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Seth Hardy

Publications and source records attributed to Seth Hardy.

5 recordsLinked to original sources
arXiv2026

Character sums over smooth numbers

Let $\Psi (x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We show that \[ \frac{1}{\varphi(q)} \sum_{\chi \bmod q} \Bigl| \sum_{\substack{n \leq x \\ P(n) \leq y}} \chi(n) \Bigr| = o \Bigl( \sqrt{\Psi(x,y)} \Bigr), \] whenever $(\log x)^6 \leq y \leq x^{\frac{1}{32 \log \log x}}$ and $q \geq x^{1 + \varepsilon}$ for some small quantifiable $\varepsilon > 0$. The saving is substantial when $\varepsilon$ is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums.

math.NT↗
arXiv2025

Helson's conjecture for smooth numbers

Let $\Psi(x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We prove that for $f$ a Steinhaus random multiplicative function, the partial sums over $y$-smooth numbers always enjoy better than squareroot cancellation, in the sense that $$ \mathbb{E} \Big|\sum_{\substack{1\leq n \leq x\\ P(n) \leq y}} f(n) \Big| = o\left( \Psi(x,y)^{1/2} \right),$$ uniformly on the entire range $ 2 \leq y \leq x$. The bounds are quantitative and give a large saving when $y$ isn't too close to $x$.

math.NT↗
arXiv2025

The distribution of partial sums of random multiplicative functions with a large prime factor

For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.

math.NT↗
arXiv2024

Bounds for exponential sums with random multiplicative coefficients

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n \theta) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N \rightarrow \infty$. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon > 0$, we prove the partial upper bound result $$ \max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n \theta) \Bigr| \ll {(\log N)}^{7/4 + \varepsilon},$$ asymptotically almost surely as $N \rightarrow \infty$, where $P(n)$ denotes the largest prime factor of $n$.

math.NT↗