Related papers: An error estimate for counting $S_3$-sextic number…

We show how the Selberg $\Lambda^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of…

We prove significant power savings for the error term when counting abelian extensions of number fields (as well as the twisted version of these results for nontrivial Galois modules). In some cases over $\mathbb{Q}$, these results reveal…

We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for…

We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the…

We prove that the smoothed counting function of the set of quartic fields, satisfying any finite set of local conditions, can be written as a linear combination of $X,X^{5/6}\log X,X^{5/6}$, upto an error term of $O(X^{13/16+o(1)})$. For…

This article considers the error term of the primes counting function. It applies some recent results on the densities of prime numbers in short intervals to derive an improvement of the error term from subexponential size to fractional…

We prove the existence of secondary terms of order $X^{3/4}$, with power saving error terms, in the counting functions of $|{\rm Sel}_2(E)|$, the 2-Selmer group of E, for elliptic curves E having height bounded by X. This is the first…

We combine a sieve method together with good uniformity estimates to prove a secondary term for the asymptotic estimate of $S_3\times A$ extensions over $\mathbb{Q}$ when $A$ is an odd abelian group with minimal prime divisor greater than…

We obtain an estimate for the main term of the counting function for numerical monoids.

We study Malle's conjecture for the group $C_2 \wr H$ where $H$ is a permutation group. Malle's conjecture for this case was proved by J\"urgen Kl\"uners in \cite{arXiv:1108.5597} under mild conditions for $H$. In this article, we provide…

We study the counting function of cubic function fields. Specifically, we derive an asymptotic formula for this counting function including a secondary term and an error term of order $\mathcal{O}\big(X^{2/3+\epsilon}\big)$, which matches…

This paper is a corrigendum to the article 'On the ideal theorem for number fields`. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.

We prove an asymptotic formula for the number of multi-quadratic number fields of bounded discriminant with a power-saving error term. Furthermore, we explicitly calculate the leading coefficient and extend our result to totally real…

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…

We study the scattering of a particle from a bound pair in an effective field theory using a distorted-wave renormalisation group method to find the power-counting for the three-body force terms. We find that three-body terms appear at…

This note fills a gap in the article with title above [1]. We provide the proof of Equation (82) of Lemma 5 in [1] and thereby complete its power counting analysis with a more precise next-to-leading-order estimate.

Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $|\Delta_K|\leq X$. Malle's conjecture predicts an asymptotic of the form $N_d(G,X)\sim CX^{\alpha}(\log X)^\beta$. Previously, Kl\"uners…

We prove an asymptotic formula with power saving error term for a certain triple divisor sum.

We hypothesize that the correct power counting for charmonia is in the parameter Lambda_QCD/m_c, but is not based purely on dimensional analysis (as is HQET). This power counting leads to predictions which differ from those resulting from…

We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a conjecture of Datskovsky-Wright and Roberts.…