We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either $f(n)=n^{s}$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$, where $a,b$ belong to some multiplicative subsets of $\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.

We establish several results concerning the expected general phenomenon

We introduce a simple approach to study partial sums of multiplicative functions which are close to their mean value. As a first application, we show that a completely multiplicative function f :with c = 0 if and only if f (p) = 1 for all but finitely many primes and |f (p)| < 1 for the remaining primes. This answers a question of Imre Ruzsa.For the case c = 0, we show, under the additional hypothesis p:|f (p)|<1 1/p < ∞, that f has bounded partial sums if and only if f (p) = χ(p)p it for some non-principal Dirichlet character χ modulo q and t ∈ R except on a finite set of primes that contains the primes dividing q, wherein |f (p)| < 1. This essentially resolves another problem of Ruzsa and generalizes previous work of the first and the second author on Chudakov's conjecture.We also consider some other applications, which include a proof of a recent conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions.

We study for bounded multiplicative functions f sums of the form n≤x n≡a (mod q) f (n), establishing a theorem stating that their variance over residue classes a (mod q) is small as soon as q = o(x), for almost all moduli q, with a nearly power-saving exceptional set of q. This substantially improves on 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 we establish that works for sums of f (n) over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki-Radziwi l l theorem on multiplicative functions in short intervals.We also consider the maximal deviation of f (n) over all residue classes a (mod q) in the square root range q ≤ x 1/2−ε , 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 the analogue of Linnik's theorem on the least prime in an arithmetic progression for products of exactly three primes, and prove the exponent 2 + o(1) for this problem for all smooth values of q.

We test M. Berry's ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions ("boundary-adapted arithmetic random waves"). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels. It is found that the precise nodal deficiency or surplus of the nodal length depends on arithmetic properties of the energy levels, in an explicit way. To obtain the said results we apply the Kac-Rice method for computing the expected nodal length of a Gaussian random field. Such an application uncovers major obstacles, e.g. the occurrence of "bad" subdomains, that, one hopes, contribute insignificantly to the nodal length. Fortunately, we were able to reduce this contribution to a number theoretic question of counting the "spectral semi-correlations", a concept joining the likes of "spectral correlations" and "spectral quasi-correlations" in having impact on the nodal length for arithmetic dynamical systems. This work rests on several breakthrough techniques of J. Bourgain, whose interest in the subject helped shaping it to high extent, and whose fundamental work on spectral correlations, joint with E. Bombieri, has had a crucial impact on the field.