You are cordially invited to the Graduate Mathematics Seminar Series organized by the Department of Mathematics.
Speaker: Muhammet Boran (Bilkent)
” BEHAVIOUR OF ARITHMETIC FUNCTIONS OVER BEATTY SEQUENCES”
Abstract: In this talk, Abercrombie, Banks, and Shparlinski investigate summatory functions of general arithmetic functions over Beatty sequences. For a real number α > 1, they study sums of the form
S_α(f, x) = ∑{n ≤ x, n ∈ B_α} f(n),
where B_α = {⌊kα⌋ : k ∈ ℕ} denotes the Beatty sequence associated with α. Using Fourier-analytic methods together with Vaaler’s trigonometric approximation to the sawtooth function, they prove that for any ε > 0 and for almost all α > 1,
S_α(f, x) = α⁻¹ ∑{n ≤ x} f(n) + O(x^{2/3+ε} M(f, x)),
where M(f, x) = 1 + max_{n ≤ x} |f(n)|. Their result requires only a mild growth condition on f and does not rely on multiplicativity, additivity, or other structural assumptions. As applications, the authors obtain improved error terms for divisor functions d_r and derive asymptotic formulae for a wide class of arithmetic functions, including the Möbius function, Euler’s totient function, the number of prime divisors, and digital sum functions.
Advisor: Assoc. Prof. Ahmet Muhtar Güloğlu
Date: 19 December 2025
Time: 13:30
Place: SA- 141