We calculate the mean and variance of sums of the Möbius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, which have a long history. In some cases there are subtle and surprising differences. The ranges over which our results hold is significantly greater than those established for the corresponding problems in the number field setting.
We prove an asymptotic formula for squarefree in arithmetic progressions with squarefree moduli, improving previous results by Prachar. The main tool is an estimate for counting solutions of a congruence inside a box that goes beyond what can be obtained by using the Weil bound.
We investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound, and lying in some arithmetic progression a (mod q). In particular, we prove an upper bound for its variance as a varies over $(\mathbb{Z}/q\mathbb{Z})^{\times}$ which considerably improves upon earlier work of Blomer.
We show that the exponent of distribution of the sequence of squarefree numbers in arithmetic progressions of prime modulus is $\geq 2/3 + 1/57$, improving a result of Prachar from 1958. Our main tool is an upper bound for certain bilinear sums of exponential sums which resemble Kloosterman sums, going beyond what can be obtained by the Polya-Vinogradov completion method.
We prove upper bounds for the error term of the distribution of squarefree numbers up to $X$ in arithmetic progressions modulo $q$ making progress towards two well-known conjectures concerning this distribution and improving upon earlier results by Hooley. We make use of recent estimates for short exponential sums by Bourgain-Garaev and for exponential sums twisted by the Möbius function by Bourgain and Fouvry-Kowalski-Michel.
We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.
We consider Mertens' function M(x,q,a) in arithmetic progression, Assuming the generalized Riemann hypothesis (GRH), we show an upper bound that is uniform for all moduli which are not too large. For the proof, a former method of K. Soundararajan is extended to L-series.
We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.
We prove estimates for the level of distribution of the Möbius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $\infty$, and the power savings converges to square-root cancellation as $q$ goes to $\infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm1$ and such that $M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go beyond $\sqrt{x}$-cancellation.
Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $μ$ be the Möbius function and $μ_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=μ^2g$ and $f=μ_2^2g$ have unbounded partial sums. Our proofs are built upon Klurman and Mangerel's proof of Chudakov's conjecture, Klurman's work on correlations of multiplicative functions and Tao's resolution of the Erdős discrepancy problem.
An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.
We give asymptotics for correlation sums linked with the distribution of squarefree numbers in arithmetic progressions over a fixed modulus. As a particular case we improve a result of Blomer concerning the variance.
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.
Let d be a squarefree integer. Does there exist four squares in arithmetic progression over Q(sqrt{d})? We shall give a partial answer to this question, depending on the value of d. In the affirmative case, we construct explicit arithmetic progressions consisting of four squares over Q(sqrt{d}).
Let $M(x)=\sum_{1\le n\le x}μ(n)$ where $μ$ is the Möbius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+ε})$ for all $ε>0$. There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of \[ M(x)\ll\sqrt{x}\exp\left((\log x)^{1/2}(\log\log x)^c\right) \] (setting $c$ to $14$, though this can be reduced). Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the Möbius function over arith…
The purpose of this paper is to give some explicit formulas involving Möbius functions, which may be known under the generalized Riemann Hypothesis, but unconditional in this paper. Concretely, we prove explicit formulas of partial sums of the Möbius function in arithmetic progressions and partial sums of the Möbius functions on an Abelian number field $K$. In addition, to obtain these explicit formulas, we study a certain finite Euler product appearing from certain relation of primitive characters and imprimitive characters in the present paper.
Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ normalized Fourier coefficient. Assuming that all the coefficients $a(n)$ are real, we study the sign of $a(n)$ when $n$ runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers $n\le x$ such that $a(n)>n^{-α}$ where $x$ and $α$ are positive and $f$ is not necessarily a Hecke eigenform.
We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields Q(sqrt(D)), where D is a squarefree integer. For this purpose, we give a characterization in terms of Q(sqrt(D))-rational points on the elliptic curve E:y^2=x^3-27. We compute the torsion subgroup of the Mordell-Weil group of this elliptic curve over Q(sqrt(D)) and we give partial answers to the finiteness of the free part of E(Q(sqrt(D))). This last task will be translated to compute if the rank of the quadratic D-twist of the modular curve X_0(36) is zero or not.
