arXiv subjects

Adam J. Harper

Publications and source records attributed to Adam J. Harper.

29 records · Page 2Linked to original sources

arXiv2014

Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

We investigate exponential sums over those numbers $\leq x$ all of whose prime factors are $\leq y$. We prove fairly good minor arc estimates, valid whenever $\log^{3}x \leq y \leq x^{1/3}$. Then we prove sharp upper bounds for the $p$-th moment of (possibly weighted) sums, for any real $p > 2$ and $\log^{C(p)}x \leq y \leq x$. Our proof develops an argument of Bourgain, showing this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$, whenever $\log^{C}x \leq y \leq x$. Previously this was only known assuming the Generalised Riemann Hypothesis. Combining them with transference machinery of Green, we prove Roth's theorem for subsets of the $y$-smooth numbers, whenever $\log^{C}x \leq y \leq x$. This provides a deterministic set, of size $\approx x^{1-c}$, inside which Roth's theorem holds.

math.NT↗

arXiv2014

Pickands' constant $H_α$ does not equal $1/Γ(1/α)$, for small $α$

Pickands' constants $H_α$ appear in various classical limit results about tail probabilities of suprema of Gaussian processes. It is an often quoted conjecture that perhaps $H_α = 1/Γ(1/α)$ for all $0 < α\leq 2$, but it is also frequently observed that this doesn't seem compatible with evidence coming from simulations. We prove the conjecture is false for small $α$, and in fact that $H_α \geq (1.1527)^{1/α}/Γ(1/α)$ for all sufficiently small $α$. The proof is a refinement of the "conditioning and comparison" approach to lower bounds for upper tail probabilities, developed in a previous paper of the author. Some calculations of hitting probabilities for Brownian motion are also involved.

math.PR↗

arXiv2013

Inverse questions for the large sieve

Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $O(X^{1/2})$, and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. We prove a variety of results and formulate various conjectures in connection with this problem, including several improvements of the large sieve bound when the residue classes occupied by $A$ have some additive structure. Unfortunately we cannot solve the problem itself.

math.NT↗

arXiv2013

Additive decompositions of sets with restricted prime factors

We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sárközy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.

math.NT↗

arXiv2013

Sharp conditional bounds for moments of the Riemann zeta function

We prove, assuming the Riemann Hypothesis, that \int_{T}^{2T} |ζ(1/2+it)|^{2k} dt \ll_{k} T log^{k^{2}} T for any fixed k \geq 0 and all large T. This is sharp up to the value of the implicit constant. Our proof builds on well known work of Soundararajan, who showed, assuming the Riemann Hypothesis, that \int_{T}^{2T} |ζ(1/2+it)|^{2k} dt \ll_{k,ε} T log^{k^{2}+ε} T for any fixed k \geq 0 and ε> 0. Whereas Soundararajan bounded \log|ζ(1/2+it)| by a single Dirichlet polynomial, and investigated how often it attains large values, we bound \log|ζ(1/2+it)| by a sum of many Dirichlet polynomials and investigate the joint behaviour of all of them. We also work directly with moments throughout, rather than passing through estimates for large values.

math.NT↗

arXiv2013

A note on the maximum of the Riemann zeta function, and log-correlated random variables

In recent work, Fyodorov and Keating conjectured the maximum size of $|ζ(1/2+it)|$ in a typical interval of length O(1) on the critical line. They did this by modelling the zeta function by the characteristic polynomial of a random matrix; relating the random matrix problem to another problem from statistical mechanics; and applying a heuristic analysis of that problem. In this note we recover a conjecture like that of Fyodorov and Keating, but using a different model for $|ζ(1/2+it)|$ in terms of a random Euler product. In this case the probabilistic model reduces to studying the supremum of Gaussian random variables with logarithmic correlations, and can be analysed rigorously.

math.NT↗

arXiv2013

Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function

We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We present an extended application to a Gaussian version of a random process studied by Halasz. This leads to much improved lower bound results for the sum of a random multiplicative function. We further illustrate our methods by improving lower bounds for some classical constants from extreme value theory, the Pickands constants $H_α$, as $α\rightarrow 0$.

math.PR↗

arXiv2012

Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers

We prove Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for the y-smooth numbers less than x, on the range log^{K}x \leq y \leq x. This improves on the range \exp{log^{2/3 + ε}x} \leq y \leq x that was previously available. Our proofs combine zero-density methods with direct applications of the large sieve, which seems to be an essential feature and allows us to cope with the sparseness of the smooth numbers. We also obtain improved individual (i.e. not averaged) estimates for character sums over smooth numbers.

math.NT↗

arXiv2011

On finding many solutions to S-unit equations by solving linear equations on average

We give improved lower bounds for the number of solutions of some $S$-unit equations over the integers, by counting the solutions of some associated linear equations as the coefficients in those equations vary over sparse sets. This method is quite conceptually straightforward, although its successful implementation involves, amongst other things, a slightly subtle use of a large sieve inequality. We also present two other results about solving linear equations on average over their coefficients.

math.NT↗

arXiv2011

On a paper of K. Soundararajan on smooth numbers in arithmetic progressions

In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to modulus q, provided that y^{4\sqrt{e}-δ} \geq q and that y is neither too large nor too small compared with x. We show that these latter restrictions on y are unnecessary, thereby proving a conjecture of Soundararajan. Our argument uses a simple majorant principle for trigonometric sums to handle a saddle point that is close to 1.

math.NT↗

arXiv2010

On the limit distributions of some sums of a random multiplicative function

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum over those n \leq x with k distinct prime factors, provided that k = o(log log x) as x \rightarrow \infty. We estimate the fourth moments of these sums, and use a conditioning argument to show that if k is of the order of magnitude of log log x then the analogous normal limit theorem does not hold. The methods extend to treat the sum over those n \leq x with at most k distinct prime factors, and in particular the sum over all n \leq x. We also treat a substantially generalised notion of random multiplicative function.

math.NT↗