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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 $h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}\Lambda(n+h_1)\cdots \Lambda(n+h_k)\lambda(n+h_{1}')\cdots \lambda(n+h_{\ell}')$$ for any $0\leq k\leq 2$ and $\ell \geq 0$. Specializing to either $\ell=0$ or $k=0$, we deduce the previously known results on the Hardy--Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to Heath-Brown and Chinis, respectively. The range of validity of our asymptotic formula is wider than in these previous results., Comment: 54 pages, no figures. To appear in J. London Math. Soc

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} \lambda(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 < \theta < 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) \Gamma)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$\int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $\lambda$ over short polynomial progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function., Comment: 107 pages; to appear in Ann. of Math

We show that almost all sectors of the disc $\{z \in \mathbb{C}: |z|^2\leq X\}$ of area $(\log X)^{15.1}$ contain products of exactly two Gaussian primes, and that almost all sectors of area $(\log X)^{1 + \varepsilon}$ contain products of exactly three Gaussian primes. The argument is based on mean value theorems, large value estimates and pointwise bounds for Hecke character sums., Comment: 54 pages

Let $\lambda$ denote the Liouville function. We show that the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\lambda(\lfloor \alpha_2n\rfloor)$ is $0$ whenever $\alpha_1,\alpha_2$ are positive reals with $\alpha_1/\alpha_2$ irrational. We also show that for $k\geq 3$ the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\cdots \lambda(\lfloor \alpha_kn\rfloor)$ has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers $\alpha_i$. 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 Crn\v{c}evi\'c--Hern\'andez--Rizk--Sereesuchart--Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets.

A conjecture of Erd\H{o}s states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer $q$, every reduced residue class $\pmod q$ can be written as $p_1p_2p_3$ with $p_1,p_2,p_3\leq q$ primes. We also show that, for any $\varepsilon > 0$ and any sufficiently large integer $q$, at least $(2/3-\varepsilon)\varphi(q)$ reduced residue classes $\pmod q$ can be represented as a product $p_1 p_2$ of two primes $p_1, p_2 \leq q$. 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 use bounds for the logarithmic density of primes in certain unions of cosets of subgroups of $\mathbb{Z}_q^\times$ of small index and study in detail the exceptional case that there exists a quadratic character $\psi \pmod{q}$ such that $\psi(p) = -1$ for almost all primes $p \leq q$., Comment: 45 pages; added Theorem 1.3(ii)

We develop an approach to study character sums, weighted by a multiplicative function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)\chi(G)\xi(G), \end{equation} where $\chi$ is a Dirichlet character and $\xi$ is a short interval character over $\mathbb{F}_q[t].$ We then deduce versions of the Matom\"aki-Radziwill theorem and Tao's two-point logarithmic Elliott conjecture over function fields $\mathbb{F}_q[t]$, where $q$ is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the M\"{o}bius function for various values of $q$. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that $q$ is a power of $2$. As an application of our results, we give a short proof of the function field version of a conjecture of K\'atai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a "corrected" form of the Erd\H{o}s discrepancy problem over $\mathbb{F}_q[t]$., Comment: 62 pages; further referee comments incorporated; to appear in Mathematika

Let $f:\mathbb{N}\to \mathbb{D}$ be a multiplicative function. Under the merely necessary assumption that $f$ is non-pretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts $h_1,h_2$ the two-point correlation $$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\overline{f}(n+h_2)}$$ tends to $0$ along a set of $x\in\mathbb{N}$ of full upper logarithmic density. We also show that the same result holds for the $k$-point correlations $$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\cdots f(n+h_k)}$$ if $k$ is odd and $f$ is a real-valued non-pretentious function. Previously, the vanishing of correlations was known only under stronger non-pretentiousness hypotheses on $f$ by the works of Tao, and Tao and the third author. We derive several applications, including: (i) A classification of $\pm 1$-valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. (ii) A proof that a class of "Liouville-like" functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. (iii) Constructing examples of multiplicative $f:\mathbb{N}\to \{-1,0,1\}$ with a given (unique) Furstenberg system, answering a question of Lema\'nczyk. (iv) A density version of the Erd\H{o}s discrepancy theorem of Tao., Comment: 55 pages; small edits

