The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on ${}^{\omega }$, the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $(a_p,s)={\prod }_p{(1-a_p{p}^{-s})}^{-1}$ for $a_p$ in ${}^{\omega }$. Among other things, using the Haar measure on ${}^{\omega }$ for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.

Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_k\left(x\right)$ where $1/k!{\sum }_{n\le x}n/\varphi \left(n\right){(1-n/x)}^k=M_k\left(x\right)+E_k\left(x\right)$. For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying $k\gg {\left(logx\right)}^{1+ϵ}$.

Let $\zeta \left(s\right)$ be the Riemann zeta-function. If $t\ge 6.8$ and $\sigma >1/2$, then it is known that the inequality $\left|\zeta \right(1-s\left)\right|>\left|\zeta \right(s\left)\right|$ is valid except at the zeros of $\zeta \left(s\right)$. Here we investigate the Lerch zeta-function $L(\lambda ,\alpha ,s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $\lambda =\alpha$ it is still possible to obtain a certain version of the inequality $|L(\lambda ,\lambda ,1-\overline{s})|>|L(\lambda ,\lambda ,s)|$.

Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that ${\sum }_{n\le x}d²\left(n\right)=xP\left(logx\right)+E\left(x\right)$, where P(y) is a cubic polynomial in y and $E\left(x\right)=O\left(x^{3/5+\epsilon }\right)$, with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E\left(x\right)=O\left(x^{1/2+\epsilon }\right)$. In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E\left(x\right)=O\left(x^{1/2}\left(logx\right)⁵loglogx\right)$. In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E\left(x\right)=O\left(x^{1/2}\left(logx\right)⁵\right)$. ...

Let $K/\mathbb{Q}$ be a nonnormal cubic extension which is given by an irreducible polynomial $g\left(x\right)=x^3+a{x}^2+bx+c$. Denote by ${\zeta }_K\left(s\right)$ the Dedekind zeta-function of the field $K$ and $a_K\left(n\right)$ the number of integral ideals in $K$ with norm $n$. In this note, by the higher integral mean values and subconvexity bound of automorphic $L$-functions, the second and third moment of $a_K\left(n\right)$ is considered, i.e., $$\sum _{n\le x}{a}_K^2\left(n\right)=x{P}_1(logx)+O\left(x^{5/7+ϵ}\right),\sum _{n\le x}{a}_K^3\left(n\right)=x{P}_4(logx)+O\left(X^{321/356+ϵ}\right),$$ where $P_1\left(t\right)$, $P_4\left(t\right)$ are polynomials of degree 1, 4, respectively, $ϵ>0$ is an arbitrarily small number.

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 $|t:=\Im s|=o\left(p^{(1-2\sigma )/(3+2\sigma )}\right)$.

We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k,l}\left(T\right)={\int }_0^T{\left|{\zeta }^{\left(l\right)}(\frac{1}{2}+it)\right|}^{2k}dt$, where $l$ is a non-negative integer and $k\ge 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k,l}\left(T\right)$.

A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with ${li}^{-1}\left(n\right)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n\&gt;s_3\left(n\right)$ where $s_3\left(n\right)$ is the sum of the first four terms of the asymptotic...