arXiv subjects

Leo Goldmakher

Publications and source records attributed to Leo Goldmakher.

16 recordsLinked to original sources

arXiv2026

A converse to a theorem of Gauss on Gauss sums

In this note we prove (under mild hypotheses) that $f$ is a nontrivial character of $\mathbb{F}_p$ if and only if the Fourier transform of $f$ has magnitude 1 somewhere in $\mathbb{F}_p^\times$. This implies a converse to a theorem of Gauss on the magnitude of the Gauss sum, in addition to other consequences. The common theme in all our results is that extremal behavior on the Fourier side imposes multiplicative structure on the physical side.

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arXiv2025

Refinements of Artin's primitive root conjecture

A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit formula for the proportion; this formula is well-supported by computations and is known to hold on a generalized Riemann hypothesis, but remains open. In this paper we propose several conjectures that capture the finer properties of the distribution of the order of $a$ (mod $p$) as $p$ varies over primes; these assertions contain Artin's original conjecture as a special case. We prove these conjectures assuming the generalized Riemann hypothesis, as well as weaker versions unconditionally.

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arXiv2021

Khovanskii's theorem and effective results on sumset structure

A remarkable theorem due to Khovanskii asserts that for any finite subset $A$ of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a polynomial for all sufficiently large $h$. Currently, neither the polynomial nor what sufficiently large means are understood. In this paper we obtain an effective version of Khovanskii's theorem for any $A \subset \mathbb{Z}^d$ whose convex hull is a simplex; previously, such results were only available for $d=1$. Our approach gives information about not just the cardinality of $hA$, but also its structure, and we prove two effective theorems describing $hA$ as a set: one answering a recent question posed by Granville and Shakan, the other a Brion-type formula that provides a compact description of $hA$ for all large $h$. As a further illustration of our approach, we derive a completely explicit formula for $|hA|$ whenever $A \subset \mathbb{Z}^d$ consists of $d+2$ points.

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arXiv2017

The frequency and the structure of large character sums

Let $M(χ)$ denote the maximum of $|\sum_{n\le N}χ(n)|$ for a given non-principal Dirichlet character $χ\pmod q$, and let $N_χ$ denote a point at which the maximum is attained. In this article we study the distribution of $M(χ)/\sqrt{q}$ as one varies over characters $\pmod q$, where $q$ is prime, and investigate the location of $N_χ$. We show that the distribution of $M(χ)/\sqrt{q}$ converges weakly to a universal distribution $Φ$, uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for $Φ$'s tail. Almost all $χ$ for which $M(χ)$ is large are odd characters that are $1$-pretentious. Now, $M(χ)\ge |\sum_{n\le q/2}χ(n)| = \frac{|2-χ(2)|}π\sqrt{q} |L(1,χ)|$, and one knows how often the latter expression is large, which has been how earlier lower bounds on $Φ$ were mostly proved. We show, though, that for most $χ$ with $M(χ)$ large, $N_χ$ is bounded away from $q/2$, and the value of $M(χ)$ is little bit larger than $\frac{\sqrt{q}}π |L(1,χ)|$.

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arXiv2017

Improving the Burgess bound via Polya-Vinogradov

We show that even mild improvements of the Polya-Vinogradov inequality would imply significant improvements of Burgess' bound on character sums. Our main ingredients are a lower bound on certain types of character sums (coming from works of the second author joint with J. Bober and Y. Lamzouri) and a quantitative relationship between the mean and the logarithmic mean of a completely multiplicative function.

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arXiv2017

Refinements of Lagrange's four-square theorem

A well-known theorem of Lagrange asserts that every nonnegative integer $n$ can be written in the form $a^2+b^2+c^2+d^2$, where $a,b,c,d \in \mathbb{Z}$. We characterize the values assumed by $a+b+c+d$ as we range over all such representations of $n$.

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arXiv2017

Mock characters and the Kronecker symbol

We introduce and study a family of functions we call the "mock characters". These functions satisfy a number of interesting properties, and of all completely multiplicative arithmetic functions seem to come as close as possible to being Dirichlet characters. Along the way we prove a few new results concerning the behavior of the Kronecker symbol.

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arXiv2015

Pólya-Vinogradov and the least quadratic nonresidue

It is well-known that cancellation in short character sums (e.g. Burgess' estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess' work, so it is desirable to find a new approach to nonresidues. The goal of this note is to demonstrate a new line of attack via long character sums, a currently active area of research. Among other results, we demonstrate that improving the constant in the Pólya-Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sums, we show that the least nonresidue for any odd primitive character (mod $k$) is bounded by $(\log k)^{1.4}$.

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arXiv2013

On the spectral distribution of large weighted random regular graphs

McKay proved that the limiting spectral measures of the ensembles of $d$-regular graphs with $N$ vertices converge to Kesten's measure as $N\to\infty$. In this paper we explore the case of weighted graphs. More precisely, given a large $d$-regular graph we assign random weights, drawn from some distribution $\mathcal{W}$, to its edges. We study the relationship between $\mathcal{W}$ and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique `eigendistribution', i.e., a weight distribution $\mathcal{W}$ such that the associated limiting spectral distribution is a rescaling of $\mathcal{W}$. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is $O(1/d^2)$). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.

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arXiv2012

A quadratic large sieve inequality over number fields

We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.

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arXiv2012

Large even order character sums

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis.

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arXiv2012

The distribution of the maximum of character sums

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

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arXiv2011

L-functions with n-th order twists

Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The main new ingredient, possibly of independent interest, is a large sieve for n-th order characters. As further applications of this tool, we derive several results concerning L(s,χ) for n-th order Hecke characters: an estimate of the second moment on the critical line, a non-vanishing result at the central point, and a zero-density theorem.

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arXiv2011

Lower bounds on odd order character sums

A classical result of Paley shows that there are infinitely many quadratic characters $χ\mod{q}$ whose character sums get as large as $\sqrt{q}\log \log q$; this implies that a conditional upper bound of Montgomery and Vaughan cannot be improved. In this paper, we derive analogous lower bounds on character sums for characters of odd order, which are best possible in view of the corresponding conditional upper bounds recently obtained by the first author.

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arXiv2010

Multiplicative mimicry and improvements of the Polya-Vinogradov inequality

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd order character sums. Furthermore, conditionally on the Generalized Riemann Hypothesis we obtain a bound for odd order character sums which is best possible.

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arXiv2008

Character sums to smooth moduli are small

Recently, Granville and Soundararajan have made fundamental breakthroughs in the study of character sums. Building on their work and using estimates on short character sums developed by Graham-Ringrose and Iwaniec, we improve the Polya-Vinogradov inequality for characters with smooth conductor.

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