We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that $\mathbb{Z}{*}_p$ has no generator in the interval [1,B]. As a consequence we prove that if $Q>exp\left[c\right(logp)/(logB\left)\right(loglogp\left)\right]$ with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that ${\nu }_q\left(or{d}_{p}b\right)={\nu }_q(p-1)$ for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.

Let $\Delta \left(x\right)$ denote the error term in the Dirichlet divisor problem, and let E(T) denote the error term in the asymptotic formula for the mean square of |ζ(1/2+it)|. If E*(t) := E(t) - 2πΔ*(t/(2π)) with Δ*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x) and ${\int }_0^{T}E*\left(t\right)dt=3/4\pi T+R\left(T\right)$, then we obtain a number of results involving the moments of |ζ(1/2+it)| in short intervals, by connecting them to the moments of E*(T) and R(T) in short intervals. Upper bounds and asymptotic formulae for integrals of the form ∫T2T(∫t-Ht+H |ζ(1/2+iu|2 duk dt$(k\in \mathbb{N},1\ll H\le T)$are...

Some problems involving the classical Hardy function $$Z\left(t\right)=\zeta \left(\frac{1}{2}+it\right){\left(\chi \left(\frac{1}{2}+it\right)\right)}^{-1/\phantom{12}2},\zeta \left(s\right)=\chi \left(s\right)\zeta \left(1-s\right)$$ , are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.

We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j\ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,{\mathrm{sym}}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $$S_j^{*}:=\sum _{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}{\lambda }_{{\mathrm{sym}}^{j}f}^2(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),$$ where $x$ is sufficiently large, and $$L(s,{\mathrm{sym}}^{j}f):=\sum _{n=1}^{\infty }\frac{{\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)}{n^s}.$$ When $j=2$, the error term which we obtain improves the earlier known result.

The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size $x/\left(q^{1/2}\varphi \left(q\right)\right)$ when logx/logq ≥ 8 and is bounded....