With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree d polynomials f that are ordered by height. This allows us to establish averaged versions of the Bateman-Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of f, both with arbitrary logarithmic power savings., 68 pages

We characterize the limiting behavior of partial sums of multiplicative functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals. Concerning the notion of short interval discrepancy, we show that a completely multiplicative $f:\mathbb{F}_q[t]\to\{-1,+1\}$ with $q$ odd has bounded short interval sums if and only if $f$ coincides with a "modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over $\mathbb{Z}$ that such modified characters are extremal with respect to the growth rate of partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of $\mathbb{F}_{q}[t]$. This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erd\H{o}s discrepancy problem admits infinitely many completely multiplicative counterexamples on $\mathbb{F}_q[t]$. Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers., Comment: 38 pages

We study higher uniformity properties of the M\"obius function $\mu$, the von Mangoldt function $\Lambda$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$. More precisely, letting $\Lambda^\sharp$ and $d_k^\sharp$ be suitable approximants of $\Lambda$ and $d_k$ and $\mu^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)\Gamma)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \] when $\theta = 5/8$ and $f \in \{\Lambda, \mu, d_k\}$ or $\theta = 1/3$ and $f = d_2$. As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,\theta$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in $L^2$. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums., Comment: 102 pages; referee comments incorporated

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., Comment: 11 pages; Referee comments incorporated; to appear in IMRN

We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of Montgomery and Vaughan that incorporates sieve weights.

We establish quantitative bounds on the $U^k[N]$ Gowers norms of the M\"obius function $\mu$ and the von Mangoldt function $\Lambda$ for all $k$, with error terms of shape $O((\log\log N)^{-c})$. As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms. We also obtain the first quantitative bounds on the size of sets containing no $k$-term arithmetic progressions with shifted prime difference., Comment: 56 pages; Theorem 1.8 strengthened; Referee comments incorporated

Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-S\'ark\"ozy asks to show that if $P(x)\in \mathbb{Z}[x]$, then the sequence $\lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial. We show that the sequence $\lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero. Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a "99% version" of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value., Comment: 43 pages; referee comments incorporated

We study for bounded multiplicative functions $f$ sums of the form \begin{align*} \sum_{\substack{n\leq x \atop n\equiv a\pmod q}}f(n), \end{align*} establishing that their variance over residue classes $a \pmod q$ is small as soon as $q=o(x)$, for almost all moduli $q$, with a nearly power-saving exceptional set of $q$. This improves and generalizes previous results of Hooley on Barban-Davenport-Halberstam-type theorems for such $f$, 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 $q$ in the cases where $q$ is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every $q$. These results are special cases of a "hybrid result" that works for sums of $f$ over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matom\"aki-Radziwill theorem on multiplicative functions in short intervals. We also consider the maximal deviation of $f$ over all residue classes $a\pmod q$ for $q\leq x^{1/2-\varepsilon}$, and show that it is small for "smooth-supported" $f$, again apart from a nearly power-saving set of exceptional $q$, 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 results relating to a ternary version of a conjecture of Erd\H{o}s on representing every element of the multiplicative group $\mathbb{Z}_p^{\times}$ as the product of two primes less than $p$., Comment: 67 pages; referee comments incorporated; to appear in Proc. London Math. Soc

We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq x+x^{\theta}}\mu(n)=o(x^{\theta}) $$ for any $\theta>0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's result corresponded to Huxley's $7/12$ exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramar\'e's identity to extract a small prime factor from the $n$-sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the M\"obius function as well as some results on multiplicative functions and almost primes in short intervals., Comment: 18 pages; referee comments incorporated