Booth Id:
MATH015T
Category:
Mathematics
Year:
2024
Finalist Names:
Hsu, Jih-An (School: National Pingtung Senior High School )
Lin, Pin-Hua (School: National Pingtung Senior High School )
Abstract:
For a fixed positive integer bєN, let the b-sight lines be defined by f(x)=ax^b, for aєQ, with the origin O as the observing point (the position of the eyes). A point in the square lattice V(m)={ (i, j) | i,jєN, 1≤i≤m, 1≤j≤m} is said to be b-visible if it is the “first” point in V(m) that can be seen from the origin O through any sight line of the form f(x)=ax^b, for some aєQ. Let H_b (m) denote the total number of b-visible points in V(m). Our goal in this project is to enumerate H_b (m) and we show that it can be expressed by Möbius function. When b=1, due to symmetry, H_1 (m) can be further reduced to a very neat formula in terms of Euler function. Moreover, by a probability result in literature, we obtain a non-trivial asymptotic limit lim_{m→∞}H_b (m)/^2 =1/ζ(b+1) where ζ(s) is the Riemann-Zeta function. Finally, assuming that we now observe lattice points in V(m) from another square S(k)={(r,t) | 0≤r≤k,0≤t≤k}, not limited to just the origin O. To see every lattice in V(m), we show that it can be done from S(k) whose side length k is no more than A·√(π(m)), where π(m) is the number of primes less than or equal to m, and A=3/√(1- 8/9 ln(2.5) )≈6.965. Our result is novel and interesting as it links counting in combinatorics with arithmetic functions in number theory and asymptotic behavior from analysis.