For a fixed integer $k\ge 3$, and fixed $\frac{1}{2}\<\sigma \<1$ we consider $${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_k^2\left(n\right)n^{-2\sigma }T+R(k,\sigma ;T),$$ where $R(k,\sigma ;T)=0\left(T\right)(T\to \infty )$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k,\sigma ;T)$ are derived in the range min $({\beta }_k,{\sigma }_k^{*})\<\sigma \<1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

In this paper, we give a new upper-bound for the discrepancy $$D\left(N\right):=\underset{0\le \gamma \le 0}{sup}|\sum _{\genfrac{}{}{0pt}{}{p\le /N}{\left\{p^{\alpha }\right\}\le \gamma }}1-\pi \left(N\right)\gamma |$$ for the sequence $\left(p^{\alpha }\right)$, when $5/3\le \alpha \>3$ and $\alpha \ne 2$.

The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{4/3}$ for the error term $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^2}x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{1+ϵ}$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical -...

A limit theorem in the space of continuous functions for the Dirichlet polynomial $$\sum _{m\le T}\frac{d_{{\kappa }_T}\left(m\right)}{m^{{\sigma }_T+it}}$$ where $d_{{\kappa }_T}\left(m\right)$ denote the coefficients of the Dirichlet series expansion of the function ${\zeta }^{{\kappa }_T}\left(s\right)$ in the half-plane $\sigma \>1$ ${\kappa }_T={(2^{-1}log{l}_T)}^{-\frac{1}{2}}$, ${\sigma }_T=\frac{1}{2}+\frac{1n^2{l}_T}{l_T}$ and $l_T\>0$, $l_T\le$ 1n $T$ and $l_T\to \infty$ as $T\to \infty$, is proved.

Several problems and results on mean values of $\zeta \left(s\right)$ are discussed. These include mean values of $\left|\zeta \right(\frac{1}{2}+it\left)\right|$ and the fourth moment of $\left|\zeta \right(\sigma +it\left)\right|$ for $1/2\<\sigma \<1$.

In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $\Lambda =b_{2}log{\alpha }_2-b_{1}log{\alpha }_1-b_{3}log{\alpha }_3\ne 0$ with $b_1,b_2,b_3$ positive integers and ${\alpha }_1,{\alpha }_2,{\alpha }_3$ positive multiplicatively independent rational numbers greater than $1$. Let ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$ with ${\alpha }_{j1},{\alpha }_{j2}$ coprime positive integers $(j=1,2,3)$. Let ${\alpha }_j\ge \text{max}\{{\alpha }_{j1},e\}$ and assume that gcd$(b_1,b_2,b_3)=1.$ Let $$b^{\text{'}}=\left(\frac{b_2}{log{\alpha }_1}+\frac{b_1}{log{\alpha }_2}\right)\left(\frac{b_2}{log{\alpha }_3}+\frac{b_3}{log{\alpha }_2}\right)$$ and assume that $B\ge \text{max}\{10,log{b}^{\text{'}}\}.$ We prove that either $\{b_1,b_2,b_3\}$ is $\left(c_4,B\right)$-linearly dependent over $\mathbb{Z}$ (with respect to $\left(a_1,a_2,a_3\right)$)...

We prove that for any real $\theta$ there are infinitely many values of $s=\sigma +it$ with $\sigma \to 1+$ and $t\to +\infty$ such that $$\Re \{exp\left(i\theta \right)logL(s,\chi )\}\ge log\frac{logloglogt}{loglogloglogt}+O\left(1\right).$$ The proof relies on an effective version of Kronecker’s approximation theorem.

We consider the $q$-ary digital expansion of the first $N$ terms of an exponential sequence $a^n$. Using a result due to Kiss and Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last $clogN$ digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first ${(logN)}^{\frac{3}{2}-ϵ}$ digits, where $ϵ$ is a positive constant. In both methods, we use estimations of exponential sums and the concept of discrepancy of real sequences...

Under the Generalized Riemann Hypothesis, it is proved that for any $k\ge 200$ there is $N_k\>0$ depending on $k$ only such that every even integer $\ge N_k$ is a sum of two odd primes and $k$ powers of $2$.

In this paper, we apply character sum estimates to study products of factorials modulo $p$.

As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $xh\left(x\right)$ where $h$ is an arithmetical function (namely $h\left(n\right)=1/n$, $h\left(n\right)=logn$, $h\left(n\right)=1/logn$) and $n$ is an integer (or a prime order) running over the interval $\left[y\right(x),x)]$. The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.