In the early 2000s, Ramakrishna asked the question: for the elliptic curve

\[E\;:\; y^2 = x^3 - x,\]

$a_p(E)$

$E/\mathbb{Q}$

Brun’s constant is the summation of the reciprocals of all twin primes, given by

$$ \begin{align*}B=\sum_{p \in P_2}{\bigg( \frac{1}{p} + \frac{1}{p+2}\bigg)}.\end{align*} $$

While rigorous unconditional bounds on B are known, we present the first rigorous bound on Brun’s constant under the assumption of GRH, yielding $B < 2.1594$.

Weighted sieves are used to detect numbers with at most S prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes p such that $p+2$ has at most two prime factors), one can utilise the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes.

As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q, any two reduced congruence classes $a_1$ and $a_2$ mod q, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod q, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod q. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any $r \ge 1$, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q.

Let $k{\geqslant} 2$ be an integer and let $\lambda$ be the Liouville function. Given k non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)=o(x)$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach. More precisely, we establish a non-trivial bound for the sums $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)$ under the existence of a Landau–Siegel zero for x in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau–Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis and Tao and Teräväinen.

A square-full number is a positive integer for which all prime divisors divide the number at least twice. For a relatively prime pair $(a,b)\in \mathbb N\times \mathbb N\cup \{0\}$ and an affine polynomial $f(x)=ax+b$, the number of $n\leqslant N$ such that $f(n)$ is square-full is of order $N^{{1}/{2}}$. For $\varepsilon>0$ and an admissible quadratic polynomial $f(x)\in \mathbb Z[x]$, we show that the number of $n\leqslant N$ such that $f(n)$ is square-full is at most $O_{\varepsilon ,f}(N^{\varpi +\varepsilon })$ for some absolute constant $\varpi <1/2$. Under the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon ,f}(N^\varepsilon )$.

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ‘arithmetic Bertini theorem’ conjectured by Poonen for ${\mathbb {P}}^1_{\mathbb {Z}}$.

Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number fields of degree n having associated Galois group $S_n$ and absolute discriminant less than X, improving the best previously known lower bound of $\gg X^{1/2+1/n}$.

Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified $A_n$-extensions of quadratic number fields of bounded discriminant.

The aim of this work is to prove a new sure upper bound in a setting that can be thought of as a simplified function field analogue. This result is comparable to a recent result of the author concerning an almost sure upper bound of random multiplicative functions. Having a simpler quantity allows us to make the proof more accessible.

Étant donnée une suite $A = (a_n)_{n\geqslant 0}$ d’entiers naturels tous au moins égaux à 2, on pose $q_0 = 1$ et, pour tout entier naturel n, $q_{n+1} = a_n q_n$. Tout nombre entier naturel $n\geqslant 1$ admet une unique représentation dans la base A, dite de Cantor, de la forme

$$ \begin{align*} m = \sum_{j\geqslant 0}\varepsilon_j(m)q_j\quad \text{avec} \quad 0\leqslant \varepsilon_j(m) < a_j = \frac{q_{j+1}}{q_j}. \end{align*} $$

$$ \begin{align*} S = \sum_{n \leqslant x}\Lambda(n) f(n) \end{align*} $$

$\Lambda $

$(a_n)_{n\geqslant 0}$

Nous étudions plusieurs exemples dans la base $A = (j+2)_{j\geqslant 0}$, dite factorielle. En particulier, si $s_A$ désigne la fonction somme de chiffres dans cette base et p parcourt la suite des nombres premiers, nous montrons que la suite $(s_A(p))_{p\in \mathcal {P}}$ est bien répartie dans les progressions arithmétiques, et que la suite $(\alpha s_A(p))_{p\in \mathcal {P}}$ est équirépartie modulo $1$ pour tout nombre irrationnel $\alpha $.

Bergelson and Richter [‘Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions’, Duke Math. J. 171(15) (2022), 3133–3200] established a new dynamical generalisation of the prime number theorem (PNT) and the PNT for arithmetic progressions. Let $h\ge 1, k\ge 2$. Mirsky [‘Note on an asymptotic formula connected with r-free integers’, Quart. J. Math. Oxford Ser. 18 (1947), 178–182] showed that the numbers n such that $n+l_1,\ldots , n+l_h$ are k-free have a natural density for any given nonnegative integers $l_1,\ldots , l_h$. In this note, we show that the Bergelson–Richter theorem holds for the numbers studied by Mirsky.

In this article, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup G contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is n less than a kth power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress toward a conjecture of Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes p, the set $\{x^2-1: x \in {\mathbb F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in ${\mathbb F}_p$ non-trivially.

We investigate the sums $(1/\sqrt {H}) \sum _{X < n \leq X+H} \chi (n)$, where $\chi $ is a fixed non-principal Dirichlet character modulo a prime q, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty $ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty $, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $\chi $, even when $H = q/\log ^{A}q$ for any fixed $A> 0$. However, we show it is true for ‘almost all’ characters on the range $q^{1-o(1)} \leq H = o(q)$.

Using Pólya’s Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs (i.e., $x,y\in {\mathbb N}$ such that $x^2\pm y^2=z^2$ for some $z\in {\mathbb N}$). We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi–Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the ‘correct’ moduli q that vary uniformly in a similar range and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the ‘incorrect’ moduli.

We prove structural results for measure-preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions, these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function, all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwiłł, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that a trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.

We show that there exists some $\delta > 0$ such that, for any set of integers B with $|B\cap[1,Y]|\gg Y^{1-\delta}$ for all $Y \gg 1$, there are infinitely many primes of the form $a^2+b^2$ with $b\in B$. We prove a quasi-explicit formula for the number of primes of the form $a^2+b^2 \leq X$ with $b \in B$ for any $|B|=X^{1/2-\delta}$ with $\delta < 1/10$ and $B \subseteq [\eta X^{1/2},(1-\eta)X^{1/2}] \cap {\mathbb{Z}}$, in terms of zeros of Hecke L-functions on ${\mathbb{Q}}(i)$. We obtain the expected asymptotic formula for the number of such primes provided that the set B does not have a large subset which consists of multiples of a fixed large integer. In particular, we get an asymptotic formula if B is a sparse subset of primes. For an arbitrary B we obtain a lower bound for the number of primes with a weaker range for $\delta$, by bounding the contribution from potential exceptional characters.

In this article, we establish a function field analog of Jacobi’s theorem on sums of squares and analyze its moments. Our approach involves employing two distinct techniques to derive the main results concerning asymptotic formulas for the moments. The first technique utilizes Dirichlet series framework to derive asymptotic formulas in the limit of large finite fields, specifically when the characteristic of $\mathbb {F}_q[T]$ becomes large. The second technique involves effectively partitioning the set of polynomials of a fixed degree, providing asymptotic formulas in the limit of large polynomial degree.

We develop a generalisation of the square sieve of Heath-Brown and use it to give an alternate proof of one of the large sieve inequalities in our previous paper [‘A large sieve inequality for characters to quadratic moduli’, Preprint, https://web.maths.unsw.edu.au/~ccorrigan/preprint6.pdf].

Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and Shusterman (2022). These results include bounds for exponential sums weighted by the Möbius function and a level of distribution for irreducible polynomials beyond 1/2, with arbitrary composite modulus. Additionally, we can do better when averaging over the modulus, to give an analogue of the Bombieri-Vinogradov Theorem with a level of distribution even further beyond 1/2.