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

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

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.

Let $(𝒪,\Sigma ,F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, $𝒮=\mathrm{Spec}𝒪$ and $(\pi :𝒳\to 𝒮;{\sigma }_1,...,{\sigma }_n)$ an $n$-pointed stable curve of genus $g$. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(𝒮\right)$. The invertible sheaf ${\omega }_{𝒳/𝒮}({\sigma }_1+\cdots +{\sigma }_n)$ inherits a hermitian structure ${\parallel ·\parallel }_{\mathrm{hyp}}$ from the dual of the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ${\omega }_{𝒳/𝒮}{({\sigma }_1+...+{\sigma }_n)}_{\mathrm{hyp}}$. The theorem is applied to modular curves $X\left(\Gamma \right)$, $\Gamma ={\Gamma }_0\left(p\right)$ or ${\Gamma }_1\left(p\right)$, $p\ge 11$ prime, with sections given by the cusps. We show $Z^{\text{'}}(Y\left(\Gamma \right),1)\sim e^a{\pi }^b{\Gamma }_2{(1/2)}^{c}L(0,{ℳ}_{\Gamma })$, with $p\equiv 11mod12$ when $\Gamma ={\Gamma }_0\left(p\right)$. Here $Z\left(Y\right(\Gamma ),s)$ is the Selberg...

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...

If $(M,\nabla )$ is a manifold with a symmetric linear connection, then $T^{*}M$ can be endowed with the natural Riemann extension $\overline{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\overline{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*}M,\overline{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\overline{g}$ reduces...

Let X be a Riemann domain over ${ℂ}^k\times {ℂ}^{\ell }$. If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set $P\subset {ℂ}^k$ such that every slice $X_a$ of X with a∉ P is a region of holomorphy with respect to the family ${f|}_{X_a}:f\in ℱ$.