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 $K/\mathbb{Q}$ 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.

For any square-free positive integer $m\equiv 10(mod16)$ with $m\ge 26$, we prove that the class number of the real cyclotomic field $\mathbb{Q}({\zeta }_{4m}+{\zeta }_{4m}^{-1})$ is greater than $1$, where ${\zeta }_{4m}$ is a primitive $4m$th root of unity.

We find the sum of series of the form $$\sum _{i=1}^{\infty }\frac{f\left(i\right)}{i^r}$$ for some special functions $f$. The above series is a generalization of the Riemann zeta function. In particular, we take $f$ as some values of Hurwitz zeta functions, harmonic numbers, and combination of both. These generalize some of the results given in Mező’s paper (2013). We use multiple zeta theory to prove all results. The series sums we have obtained are in terms of Bernoulli numbers and powers of $\pi$.

Let $\Gamma$ be a group and $r_n\left(\Gamma \right)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function ${𝒵}_{\Gamma }\left(s\right)={\sum }_{n=1}^{\infty }{r}_n\left(\Gamma \right)n^{-s}$. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then ${𝒵}_{\Gamma }\left(s\right)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp₂(ℤ) of genus 2. The coefficients of the spinor zeta function $Z_F\left(s\right)$ are denoted by cₙ. Let $D_{\rho }(x;Z_F)$ be the Riesz mean of cₙ. Kohnen and Wang obtained the truncated Voronoï-type formula for $D_{\rho }(x;Z_F)$ under the Ramanujan-Petersson conjecture. In this paper, we study the higher power moments of $D_{\rho }(x;Z_F)$, and then derive an asymptotic formula for the hth (h=3,4,5) power moments of $D₁(x;Z_F)$ by using Ivić’s large value arguments and other techniques. ...

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 $E\left(x\right)=O\left(x^{1/2}\left(logx\right)⁵\right)$. ...