Conditional bounds for the least quadratic non-residue and related problems

This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood for $L$-functions at $s=1$. In particular, we derive explicit upper and lower bounds for $L(1,\chi)$ and $\zeta(1+it)$, and deduce explicit bounds for the class number of imaginary quadratic fields. Finally, we improve the best known theoretical bounds for the least quadratic non-residue, and more generally, the least $k$-th power non-residue.Comment: We thank Emanuel Carneiro and Micah Milinovich for drawing our attention to an error in Lemma 6.1 of the previous version, which affects the asymptotic bounds in Theorems 1.2 and 1.3 there. These results are corrected in this updated versio

Squarefree smooth numbers and Euclidean prime generators

We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.Comment: 8 pages, to appear in Proceedings of the AM

Bounding zeta on the 1-line under the partial Riemann hypothesis

We provide explicit bounds in the theory of the Riemann zeta-function at the line $\Re{s}=1$, assuming that the Riemann hypothesis holds until the height $T$. In particular, we improve some bounds, in finite regions, for the logarithmic derivative and the reciprocal of the Riemann zeta-function.Comment: Typos corrected. To appear in the Bulletin of the Australian Mathematical Societ

On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

For an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes l, on average, over all good reductions of E modulo primes p. We show that, under the Generalised Riemann Hypothesis, for almost all primes p there are enough small Elkies primes l to ensure that the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1)) expected time.Comment: 20 pages, to appear in LMS J. Comput. Mat

An explicit upper bound for L(1,χ)L(1, \chi) when χ\chi is quadratic

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, \chi)|\leq \frac{1}{2}\log q$ for all $q\geq 2\cdot 10^{23}$ and $|L(1, \chi)|\leq \frac{9}{20}\log q$ for all $q\geq 5\cdot 10^{50}$.Comment: 17 page

Supersolvability and nilpotency in terms of the commuting probability and the average character degree

Let $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $|G|$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is nilpotent or supersolvable.Comment: 15 page