## 134 Citations

On the random Chowla conjecture

- Mathematics
- 2022

We show that for a Steinhaus random multiplicative function f : N → D and any polynomial P (x) ∈ Z[x] of degP ≥ 2 which is not of the form w(x+ c) for some w ∈ Z, c ∈ Q, we have 1 √ x ∑ n≤x f(P (n))…

The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero

- MathematicsJournal of the London Mathematical Society
- 2022

Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy–Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers x, and any distinct integers…

Sums of two squares are strongly biased towards quadratic residues

- Mathematics
- 2021

Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than of the form 1 mod 4. This was conﬁrmed theoretically much later by Rubinstein…

Averages of the M\"obius function on shifted primes

- Mathematics
- 2020

It is a folklore conjecture that the M\"obius function exhibits cancellation on shifted primes; that is, $\sum_{p\le X}\mu(p+h) \ = \ o(\pi(X))$ as $X\to\infty$ for any fixed shift $h>0$. We prove…

VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES

- MathematicsForum of Mathematics, Sigma
- 2019

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is ‘approximately…

Correlation of arithmetic functions over $$\mathbb {F}_q[T]$$

- MathematicsMathematische Annalen
- 2019

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin…

THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS

- MathematicsForum of Mathematics, Pi
- 2016

Let $\unicode[STIX]{x1D706}$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$\begin{eqnarray}\mathop{\sum }_{n\leqslant…

An averaged form of Chowla's conjecture

- Mathematics
- 2015

Let $\lambda$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm…

A dynamical point of view on the set of B-free integers

- Mathematics
- 2013

We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise…

On the Hardy–Littlewood–Chowla conjecture on average

- MathematicsForum of Mathematics, Sigma
- 2022

Abstract There has been recent interest in a hybrid form of the celebrated conjectures of Hardy–Littlewood and of Chowla. We prove that for any $k,\ell \ge 1$ and distinct integers $h_2,\ldots…