arXiv subjects

Adam J. Harper

Publications and source records attributed to Adam J. Harper.

At least 19 recordsLinked to original sources

arXiv2026

Lower bounds for low moments of character sums, I: Short sums with general multiplicative weights

We establish sharp lower bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r^{0.499}$, and $0 \leq q \leq 1$ is real. These match the better than squareroot cancellation upper bounds obtained in previous work of the author. We prove the same sharp lower bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of zeta sums, and more generally for moments of character sums $\sum_{n \leq x} h(n) \chi(n)$ with suitably bounded multiplicative twist $h(n)$. The proofs are based on a comparison of the sizes of $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} (\sum_{n \leq x} \chi(n)) \overline{I(\chi)}$, $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |I(\chi)|^2$ and $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |I(\chi)|^4$, where $I(\chi)$ is a certain ``barrier adjusted'' Perron integral inspired by the analogous results for random multiplicative functions. In a companion paper, we extend these arguments to the full interesting range $x \leq 0.99r$ for the unweighted character sum moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$. This leads to a positive proportion non-vanishing result for Dirichlet theta functions $\theta(1,\chi)$.

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arXiv2026

Distribution of random multiplicative functions in short intervals, with proper normalization

We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function $\sum_{x\le n \le x+y} f(n)$ over short intervals $[x, x+y]$, where $y \rightarrow \infty$ but $y=o(x)$. We show that with appropriate normalization, the limiting distribution is Gaussian for all such $y$. A key new feature of our result is that the normalization factor is different from the standard deviation $\sqrt{y}$ when $y$ is very close to $x$. In contrast, when $y \asymp x$ there is no normalization for which the limiting distribution is a non-degenerate Gaussian.

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arXiv2025

Better than squareroot cancellation in number theory

We give a short survey of the phenomenon of better than squareroot cancellation, specifically as it applies to averages of multiplicative character sums (such as $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$) thanks to their connection with so-called multiplicative chaos. We focus on the number theoretic aspects of the arguments, and also touch on some possible applications.

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arXiv2024

Simple Barban--Davenport--Halberstam type asymptotics for general sequences

We prove two estimates for the Barban--Davenport--Halberstam type variance of a general complex sequence in arithmetic progressions. The proofs are elementary, and our estimates are capable of yielding an asymptotic for the variance when the sequence is sufficiently nice, and is either somewhat sparse or is sufficiently like the integers in its divisibility by small moduli. As a concrete application, we deduce a Barban--Davenport--Halberstam type variance asymptotic for the $y$-smooth numbers less than $x$, on a wide range of the parameters. This addresses a question considered by Granville and Vaughan.

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arXiv2024

Moments of random multiplicative functions, III: A short review

We give a short review of recent progress on determining the order of magnitude of moments $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ of random multiplicative functions, and of closely related issues. We hope this can serve as a concise introduction to some of the ideas involved, for those who may not have too much background in the area.

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arXiv2023

The typical size of character and zeta sums is $o(\sqrt{x})$

We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)| = o(\sqrt{x})$, and so the sums $\sum_{n \leq x} \chi(n)$ typically exhibit "better than squareroot cancellation". We prove analogous better than squareroot bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of zeta sums; of Dirichlet theta functions $\theta(1,\chi)$; and of the sums $\sum_{n \leq x} h(n) \chi(n)$, where $h(n)$ is any suitably bounded multiplicative function (for example the M\"{o}bius function $\mu(n)$). The proofs depend on similar better than squareroot cancellation phenomena for low moments of random multiplicative functions. An important ingredient is a reorganisation of the conditioning arguments from the random case, so that one only needs to "condition" on a small collection of fairly short prime number sums. The conditioned quantities arising can then be well approximated by twisted second moments, whose behaviour is the same for character and zeta sums as in the random case.

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arXiv2022

A note on character sums over short moving intervals

We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} \chi(n)$, where $\chi$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erd\H{o}s, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty$ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty$, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $\chi$, even when $H = q/\log^{A}q$ for any fixed $A > 0$. On the other hand, we show it is true for "almost all" characters on the range $q^{1-o(1)} \leq H = o(q)$. Using P\'{o}lya's Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.

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arXiv2020

Almost sure large fluctuations of random multiplicative functions

We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such bound that grows faster than $\sqrt{x}$, answering a question of Hal\'asz and proving a conjecture of Erd\H{o}s. It is plausible that the exponent $1/4$ is sharp in this problem. The proofs work by establishing a multivariate Gaussian approximation for the sums $\sum_{n \leq x} f(n)$ at a sequence of $x$, conditional on the behaviour of $f(p)$ for all except the largest primes $p$. The most difficult aspect is showing that the conditional covariances of the sums are usually small, so the corresponding Gaussians are usually roughly independent. These covariances are related to an Euler product (or multiplicative chaos) type integral twisted by additive characters, which we study using various tools including mean value estimates for Dirichlet polynomials, high mixed moment estimates for random Euler products, and barrier arguments with random walks.