Let $μ$ be the Möbius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $μ$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-σ}$ for any $σ> 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $μ$, which corresponds to the special case $k=1$.
We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, obtaining an exponent of distribution $\frac{1}{2} + \frac{7}{300}$.
This paper concerns the lattice $L_n$ of subsets of $\{1,\ldots,n\}$ that are arithmetic progressions, under the inclusion order. For $n\geq 4$, this poset is not graded and thus not semimodular. We give three independent proofs of the fact that for $n\geq 2$, $μ_n(L_n) = μ(n-1)$, where $μ_n$ is the Möbius function of $L_n$ and $μ$ is the classical (number-theoretic) Möbius function. We also show that $L_n$ is comodernistic, which implies that $L_n$ is EL-labelable. Comodernism is then used to prove that the order complex $Δ_n$ of the lattice is either contractible or homotopy equivalent to a …
Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb F_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the Möbius function in arithmetic progressions, and resolve Chowla's $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the Möbius function, we obtain a level of distribution beyond $1/2$ for…
In this paper we address the problem of computing asymptotic formulae for the expected values and second moments of central values of primitive Dirichlet $L$-functions $L(1/2,χ_{8d}\otimesψ)$ when $ψ$ is a fixed even primitive non-quadratic character of odd modulus $q$, $χ_{8d}$ is a primitive quadratic character, $d\equiv h\pmod r$ is odd and squarefree and $r\equiv0\pmod q$ is even. Restricting to these arithmetic progressions ensures that the resulting sets of $L$-functions each form a ``family of primitive $L$-functions" in the specific sense defined by Conrey, Farmer, Keating, Rubinstein …
This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In particular, the results of this paper may be used, together with the machinery of [LEP], to establish an asymptotic for the number of four-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally any problem counting prime points inside a ``non-degenerate'' affine lattice of codimension at most 2. The main result of this paper is a proof o…
Let Q(N;q,a) denotes the number of squares in the arithmetic progression qn+a, for n=0, 1,...,N-1, and let Q(N) be the maximum of Q(N;q,a) over all non-trivial arithmetic progressions qn + a. Rudin's conjecture asserts that Q(N)=O(Sqrt(N)), and in its stronger form that Q(N)=Q(N;24,1) if N=> 6. We prove the conjecture above for 6<=N<=52. We even prove that the arithmetic progression 24n+1 is the only one, up to equivalence, that contains Q(N) squares for the values of N such that Q(N) increases, for 7<=N<=52 (hence, for N=8,13,16,23,27,36,41 and 52). This allow us to assert, what we have calle…
The lecture, given at the ICM 2014 in Seoul, explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic number theory. The resulting theorems can be used to check existing conjectures over the integers, and to generate new ones. Among the problems discussed are: Counting primes in short intervals and in arithmetic progressions; Chowla's conjecture on the autocorrelation of the Mobius function; and the additive divisor problem.
Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $ϕ$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the only such functions are those of the form $cx^j(ψ(x))^m$ with $ψ\in K(x)$, and for $m \leq 4$ we show the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set $\{n \geq 0 : ϕ^{n}(a) \in λ(\mathbb{P}^1(K))\}$ is a…
Let $\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\{f(p)\}_{p\in\mathcal{P}}$ form a sequence of $\pm1$ valued independent random variables with $\mathbb{E} f(p)<0$, $\forall p\in \mathcal{P}$. The function $f$ is called strongly biased (towards classical Möbius function), if $\sum_{p\in\mathcal{P}}\frac{f(p)}{p}=-\infty$ a.s., and it is weakly biased if $\sum_{p\in\mathcal{P}}\frac{f(p)}{p} $ converges a.s. Let $M_f(x):=\sum_{n\leq x}f(n)$. We establish a number of necessary and sufficient co…
