Seth Hardy

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math.NT2026

Character sums over smooth numbers

Seth Hardy, Max Wenqiang Xu

Let $Ψ (x, y)$ denote the count of $y$ -smooth numbers below $x$ and $P (n)$ denote the largest prime factor of $n$ . We show that \[ \frac{1}{φ(q)} \sum_{χ\bmod q} \Bigl| \sum_{\substa…

math.NT2025

Helson's conjecture for smooth numbers

Seth Hardy, Max Wenqiang Xu

Let $Ψ (x, y)$ 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, t…

math.NT2025

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

Seth Hardy

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…

math.NT2024

Bounds for exponential sums with random multiplicative coefficients

Seth Hardy

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{θ\in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n θ) \Bigr| \gg \sqr…