by proving that g(p) ≤ √p -2 for all $409<p<2.5\times {10}^{15}$ and for all $p>3.38\times {10}^{71}$.

We study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form [g(n)].

We consider some applications of the singular integral equation of the second kind of Fox. Some new solutions to Fox’s integral equation are discussed in relation to number theory.

We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty$ and any constant $\epsilon >0$, we can show that $$\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{[\alpha n+\beta ]\in {𝒬}_k}}1-\frac{x}{\zeta \left(k\right)}\ll x^{k/(2k-1)+\epsilon }+x^{1-1/(\tau +1)+\epsilon },$$ where ${𝒬}_k$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$$\epsilon ,$$k$...