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arXiv2019

On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture

We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2} |\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as $T \leq t \leq 2T$ varies. In particular, we prove that for $(1+o(1))T$ values of $T \leq t \leq 2T$ we have $\max_{|h| \leq 1/2} \log|\zeta(1/2+it+ih)| \leq \log\log T - (3/4 + o(1))\log\log\log T$, matching for the first time with both the leading and second order terms predicted by a conjecture of Fyodorov, Hiary and Keating. The proofs work by approximating the zeta function in mean square by the product of a Dirichlet polynomial over smooth numbers and one over rough numbers. They then apply ideas and results from corresponding random model problems to compute averages of this product, under size restrictions on the smooth part that hold for most $T \leq t \leq 2T$ (but reduce the size of the averages). There are connections with the study of critical multiplicative chaos. Unlike in some previous work, our arguments never shift away from the critical line by more than a tiny amount $1/\log T$, and they don't require explicit calculations of Fourier transforms of Dirichlet polynomials.

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arXiv2019

The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwi\l\l, and Soundararajan]

This is the text to accompany my Bourbaki seminar from 30th March 2019, on the maximum size of the Riemann zeta function in "almost all" intervals of length 1 on the critical line. It surveys the conjecture of Fyodorov--Hiary--Keating on the behaviour of this typical maximum, as well as recent progress towards the conjecture by Najnudel and by Arguin--Belius--Bourgade--Radziwi\l\l--Soundararajan. There is also some general background discussion of the value distribution and large values of zeta.

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arXiv2018

Moments of random multiplicative functions, II: High moments

We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$. In the Steinhaus case, we show that $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when $q \approx (1+\sqrt{5})/2$, where the size starts to be dominated by "orthogonal" rather than "unitary" behaviour. We also deduce some consequences for the large deviations of $\sum_{n \leq x} f(n)$. The proofs use various tools, including hypercontractive inequalities, to connect $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ with the $q$-th moment of an Euler product integral. When $q$ is large, it is then fairly easy to analyse this integral. When $q$ is close to 1 the analysis seems to require subtler arguments, including Doob's $L^p$ maximal inequality for martingales.

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arXiv2017

Extreme biases in prime number races with many contestants

We continue to investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. We show that provided $n/\log q \rightarrow \infty$ as $q \rightarrow \infty$, we can find $n$ competitor classes modulo $q$ so that the corresponding $n$-way prime number race is extremely biased. This improves on the previous range $n \geq \varphi(q)^{\epsilon}$, and (together with an existing result of Harper and Lamzouri) establishes that the transition from all $n$-way races being asymptotically unbiased, to biased races existing, occurs when $n = \log^{1+o(1)}q$. The proofs involve finding biases in certain auxiliary races that are easier to analyse than a full $n$-way race. An important ingredient is a quantitative, moderate deviation, multi-dimensional Gaussian approximation theorem, which we prove using a Lindeberg type method.

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arXiv2017

Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos

We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0 \leq q \leq 1$. In the Steinhaus case, this is equivalent to determining the order of $\lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{-it}|^{2q} dt$. In particular, we find that $\mathbb{E}|\sum_{n \leq x} f(n)| \asymp \sqrt{x}/(\log\log x)^{1/4}$. 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 $\sum_{n \leq x} f(n)$. The proofs develop a connection between $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ 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.

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arXiv2016

Lower bounds for the variance of sequences in arithmetic progressions: primes and divisor functions

We develop a general method for lower bounding the variance of sequences in arithmetic progressions mod $q$, summed over all $q \leq Q$, building on previous work of Liu, Perelli, Hooley, and others. The proofs lower bound the variance by the minor arc contribution in the circle method, which we lower bound by comparing with suitable auxiliary exponential sums that are easier to understand. As an application, we prove a lower bound of $(1-\epsilon) QN\log(Q^2/N)$ for the variance of the von Mangoldt function $(\Lambda(n))_{n=1}^{N}$, on the range $\sqrt{N} (\log N)^C \leq Q \leq N$. Previously such a result was only available assuming the Riemann Hypothesis. We also prove a lower bound $\gg_{k,\delta} Q N (\log N)^{k^2 - 1}$ for the variance of the divisor functions $d_k(n)$, valid on the range $N^{1/2+\delta} \leq Q \leq N$, for any natural number $k \geq 2$.

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arXiv2015

Orderings of weakly correlated random variables, and prime number races with many contestants

We investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. Among our results we exhibit, for the first time, prime races modulo $q$ with $n$ competitor classes where the biases do not dissolve when $n, q\to \infty$. We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of $q$, whereas previous methods could only allow a power of $\log q$. The proofs use harmonic analysis related to the Hardy--Littlewood circle method to control the average size of correlations in prime number races. They also use various probabilistic tools, including an exchangeable pairs version of Stein's method, normal comparison tools, and conditioning arguments. In the process we derive some general results about orderings of weakly correlated random variables, which may be of independent interest.

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arXiv2015

Maxima of a randomized Riemann zeta function, and branching random walks

A recent conjecture of Fyodorov--Hiary--Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is $\exp\{\log \log T -\frac{3}{4}\log \log \log T+O(1)\}$, for an interval at (large) height $T$. In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.

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arXiv2015

A note on Helson's conjecture on moments of random multiplicative functions

We give lower bounds for the small moments of the sum of a random multiplicative function, which improve on some results of Bondarenko and Seip and constitute further progress towards (dis)proving a conjecture of Helson. We also prove asymptotics for the even integer moments. The latter have also been obtained very recently and independently by Heap and Lindqvist. Our proofs involve general lower bound techniques for random multiplicative functions, mean value results for multiplicative functions in several variables, and some calculations with Birkhoff polytopes.

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arXiv2015

Mean values of multiplicative functions over function fields

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.

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