T. Trudgian

activity

20082023

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

20 citations · 104 across the 42 of their papers we have counts for

collaborators

Showing 2019Show all

7 papers · 1 filter

math.NT2019

The distribution of $k$ -free numbers

Michael J. Mossinghoff, Tomás Oliveira e Silva, Tim Trudgian

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

math.NT2019★ 5 cited

Two explicit divisor sums

Michaela Cully-Hugill, Timothy Trudgian

We give explicit bounds on sums of $d (n)^{2}$ and $d_{4} (n)$ , where $d (n)$ is the number of divisors of $n$ and $d_{4} (n)$ is the number of ways of writing $n$ as a product of four numbe…

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.NT2019

The least primitive root modulo $p^{2}$

Bryce Kerr, Kevin McGown, Tim Trudgian

We provide an explicit estimate on the least primitive root mod $p^{2}$ . We show, in particular, that every prime $p$ has a primitive root mod $p^{2}$ that is less than $p^{0.99}$ .

math.NT2019

A tale of two omegas

Michael J. Mossinghoff, Timothy S. Trudgian

We consider $ω (n)$ and $Ω (n)$ , which respectively count the number of distinct and total prime factors of $n$ . We survey a number of similarities and differences between these two…

math.NT2019

Explicit upper bounds on the least primitive root

Kevin J. McGown, Tim Trudgian

We give a method for producing explicit bounds on $g (p)$ , the least primitive root modulo $p$ . Using our method we show that $g (p) < 2 r 2^{r ω (p - 1)} p^{\frac{1}{4} + \frac{1}{4 r}}$ fo…

The distribution of k-free numbers

Two explicit divisor sums

Sign changes in the prime number theorem

The least primitive root modulo p2

A tale of two omegas

Explicit upper bounds on the least primitive root

The distribution of $k$ -free numbers

The least primitive root modulo $p^{2}$