Extremal metrics for Laplace eigenvalues in perturbed conformal classes on products

In this short note, we prove that conformal classes which are small perturbations of a product conformal class on a product with a standard sphere admit a metric extremal for some Laplace eigenvalue. As part of the arguments, we obtain perturbed harmonic maps with constant density

Linear correlations amongst numbers represented by positive definite binary quadratic forms

Given a positive definite binary quadratic form f, let r(n) = |{(x,y): f(x,y)=n}| denote its representation function. In this paper we study linear correlations of these functions. For example, if r_1, ..., r_k are representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d) ... r_k(n+ (k-1)d).Comment: 60 pages. Small correction

Generalized Fourier coefficients of multiplicative functions

We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto \delta^{\omega (n)}$, where $\delta \neq 0$ and where $\omega$ counts the number of distinct prime factors of $n$, as well as the function $n \mapsto |\lambda_f(n)|$, where $\lambda_f(n)$ denotes the Fourier coefficients of a primitive holomorphic cusp form. For this class of functions we show that after applying a `$W$-trick' their elements become orthogonal to polynomial nilsequences. The resulting functions therefore have small uniformity norms of all orders by the Green--Tao--Ziegler inverse theorem, a consequence that will be used in a separate paper in order to asymptotically evaluate linear correlations of multiplicative functions from our class. Our result generalises work of Green and Tao on the M\"obius function.Comment: 95 pages; final versio

Correlations of the divisor function

In this paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d).Comment: 33 pages. Corrections and journal referenc

Linear correlations of multiplicative functions

We prove a Green--Tao theorem for multiplicative functions.Comment: 60 pages; final versio

On the square-free representation function of a norm form and nilsequences

We show that the restriction to square-free numbers of the representation function attached to a norm form does not correlate with nilsequences. By combining this result with previous work of Browning and the author, we obtain an application that is used in recent work of Harpaz and Wittenberg on the fibration method for rational points.Comment: 26 pages; revision taking into account referee's comment