arXiv subjects

C. L. Turnage-Butterbaugh

Publications and source records attributed to C. L. Turnage-Butterbaugh.

4 recordsLinked to original sources

arXiv2020

Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function

We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $ζ(s)$. In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where $S(t)$ denotes the argument of $ζ(s)$ on the critical line and $h \ll 1 / \log T$. We also use these moments to prove explicit results on the density of the nontrivial zeroes of $ζ(s)$ of a given multiplicity.

math.NT↗

arXiv2019

A note on small gaps between zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we improve on previous results by proving there are infinitely many zeros of the Riemann zeta-function whose differences are smaller than 0.50412 times the average spacing. To obtain this result, we generalize a set of weights that were developed by Xiaosheng Wu, who used them to find a positive proportion of large and small gaps between zeros of the Riemann zeta-function.

math.NT↗

arXiv2019

Extremal primes of elliptic curves without complex multiplication

Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. Assuming that all the symmetric power L-functions associated to E are automorphic and satisfy the Generalized Riemann Hypothesis, we give the first non-trivial upper bounds for the number of such primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner (arXiv:1305.5283) and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

math.NT↗

arXiv2017

On $r$-gaps between zeros of the Riemann zeta-function

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(γ_{n+r}-γ_n)/(2π/\log γ_n) > r + Θ\sqrt{r}$ and $(γ_{n+r}-γ_n)/(2π/\log γ_n) < r - \vartheta\sqrt{r}$ for explicit absolute positive constants $Θ$ and $\vartheta$, where $γ$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontrivial zeros of the Riemann zeta-function.

math.NT↗