Square Root Cancellation Research Articles

Better than square-root cancellation for random multiplicative functions

We investigate when the better than square-root cancellation phenomenon exists for ∑ n ≤ N a ( n ) f ( n ) \sum _{n\le N}a(n)f(n) , where a ( n ) ∈ C a(n)\in \mathbb {C} and f ( n ) f(n) is a random multiplicative function. We focus on the case where a ( n ) a(n) is the indicator function of R R rough numbers. We prove that log ⁡ log ⁡ R ≍ ( log ⁡ log ⁡ x ) 1 2 \log \log R \asymp (\log \log x)^{\frac {1}{2}} is the threshold for the better than square-root cancellation phenomenon to disappear.

Square-root cancellation for sums of factorization functions over squarefree progressions in $\mathbb{F}_q[t

Square-root cancellation for sums of factorization functions over squarefree progressions in $\mathbb{F}_q[t

Extending the support of 1- and 2-level densities for cusp form $L$-functions under square-root cancellation hypotheses

Extending the support of 1- and 2-level densities for cusp form $L$-functions under square-root cancellation hypotheses

Two-dimensional Weyl sums failing square-root cancellation along lines

Two-dimensional Weyl sums failing square-root cancellation along lines

Double square moments and bounds for resonance sums of cusp forms

Let f and g be holomorphic cusp forms for the modular group SL2(Z) of weight k1 and k2 with Fourier coefficients λf(n) and λg(n), respectively. For real α≠0 and 0<β≤1, consider a smooth resonance sum SX(f,g;α,β) of λf(n)λg(n) against e(αnβ) over X≤n≤2X. Double square moments of SX(f,g;α,β) over both f and g are nontrivially bounded when their weights k1 and k2 tend to infinity together. By allowing both f and g to move, these double moments are indeed square moments associated with automorphic forms for GL(4). By taking out a small exceptional set of f and g, bounds for individual SX(f,g;α,β) shall then be proved. These individual bounds break the resonance barrier of X5/8 for 1/6<β<1 and achieve a square-root cancellation for 1/3<β<1 for almost all f and g as an evidence for Hypothesis S for cusp forms over integers. The methods used in this study include Petersson's formula, Poisson's summation formula, and stationary phase integrals.

Subconvexity in Inhomogeneous Vinogradov Systems

Abstract When k and s are natural numbers and ${\mathbf h}\in {\mathbb Z}^k$, denote by $J_{s,k}(X;\,{\mathbf h})$ the number of integral solutions of the system $$ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\leqslant j\leqslant k), $$ with $1\leqslant x_i,y_i\leqslant X$. When $s\lt k(k+1)/2$ and $(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$, Brandes and Hughes have shown that $J_{s,k}(X;\,{\mathbf h})=o(X^s)$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov’s mean value theorem, we obtain an asymptotic formula for $J_{s,k}(X;\,{\mathbf h})$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation.

A model problem for multiplicative chaos in number theory

Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper’s work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov–Hiary–Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version of Helson’s conjecture. We develop and simplify the ideas of Harper in this context, with the hope that the simplified proof would be of use to readers seeking a gentle entry-point to this fascinating area.

Möbius cancellation on polynomial sequences and the quadratic Bateman–Horn conjecture over function fields

We establish cancellation in short sums of certain special trace functions over \({\mathbb {F}}_q[u]\) below the Pólya–Vinogradov range, with savings approaching square-root cancellation as q grows. This is used to resolve the \({\mathbb {F}}_q[u]\)-analog of Chowla’s conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman–Horn conjecture in degree 2, for some values of q. A final application is to sums of trace functions over primes in \({\mathbb {F}}_q[u]\).

Square-root cancellation for sums of factorization functions over short intervals in function fields

We present new estimates for sums of the divisor function and other similar arithmetic functions in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that approaches square-root cancellation as long as the characteristic of the finite field is relatively large. This is done by a geometric method, inspired by work of Hast and Matei, where we calculate the singular locus of a variety whose Fq-points control this sum. This has applications to highly unbalanced moments of L-functions.

Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function

We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T \log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer--Gonek--Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We determine upper bounds on the level of squareroot cancellation and lower bounds which suggest a degree of cancellation much greater than this which we speculate is in accordance with the influence of the Euler product.

MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS

We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ , where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$ . In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$ . In particular, we find that $\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$ . This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of $\sum _{n\leqslant x}f(n)$ . The proofs develop a connection between $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ and the $q$ th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.

Square-root cancellation for the signs of Latin squares

Let L(n) be the number of Latin squares of order n, and let Leven(n) and Lodd(n) be the number of even and odd such squares, so that L(n)=Leven(n)+Lodd(n). The Alon-Tarsi conjecture states that Leven(n) ź Lodd(n) when n is even (when n is odd the two are equal for very simple reasons). In this short note we prove that $$\left| {{L^{even}}\left( n \right) - {L^{odd}}\left( n \right)} \right| \leqslant L{\left( n \right)^{\frac{1}{2} + o\left( 1 \right)}}$$|Leven(n)źLodd(n)|źL(n)12+o(1), thus establishing the conjecture that the number of even and odd Latin squares, while conjecturally not equal in even dimensions, are equal to leading order asymptotically. Two proofs are given: both proceed by applying a differential operator to an exponential integral over SU(n). The method is inspired by a recent result of Kumar-Landsberg.

Surpassing the ratios conjecture in the 1-level density of DirichletL-functions

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions in the family of all characters modulo $q$, with $Q/2 < q\leq Q$. For test functions whose Fourier transform is supported in $(-3/2, 3/2)$, we calculate this quantity beyond the square-root cancellation expansion arising from the $L$-function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the Ratios Conjecture, and proves that the exponent of the error term $Q^{-\frac 12 +\epsilon}$ in the Ratios Conjecture is best possible. We also give more precise results when the support of the Fourier Transform of the test function is restricted to the interval $[-1,1]$. Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery's, which implies that the Ratios Conjecture's prediction holds for any finite support up to an error $Q^{-\frac 12 +\epsilon}$.

An orthogonal test of the L-functions Ratios Conjecture, II

Recently Conrey, Farmer, and Zirnbauer developed the L-functions Ratios conjecture, which gives a recipe that predicts a wealth of statistics, from moments to spacings between adjacent zeros and values of L-functions. The problem with this method is that several of its steps involve ignoring error terms of size comparable to the main term; amazingly, the errors seem to cancel and the resulting prediction is expected to be accurate up to square-root cancellation. We prove the accuracy of the Ratios Conjecture's prediction for the 1-level density of families of cuspidal newforms of constant sign (up to square-root agreement for support in (-1,1), and up to a power savings in (-2,2)), and discuss the arithmetic significance of the lower order terms. This is the most involved test of the Ratios Conjecture's predictions to date, as it is known that the error terms dropped in some of the steps do not cancel, but rather contribute a main term! Specifically, these are the non-diagonal terms in the Petersson formula, which lead to a Bessel-Kloosterman sum which contributes only when the support of the Fourier transform of the test function exceeds (-1, 1).

Rational points on complete intersections of higher degree, and mean values of Weyl sums

We establish upper bounds for the number of rational points of bounded height on complete intersections. When the degree of the intersection is sufficiently large in terms of its dimension, and the contribution arising from appropriate linear spaces is removed, these bounds are smaller than those arising from the expectation of ‘square-root cancellation’. In particular, there is a paucity of non-diagonal solutions to the equation x 1 d + ⋯ + x s d = x s + 1 d + ⋯ + x 2 s d , provided that d ⩾ (2s)4s. There are consequences for the approximate distribution function of Weyl sums of higher degree, and also for quasi-diagonal behaviour in mean values of smooth Weyl sums.

Two-Dimensional Analogues of an Inequality of Esseen with Applications to the Central Limit Theorem

Two-Dimensional Analogues of an Inequality of Esseen with Applications to the Central Limit Theorem