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 $p>3.38\times {10}^{71}$.

Let $p$ be an odd prime and $c$ a fixed integer with $(c,p)=1$. For each integer $a$ with $1\le a\le p-1$, it is clear that there exists one and only one $b$ with $0\le b\le p-1$ such that $ab\equiv c$ (mod $p$). Let $N(c,p)$ denote the number of all solutions of the congruence equation $ab\equiv c$ (mod $p$) for $1\le a$, $b\le p-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1(modp)$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value problem involving...