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We study for bounded multiplicative functions f$f$ sums of the form ∑n⩽xn≡a(modq)f(n),$$\begin{align*} \hspace*{7pc}\sum _{\substack{n\leqslant x\\ n\equiv a\ (\mathrm{mod}\ q)}}f(n), \end{align*}$$establishing that their variance over residue classes a(modq)$a \ (\mathrm{mod}\ q)$ is small as soon as q=o(x)$q=o(x)$, for almost all moduli q$q$, with a nearly power‐saving exceptional set of q$q$. This improves and generalizes previous results of Hooley on Barban–Davenport–Halberstam type theorems for such f$f$, 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$q$ in the cases where q$q$ is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every q$q$. These results are special cases of a "hybrid result" that we establish that works for sums of f$f$ 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 f$f$ over all residue classes a(modq)$a\ (\mathrm{mod}\ q)$ in the square root range q⩽x1/2−ε$q\leqslant x^{1/2-\varepsilon }$, and show that it is small for "smooth‐supported" f$f$, again apart from a nearly power‐saving set of exceptional q$q$, 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 a ternary approximation to a conjecture of Erdős on representing every element of the multiplicative group Zp×$\mathbb {Z}_p^{\times }$ as the product of two primes less than p$p$. [ABSTRACT FROM AUTHOR]

We develop an approach to study character sums, weighted by a multiplicative function f:Fq[t]→S1$f\colon \mathbb {F}_q[t]\rightarrow S^1$, of the form ∑deg(G)=NGmonicf(G)χ(G)ξ(G),$$\begin{align*}\hskip7pc \sum _{\substack{\textnormal {deg}(G) = N \\ G \text{ monic}}}f(G)\chi (G)\xi (G), \end{align*}$$where χ is a Dirichlet character and ξ is a short interval character over Fq[t]$\mathbb {F}_q[t]$. We then deduce versions of the Matomäki–Radziwiłł theorem and Tao's two‐point logarithmic Elliott conjecture over function fields Fq[t]$\mathbb {F}_q[t]$, 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 Möbius 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 Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the existing one in 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 Erdős discrepancy problem over Fq[t]$\mathbb {F}_q[t]$. [ABSTRACT FROM AUTHOR]

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$x$, and any distinct integers h1,⋯,hk,h1′,⋯,hℓ′$h_1,\dots ,h_k,h^{\prime }_1,\dots ,h^{\prime }_\ell$, we establish an asymptotic formula for ∑n⩽xΛ(n+h1)⋯Λ(n+hk)λ(n+h1′)⋯λ(n+hℓ′)$$\begin{equation*} \hspace*{13pt}\sum _{n\leqslant x}\Lambda (n+h_1)\cdots \Lambda (n+h_k)\lambda (n+h_{1}^{\prime })\cdots \lambda (n+h_{\ell }^{\prime })\hspace*{-13pt} \end{equation*}$$for any 0⩽k⩽2$0\leqslant k\leqslant 2$ and ℓ⩾0$\ell \geqslant 0$. Specializing to either ℓ=0$\ell =0$ or k=0$k=0$, 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. [ABSTRACT FROM AUTHOR]