You are cordially invited to the Graduate Mathematics Seminar Series organized by the Department of Mathematics.<\/p>\n
Speaker: Muhammet Boran (Bilkent)<\/p>\n
\u201d BEHAVIOUR OF ARITHMETIC FUNCTIONS OVER BEATTY SEQUENCES\u201d<\/p>\n
Abstract: In this talk, Abercrombie, Banks, and Shparlinski investigate summatory functions of general arithmetic functions over Beatty sequences. For a real number \u03b1 > 1, they study sums of the form Advisor: Assoc. Prof. Ahmet Muhtar G\u00fclo\u011flu<\/p>\n Date: 19 December 2025 You are cordially invited to the Graduate Mathematics Seminar Series organized by the Department of Mathematics. Speaker: Muhammet Boran (Bilkent) \u201d BEHAVIOUR OF ARITHMETIC FUNCTIONS OVER BEATTY SEQUENCES\u201d Abstract: In this talk, Abercrombie, Banks, and Shparlinski investigate summatory functions of general arithmetic functions over Beatty sequences. For a real number [\u2026] <\/p>\n
\nS_\u03b1(f, x) = \u2211{n \u2264 x, n \u2208 B_\u03b1} f(n),
\nwhere B_\u03b1 = {\u230ak\u03b1\u230b : k \u2208 \u2115} denotes the Beatty sequence associated with \u03b1. Using Fourier-analytic methods together with Vaaler\u2019s trigonometric approximation to the sawtooth function, they prove that for any \u03b5 > 0 and for almost all \u03b1 > 1,
\nS_\u03b1(f, x) = \u03b1\u207b\u00b9 \u2211<\/em>{n \u2264 x} f(n) + O(x^{2\/3+\u03b5} M(f, x)),
\nwhere M(f, x) = 1 + max_{n \u2264 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 M\u00f6bius function, Euler\u2019s totient function, the number of prime divisors, and digital sum functions.<\/p>\n
\nTime: 13:30
\nPlace: SA- 141<\/p>\n","protected":false},"excerpt":{"rendered":"