Related papers: On the lowest zero of Dedekind zeta function

Let $S(\sigma,t)=\frac{1}{\pi}\arg\zeta(\sigma+it)$ be the argument of the Riemann zeta-function at the point $\sigma+it$ in the critical strip. For $n\geq 1$ and $t>0$, we define \begin{equation*} S_{n}(\sigma,t) = \int_0^t…

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…

Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/\zeta(k)$. It is well known that $R_k(x)=\Omega(x^{\frac{1}{2k}})$, and widely conjectured that…

In this paper, we derive new lower bounds for the normalized distances between consecutive maxima of the Riemann zeta-function on the critical line subject to the truth of the Riemann hypothesis. The method of our proofs relies on a Sobolev…

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…

The aim of this work is to improve some elementary results regarding both the Deuring-Phenomenon and the Heilbronn-Phenomenon. We will give better estimates regarding both the influence of zeros of the Riemann zeta function on the…

We consider the real part $\Re(\zeta(s))$ of the Riemann zeta-function $\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $\sigma_0 = 1.19\ldots$ which is defined to be the supremum of $\sigma$ such…

We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute…

The greatest lower bound of the real parts of the roots of a partial sum of the Dirichlet series of Riemann's zeta function is asymptotically equivalent to the opposite of the number of terms of this sum, multiplied by the Napierian…

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 paper, we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds…

In this paper, we will employ the Opial and Wirtinger type inequalities to derive some conditional and unconditional lower bounds for the gaps between the zeros of the Riemann zeta-function. First, we prove (unconditionally) that the…

This paper studies a zeta function of two complex variables (w, s) attached to an algebraic number field K, introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov…

The Dirichlet eta function can be divided into $n$-th partial sum $\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term which can be approximated by the expression for $n$. And then, to increase reliability, we make…

The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from…

This paper is a summary of the general approach outlined in my previous papers toward proving the riemann hypothesis. Numerical and graphical proof of the Riemann Hypothesis is presented with analytical arguments although more work needs…

Hypothesis of Riemann is rejected by definition, because {\zeta}(s), where s zeros of {\zeta}(s)=0, is not be equal by definition to the particular sum, which it assumes to be equal. R(s) = 1/2 holds only for the zeros of {\zeta}(s) = 0 and…

Adapting a technique of Pintz, we give an elementary demonstration of the Deuring phenomenon: a zero of \zeta(s) off the critical line gives a lower bound on L(1,\chi). The necessary tools are Dirichlet's 'method of the hyperbola', Euler…

Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been…

For $0 < a \le 1/2$, we define the quadrilateral zeta function $Q(s,a)$ using the Hurwitz and periodic zeta functions and show that $Q(s,a)$ satisfies Riemann's functional equation studied by Hamburger, Heck and Knopp. Moreover, we prove…