Document Type
Lecture
Publication Date
3-20-2025
Abstract
Let r1(n) be the number of representations of n as the sum of a square and a square of a prime. We discuss the erratic behavior of r1, which is similar to the one of the divisor function. We will show that the number of integers up to x that have at least one such representation is asymptotic to (π/2)(x/(log x)) minus a secondary term of size x/(log x)1+d+o(1), where d is the multiplication table constant. Detailed heuristics suggest very precise asymptotic for the secondary term as well. In particular, our proofs imply that the main contribution to the mean value of r1(n) comes from integers with “unusual” number of prime factors, i.e., those with ω(n) ∼ 2 log log x (for which r1(n) ∼ (log x)log 4−1), where ω(n) is the number of district prime factors of n.
In the talk we will review the results of two works: my paper from 2022 and a recent joint preprint with Andrew Granville and Cihan Sabuncu.
Relational Format
presentation
Recommended Citation
Sedunova, Alisa, "The multiplication table constant and sums of two squares" (2025). Algebra/Number Theory Seminar. 55.
https://egrove.olemiss.edu/math_algebra_number_theory/55
Accessibility Status
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