Related papers: A modest improvement on the function $S(T)$

Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x)…

In this article, we prove an explicit bound for $N(\sigma,T)$, the number of zeros of the Riemann zeta function satisfying $\sigma < \Re s <1 $ and $0 < \Im s < T$. This result provides a significant improvement over Rosser's bound for…

In this article, we give explicit bounds of order $\log t$ for $\sigma$ close to $1$, for two quantities: $|\zeta'(\sigma +it)/\zeta(\sigma +it)|$ and $|1/\zeta(\sigma +it)|$. We correct an error in the literature, and especially in the…

We find an asymptotic expansion of Selberg's central limit theorem for the Riemann zeta function on $\sigma = \frac12 + ( \log T)^{-\theta}$ and $t \in [T, 2T]$, where $ 0 < \theta < \frac12$ is a constant.

We prove a lower bound on the supremum of the function $S_1(T)$ on short intervals, defined by the integration of the argument of the Riemann zeta-function. The same type of result on the supremum of $S(T)$ have already been obtained by…

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for…

We prove an explicit upper bound of the function $S(t,\chi)$, defined by the argument of Dirichlet $L$-functions. An explicit upper bound of the function $S_1(t)$, defined by the integral of the argument of the Riemann zeta-function, have…

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…

We consider the distribution of $\arg\zeta(\sigma+it)$ on fixed lines $\sigma > \frac12$, and in particular the density \[d(\sigma) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\arg\zeta(\sigma+it)| > \pi/2\}|\,,\] and the…

We prove a totally explicit bound for short sums of certain non-negative arithmetic functions satisfying a general growth condition, and apply this result to derive two explicit estimates for the Erd\H{o}s-Hooley $\Delta$-function in short…

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Re s| \leq K$, where…

Let $S(t) = \frac{1}{\pi}\Im \log\zeta\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log…

We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the…

We prove several results on the distribution function of $\zeta(1+it)$ in the complex plane, that is the joint distribution function of $\arg\zeta(1+it)$ and $|\zeta(1+it)|$. Similar results are also given for $L(1,\chi)$ (as $\chi$ varies…

For a certain function $J(s)$ we prove that the identity $$\frac{\zeta(2s)}{\zeta(s)}-\left(s-\frac{1}{2}\right)J(s)=\frac{\zeta(2s+1)}{\zeta(s+1/2)}, $$ holds in the half-plane Re$(s)>1/2$ and both sides of the equality are analytic in…

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…

Given any complex number $a$, we prove that there are infinitely many simple roots of the equation $\zeta(s)=a$ with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the…

We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and…

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short…