2025/10/23 by Mayank Pandey, Maksym Radziwiłł, Pandey, Mayank +1
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2510.20194
openalex publication_date 2025/10/23 · openalex created_date 2025/10/25 · openalex updated_date 2026/07/28
Let f be a real-valued 1-bounded multiplicative function. Suppose that the mean-value of f2 exists, and ∫01 | ∑n ≤ N f(n)e2πi n α | d α≤ No(1) as N → ∞, then there exists a quadratic character χ such that for every δ> 0 the (logarithmic) proportion of primes p ≤ N such that |f(p) - χ(p)| < δ tends to 1 as N → ∞. More generally we show that for all N, Δ≥ 1 and 1-bounded multiplicative functions f, if ∫01 | ∑n ≤ N f(n) e2πi n α | d α≤ Δ and the L2 norm of f over [1, N] is ≥ N / 100, then f pretends to be a multiplicative character of conductor ≤ Δ2 on primes in [Δ2, N]. We highlight that the result is uniform in f, N and Δ and sharp as far as the size of the conductor goes. Moreover, the restriction to primes p ∈ [Δ2, N] turns out to be sharp in a suitably generalized version of this result, concerning sequences f that are close 1% of the time to multiplicative functions.