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Abstract Let$$\lambda $$ $\lambda$denote the Liouville function. We show that the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\lambda (\lfloor \alpha _2n\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\lambda \left(\lfloor {\alpha }_{2}n\rfloor \right)$is 0 whenever$$\alpha _1,\alpha _2$$ ${\alpha }_1,{\alpha }_2$are positive reals with$$\alpha _1/\alpha _2$$ ${\alpha }_1/{\alpha }_2$irrational. We also show that for$$k\geqslant 3$$ $k⩾3$the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\cdots \lambda (\lfloor \alpha _kn\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\cdots \lambda \left(\lfloor {\alpha }_{k}n\rfloor \right)$has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers$$\alpha _i.$$ ${\alpha }_i.$Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crnčević–Hernández–Rizk–Sereesuchart–Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets. 

Abstract A conjecture of Erdős states that, for any large primeq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integerq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be written as $p_1{p}_2{p}_3${p_{1}p_{2}p_{3}}with $p_1,p_2,p_3\le q${p_{1},p_{2},p_{3}\leq q}primes. We also show that, for any $\epsilon >0${\varepsilon>0}and any sufficiently large integerq, at least $\left(\frac{2}{3}-\epsilon \right)\phi \left(q\right)${(\frac{2}{3}-\varepsilon)\varphi(q)}reduced residue classes $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}.The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ${\mathbb{Z}}_q^{\times }${\mathbb{Z}_{q}^{\times}}of small index and study in detail the exceptional case that there exists a quadratic character $\psi \left(\mathrm{mod}q\right)${\psi~{}(\operatorname{mod}\,q)}such that $\psi \left(p\right)=-1${\psi(p)=-1}for very many primes $p\le q${p\leq q}. 

We show that, for almost all x x , the interval ( x , x + ( log ⁡ x ) 2.1 ] (x, x+(\log x)^{2.1}] contains products of exactly two primes. This improves on a work of the second author that had 3.51 3.51 in place of 2.1 2.1 . To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown’s mean value theorem for sparse Dirichlet polynomials. 

Abstract We study for bounded multiplicative functions sums of the formestablishing that their variance over residue classes is small as soon as , for almost all moduli , with a nearly power‐saving exceptional set of . This improves and generalizes previous results of Hooley on Barban–Davenport–Halberstam type theorems for such , and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well‐known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli in the cases where is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every . These results are special cases of a “hybrid result” that we establish that works for sums of over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki–Radziwiłł theorem on multiplicative functions in short intervals. We also consider the maximal deviation of overallresidue classes in the square root range , and show that it is small for “smooth‐supported” , again apart from a nearly power‐saving set of exceptional , thus providing a smaller exceptional set than what follows from Bombieri–Vinogradov type theorems. As an application of our methods, we consider Linnik‐type problems for products of exactly three primes, and in particular prove a ternary approximation to a conjecture of Erdős on representing every element of the multiplicative group as the product of two primes less than . 

Abstract We show that for all $$n\leq X$$ apart from $$O(X\exp (-c(\log X)^{1/2}(\log \log X)^{1/2}))$$ exceptions, the alternating group $$A_{n}$$ is invariably generated by two elements of prime order. This answers (in a quantitative form) a question of Guralnick, Shareshian, and Woodroofe.