Dave Platt

activity

20122020

most citedAccurate estimation of sums over zeros of the Riemann zeta-function

20 citations · 52 across the 14 of their papers we have counts for

collaborators

19 papers

math.NT2020★ 20 cited

Accurate estimation of sums over zeros of the Riemann zeta-function

Richard P. Brent, David J. Platt, Timothy S. Trudgian

We consider sums of the form $\sum ϕ (γ)$ , where $ϕ$ is a given function, and $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We…

math.NT2020★ 2 cited

A harmonic sum over nontrivial zeros of the Riemann zeta-function

Richard P. Brent, David J. Platt, Timothy S. Trudgian

We consider the sum $\sum 1/ γ$ , where $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0, T]$ , and consider the behaviour of the sum a…

math.NT2020

The mean square of the error term in the prime number theorem

Richard P. Brent, David J. Platt, Timothy S. Trudgian

We show that, on the Riemann hypothesis, $lim sup_{X \to \infty} I (X) / X^{2} \leq 0.8603$ , where $I (X) = \int_{X}^{2 X} (ψ (x) - x)^{2} d x .$ This proves (and improves on) a claim by Pintz fro…

math.NT2020★ 1 cited

The Riemann hypothesis is true up to $3 \cdot 1 0^{12}$

Dave Platt, Tim Trudgian

We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3 \cdot 1 0^{12}$ . That is, all zeroes $β + iγ$ of the Riemann zet…

math.NT2019

Sign changes in the prime number theorem

Thomas Morrill, Dave Platt, Tim Trudgian

Let $V (T)$ denote the number of sign changes in $ψ (x) - x$ for $x \in [1, T]$ . We show that $lim inf_{T \to \infty} V (T) / lo g T \geq γ_{1} / π + 1.867 \cdot 1 0^{- 30}$ , where $γ_{…

math.NT2018★ 3 cited

Robin's inequality for 20-free integers

Thomas Morrill, David Platt

In 1984, Robin showed that the Riemann Hypothesis for $ζ$ is equivalent to demonstrating $σ (n) < e^{γ} n lo g lo g n$ for all $n > 5040$ . Robin's inequality has since been proven for…

Accurate estimation of sums over zeros of the Riemann zeta-function

A harmonic sum over nontrivial zeros of the Riemann zeta-function

The mean square of the error term in the prime number theorem

The Riemann hypothesis is true up to 3⋅1012

Sign changes in the prime number theorem

Robin's inequality for 20-free integers

The Riemann hypothesis is true up to $3 \cdot 1 0^{12}$