Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let {λf(n)}n be its sequence of normalised Fourier coefficients. We show that if K/Q is any number field, and NK denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers n ∈ NK yield a sign change for thesequence {λf(n)}n∈NK . More precisely, for a positive proportion of n ∈ NK ∩ [1, X] we have λf(n)λf(n ) < 0, where n is the first element of NK greater than n for which λf(n) = 0. For example, for K = Q(i) and NK = {m2 + n2 : m, n ∈ Z} the set of sums of two squares, we obtain f X/√log X such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums oftwo squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed a = 0 there are f ,ε X1/2−ε sign changes for λf along the sequence of integers of the form a + m2 + n2 ≤ X

Divisor-bounded multiplicative functions in short intervals

We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h(logX)c , with h=h(X)→∞ and where c=cf≥0 is determined by the distribution of {|f(p)|}p in an explicit way. We give three applications. First, we show that the classical Rankin–Selberg-type asymptotic formula for partial sums of |λf(n)|2 , where {λf(n)}n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length hlogX , if h=h(X)→∞ . We also generalize this result to sequences {|λπ(n)|2}n , where λπ(n) is the nth coefficient of the standard L-function of an automorphic representation π with unitary central character for GLm , m≥2 , provided π satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments {|λf(n)|α}n over intervals of length h(logX)cα , with cα>0 explicit, for any α>0 , as h=h(X)→∞ . Finally, we show that the (non-multiplicative) Hooley Δ -function has average value ≫loglogX in typical short intervals of length (logX)1/2+η , where η>0 is fixed

On Integer-Valued Multiplicative Functions at Consecutive Integers

Let $f: \mathbb{N} \rightarrow \mathbb{Z}$ be an integer-valued multiplicative function, for which $$\sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty, \quad \sum_{p : \, f(p) = 0} \frac{1}{p} < \infty.$$ We show under these conditions alone (and no further assumptions on the growth of $f$) that for any $h \geq 1$, the set $$\{n \in \mathbb{N} : f(n) = f(n+h), \, f(n)f(n+h) \neq 0\}.$$ has logarithmic density 0. We present an application of (an extension of) this result to the sequence of Fourier coefficients of a non-CM holomorphic eigencusp form for $\text{SL}_2(\mathbb{Z})$. The proof employs a novel technique that allows us to analyse the value distribution of $f$ by means of the compositions $|f|^{it}$ and $\chi(f)$ for varying, suitably chosen $t \in \mathbb{R}$ and Dirichlet characters $\chi$, both of which are multiplicative functions taking values in the closed unit disc. Key inputs arise from the inverse theory of sumsets in discrete and continuous additive combinatorics.Comment: 27 pages, comments welcome; minor typos fixe

Three conjectures about character sums

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L(1, χ) and L(1+it, χ)) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions

Correlations of multiplicative functions in function fields

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 Mathematik