We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function $a_n$ with the periodic Bernoulli polynomial weight ${\overline{B}}_{\varkappa }\left(nx\right)$ and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order $0$ or $1$ gives the well-known explicit formula for respectively the partial...

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.

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $\sigma \left(n\right):={\sum }_{d|n}d\<e^{\gamma }nloglogn$ is satisfied for $n\ge 5041$, where $\gamma$ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if $n\ge 37$ does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that $n$ must be divisible by a fifth power $\>1$. As consequence we obtain that RH holds true iff every natural number divisible by a fifth power...

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.