相关论文: On the Riemann zeta-function and the divisor probl…

We combine our version of the resonance method with certain convolution formulas for $\zeta(s)$ and $\log\, \zeta(s)$. This leads to a new $\Omega$ result for $|\zeta(1/2+it)|$: The maximum of $|\zeta(1/2+it)|$ on the interval $1 \le t \le…

Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\sum_{n\leq x}d^2(n)=xP(\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^{{3\over 5}+\epsilon}), $$ where $\epsilon$ is a…

For a fixed irrational $\theta > 0$ with a prescribed irrationality measure function, we study the correlation $\int_1^X \Delta(x) \Delta(\theta x) dx$, where $\Delta$ is the Dirichlet error term in the divisor problem. When $\theta$ has a…

Let Zn denote the additive group of residue classes modulo n. Let c(l,m,n) denote the number of cyclic subgroups of Zl *Zm *Zn. For any x > 1, we consider the asymptotic behavior of D3c(x):= \sum_{lmn\leq x} c(l,m,n), obtain an asymptotic…

We quantify the set of known exponent pairs $(k, \ell)$ and develop a framework to compute the optimal exponent pair for an arbitrary objective function. Applying this methodology, we make progress on several open problems, including bounds…

In this paper, we present an improved explicit subconvexity result for the Riemann zeta function $\zeta\left( s\right)$ along the critical line $s=1/2+it$, given by Hiary, Patel and Yang in 2024. This new bound is derived by combining a…

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic…

Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the…

Let $T>0$ and consider the random Dirichlet polynomial $S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$, where $(X_n)_{n}$ are i.i.d. Gaussian random variables with mean $0$ and variance $1$. We prove that the expected number of roots of…

Let $\tau(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} \tau\left( \left[ \frac{x}{n_1 n_2} \right] \right) =…

We establish a smoothed asymptotic formula for the third moment of quadratic {D}irichlet $L$-functions at the central value. In addition to the main term, which is known, we prove the existence of a secondary term of size $x^{\frac{3}{4}}$.…

Let $d(n)$ denote the Dirichlet divisor function. Define \begin{equation*} \mathcal{S}_{k}(x)=\sum_{\substack{1\leqslant n_1,n_2,n_3 \leqslant x^{1/2} \\ 1\leqslant n_4\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k), \qquad 3\leqslant k\in…

In this paper we prove that the Dirichlet $L$-functions $L(1/2+ix,\chi_q)$, where $\chi_q$ is uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converges to a random Schwartz distribution $\zeta_{\mathrm{rand}}$, which…

Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^\theta$ with fixed $0<\theta \le 1$, $$ \int_T^{T+H}|\zeta(1/2+it)|^{2k} dt \ll H(\log T)^{k^2(1+O(1/\log_3T))}, $$ where $\log_jT = \log(\log_{j-1}T)$. The proof…

We obtain a new upper bound for $\sum_{h\le H}\Delta_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $\Delta_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the…

We establish an unconditional asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For $\Re(s)=\sigma$ satisfying $(\log T)^{-1/3+\epsilon} \leq (2\sigma-1) \leq (\log \log…

Let R(t) be the remainder term in Weyl's law for a 3-dimensional Riemannian Heisenberg manifold with a certain arithmetic metric. We prove a third moment result stating that \int_1^T R(t)^3 dt =d_3 T^(13/4)+O_\delta(T^(45/14+\delta)), where…

Let $F({\bf x})={\bf x}^tQ_{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge…

We introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share…

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x +…