arXiv subjects

S. J. Lester

Publications and source records attributed to S. J. Lester.

4 recordsLinked to original sources

arXiv2014

$a$-Points of the Riemann zeta-function on the critical line

We investigate the proportion of the nontrivial roots of the equation $ζ(s)=a$, which lie on the line $\Re s=1/2$ for $a \in \mathbb C$ not equal to zero. We show that at most one-half of these points lie on the line $\Re s=1/2$. Moreover, assuming a spacing condition on the ordinates of zeros of the Riemann zeta-function, we prove that zero percent of the nontrivial solutions to $ζ(s)=a$ lie on the line $\Re s=1/2$ for any nonzero complex number $a$.

math.NT↗

arXiv2013

On the distribution of the zeros of the derivative of the Riemann zeta-function

We establish an unconditional asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For $\Re(s)=σ$ satisfying $(\log T)^{-1/3+ε} \leq (2σ-1) \leq (\log \log T)^{-2}$, we show that the number of zeros of $ζ'(s)$ with imaginary part between zero and $T$ and real part larger than $σ$ is asymptotic to $T/(2π(σ-1/2))$ as $T \rightarrow \infty$. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for $σ$ in this range the zeros of $ζ'(s)$ are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

math.NT↗

arXiv2013

The distribution of the logarithmic derivative of the Riemann zeta-function

We investigate the distribution of the logarithmic derivative of the Riemann zeta-function on the line Re(s)=σ, where σ, lies in a certain range near the critical line σ=1/2. For such σ, we show that the distribution of ζ'/ζ(s) converges to a two-dimensional Gaussian distribution in the complex plane. Upper bounds on the rate of convergence to the Gaussian distribution are also obtained.

math.NT↗

arXiv2013

On Balazard, Saias, and Yor's equivalence to the Riemann Hypothesis

Balazard, Saias, and Yor proved that the Riemann Hypothesis is equivalent to a certain weighted integral of the logarithm of the Riemann zeta-function along the critical line equaling zero. Assuming the Riemann Hypothesis, we investigate the rate at which a truncated version of this integral tends to zero, answering a question of Borwein, Bradley, and Crandall and disproving a conjecture of the same authors. A simple modification of our techniques gives a new proof of a classical Omega theorem for the function S(t) in the theory of the Riemann zeta-function.

math.NT↗