Related papers: Lehmer pairs and binomial series

In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two…

This paper describes a practical methodology for computing the Hardy function Z(t), using just O(((t/epsilon)^(1/3))*(log(t))^(2+o(1)))) standard computational operations, to a tolerance of epsilon in the relative error. The methodology is…

In this paper we perform a detailed analysis of Riemann's hypothesis, dealing with the zeros of the analytically-extended zeta function. We use the functional equation $\zeta(s) = 2^{s}\pi^{s-1}\sin{(\displaystyle \pi…

In recent work, Fyodorov and Keating conjectured the maximum size of $|\zeta(1/2+it)|$ in a typical interval of length O(1) on the critical line. They did this by modelling the zeta function by the characteristic polynomial of a random…

Various properties of the Mellin transform function $$ {\cal M}_k(s) := \int_1^\infty Z^k(x)x^{-s}dx $$ are investigated, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is Hardy's…

Hardy's theorem for the Riemann zeta-function $\zeta(s)$ says that it admits infinitely many complex zeros on the line $\Re({s}) = \frac{1}{2}$. In this note, we give a simple proof of this statement which, to the best of our knowledge, is…

The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $\log{\zeta(s)}$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the…

Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|\zeta(1/2+it)|$ on the critical line, and zeros…

An explicit subconvex bound for the Riemann zeta function $\zeta(s)$ on the critical line $s=1/2+it$ is proved. Previous subconvex bounds relied on an incorrect version of the Kusmin-Landau lemma. After accounting for the needed correction…

A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its…

Let $\Delta(x)$ denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - \frac{1}{2}\Delta(4x)$. We show that $$…

Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not…

For $0<\alpha, \lambda \leq 1$, the Lerch zeta-function is defined by $L(s;\alpha, \lambda)$$:= \sum_{n=0}^\infty e^{2\pi i\lambda n} (n+\alpha)^{-s}$, where $\sigma>1$. In this paper, we prove joint universality for Lerch zeta-functions…

Assuming the Riemann Hypothesis we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $\zeta(s)$. For example, integrating $|\zeta(1/2+\alpha+it)|^{-2k}$ with respect to $t$…

Using the $\zeta$ functional equation and the Hadamard product, an analytical expression for the sum of the reciprocal of the $\zeta$ zeros is established. We then demonstrate that on the critical line, $|\zeta|$ is convex, and that in the…

We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $,…

An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is…

The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its…

The $2$kth pseudomoments of the Riemann zeta function $\zeta(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $\zeta(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like…

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…