Related papers: Bounding the log-derivative of the zeta-function

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 show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta…

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…

We prove Riemann hypothesis. Method is to show the convexity of function which has zeros on open critical strip the same as zeta function.

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…

In our previous work (https://doi.org/10.1002/mana.202000268, Math. Nachr., 2021), we proposed an upper bound of the logarithmic derivative of Selberg's zeta function for the modular groups in the critical strip. The present paper studies…

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann's $\xi(s)$ function, and thus $\zeta(s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the…

Make an exponential transformation in the integral formulation of Riemann's zeta-function zeta(s) for Re(s) > 0. Separately, in addition make the substitution s -> 1 - s and then transform back to s again using the functional equation.…

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods…

We estimate large and small values of $|\zeta(\rho')|$, where $\rho'$ runs over critical points of the zeta function in the right half of the critical strip, that is, the points where $\zeta'(\rho')=0$ and $1/2<\Re \rho'<1$.

We investigate the distribution of the logarithmic derivative of the Riemann zeta-function on the line Re(s)=\sigma, where \sigma, lies in a certain range near the critical line \sigma=1/2. For such \sigma, we show that the distribution of…

We use expansions with functions related to some special functions such as Hermite or Laguerre to get some conjectural expansions of the Riemann Zeta function in the critical strip involving a set of polynomials which have their zeros on…

We provide explicit upper bounds of the order $\log t/\log\log t$ for $|\zeta'(s)/\zeta(s)|$ and $|1/\zeta(s)|$ when $\sigma$ is close to $1$. These improve existing bounds for $\zeta(s)$ on the $1$-line.

In the paper the well known Riemann Hypothesis is proven. The proof is based on uniform approximation of the zeta function discs of the critical strip placed to the right from the critical line.The basic moment is a use of a new mesure…

We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta-function averaged over a subfamily of zeros of the zeta function which is expected to have full density inside the set of all zeros. For…

Assuming the Riemann hypothesis, we establish an upper bound for the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where $k$ is a positive real number. Our upper bound agrees with conjectures of…

Building on work in \cite{AB24} on the Riemann zeta function at height $T$ off the critical line, we prove an unconditional lower bound on the critical line for real large deviations of the order $V\sim\alpha\log\log T$ for any $\alpha>0.$…

In this work, we investigate the positivity of the real part of the log-derivative of the Riemann $\xi$-function in the region $1/2+1/\sqrt{\log t}<\sigma<1$, where $t$ is sufficiently large. We provide an explicit lower bound for…

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.