Related papers: Strong orthogonality between the Mobius function a…

We show that the M\"{o}bius function is orthogonal to the Thue-Morse sequence $t(n)$ taken along the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ for any $1 < c < 2$. Previously this property was established for the subsequence along the…

We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power…

We prove a kind of "almost all symmetry" result for the Liouville function $\lambda(n):=(-1)^{\Omega(n)}$, giving non-trivial bounds for its "symmetry integral", say $I_{\lambda}(N,h)$ : we get $I_{\lambda}(N,h)\ll NhL^3+Nh^{21/20}$, with…

In 2014, Michal Lewicki and Andrzej Olbry\'s proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation \[ f(tx + (1-t)y) = t f(x) + (1-t) f(y),\qquad x\leq y, \] called…

The purpose of this note is to improve the current theoretical results for the correlation functions of the Mobius sequence $\{\mu(n): n\geq 1 \}$ and the Liouville sequence $\{\lambda(n): n\geq 1 \}$.

This work provides a complete characterization of congruent numbers in terms of Pythagorean triples. Specifically, we show that every congruent number can be written as $$\frac{nm\left(m-n\right)\left(m+n\right)}{\sigma^2}$$ were as…

For a nilmanifold $G/\Gamma$, a $1$-Lipschitz continuous function $F$ and the M\"obius sequence $\mu(n)$, we prove a bound on the decay of the averaged short interval correlation $$\frac1{HN}\sum_{n\leq N}\Big|\sum_{h\leq H}…

We show that the (morphic) sequence $(-1)^{s_\varphi(n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_\varphi$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum_{n<N}…

We show that Sarnak's conjecture on M\"obius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational…

We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type $e_q(f(n))$, where $f$ is a rational fraction, in the P\'olya-Vinogradov range. This applies to Kloosterman sums, and may be used to study…

Two normal functionals on a JBW$^*$-triple are known to be orthogonal if and only if they are $L$-orthogonal (meaning that they span an isometric copy of $\ell_1(2)$). This is shown to be stable under small norm perturbations in the…

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $\Omega \subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the…

This paper is aimed to prove a quantitative estimate (in terms of the modulus of continuity) for the convergence in the nonlinear version of Korovkin's theorem for sequences of weakly nonlinear and monotone operators defined on spaces of…

Let $ G $ be a connected, simply connected nilpotent Lie group and $ \Gamma < G $ a lattice. We prove that each ergodic diffeomorphism $ \phi(x\Gamma)=uA(x)\Gamma $ on the nilmanifold $ G/\Gamma $, where $ u\in G $ and $ A:G\to G $ is a…

In this paper, we primarily deal with approximately monotone and convex sequences. We start by showing that any sequence can be expressed as the difference between two nondecreasing sequences. One of these two monotone sequences act as the…

This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In…

We show that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression then for every $N\ge 2$ there exist (unilateral or bilateral) subshifts $\Sigma$ over $N$ symbols, with entropy…

The orders of magnitudes of the summatory Liouville function L(x), and the summatory Mobius function M(x), are unconditionally proven to be of the forms L(x) = O(x^.5)), and M(x) = O(x^.5) respectively. Furthermore, applications of these…

We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid…

The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be $[y',y]$-smooth if all of its prime factors belong to the interval $[y',y]$. We identify suitable weights…