Related papers: Many odd zeta values are irrational

In this paper, we study lower bounds of a general family of $L$-functions on the $1$-line. More precisely, we show that for any $F(s)$ in this family, there exists arbitrary large $t$ such that $F(1+it)\geq e^{\gamma_F} (\log_2 t + \log_3…

Let $S(t)$ denote the argument of the Riemann zeta-function, defined as $$ S(t)=\dfrac{1}{\pi}\,\Im\log\zeta(1/2+it). $$ Assuming the Riemann hypothesis, we prove that $$ S(t)=\Omega_{\pm}\bigg(\dfrac{\log t\log\log\log t}{\log\log…

In the present paper, we show that under the Riemann hypothesis, and for fixed $h, \epsilon > 0$, the supremum of the real and the imaginary parts of $\log \zeta (1/2 + it)$ for $t \in [UT -h, UT + h]$ are in the interval $[(1-\epsilon)…

This note proves that the first odd zeta value does not have a closed form formula $\zeta(3)\ne r \pi^3$ for any rational number $r \in \mathbb{Q}$. Furthermore, assuming the irrationality of the second odd zeta value $\zeta(5)$, it is…

The Riemann hypothesis, stating that the real part of all non-trivial zero points fo the zeta function must be $\frac{1}{2}$, is one of the most important unproven hypothesises in number theory. In this paper we will proof the Riemann…

Let $d(n)$ be the number of divisors of $n$, let $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several…

Several identities for the Riemann zeta-function $\zeta(s)$ are proved. For example, if $s = \sigma + it$ and $\sigma > 0$, then $$ \int_{-\infty}^\infty |{(1-2^{1-s})\zeta(s)\over s}|^2dt = {\pi\over\sigma}(1 -…

Assuming the Riemann Hypothesis, we show that for $k>0$ $$ \frac{1}{T}\text{meas}\Big\{t\in [T,2T]:|\zeta(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}}, $$ where $C_k=\exp(e^{ck})$ for some absolute…

In this note, I develop step-by-step proofs of irrationality for $\,\zeta{(2)}\,$ and $\,\zeta{(3)}$. Though the proofs follow closely those based upon unit-square integrals proposed originally by Beukers, I introduce some modifications…

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…

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…

In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing \[ \left| N (T) - \frac{T}{ 2 \pi} \log \left( \frac{T}{2\pi e}\right) \right|\le 0.10076\log T+0.24460\log\log T+8.08344, \] for every $T\ge e$,…

It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant…

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit…

Under the Riemann Hypothesis, we show that as $t$ varies in $T\leq t \leq 2T$, the distribution of $\log|\zeta(1/2+it)|$ with respect to the measure $|\zeta(1/2+it)|^2dt$ is approximately normal with mean $\log\log T$ and variance…

We prove that $|\zeta(\sigma+it)|\le 70.7 |t|^{4.438 (1-\sigma)^{3/2}}\log^{2/3}|t|$ for $1/2\le\sigma\le 1$ and $|t|\ge 3$. As a consequence, we improve the explicit zero-free region for $\zeta(s)$, showing that $\zeta(\sigma+it)$ has no…

The irrationality exponent $\mu(t)$ of an irrational number t, defined using the irrationality measure $1/q^\mu$, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure $1/\beta^q$,…

In this paper we provide an explicit bound for $|\zeta(1+it)|$ in the form of $|\zeta(1+it)|\leq \min\left(\log t, \frac{1}{2}\log t+1.93, \frac{1}{5}\log t+44.02 \right)$. This improves on the current best-known explicit bound of…

We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the…

We establish an omega theorem for logarithmic derivative of the Riemann zeta function near the 1-line by resonance method. We show that the inequality $\left| \zeta^{\prime}\left(\sigma_A+it\right)/\zeta\left(\sigma_A+it\right) \right|…