Asymptotic formulae for arithmetic functions
Vishaal Kapoor · cIRcle (University of British Columbia) · 2011
In this work we will consider several questions concerning the asymptotic nature of arithmetic functions. First, we conduct a finer analysis on the behavior of λ(Euler's totient function(n)) in relation to λ(λ(n)), proving that log(λ(Euler's totient function(n))/λ(λ(n))) is asymptotic to (log log n)(log log log n)for almost all n. Second, we establish an asymptotic formula for sums of a generalized divisor function on the Gaussian numbers. And third, for complex-valued multiplicative functions that are suffciently close to 1 on the primes and bounded on prime powers, we determine the average value over a short interval x < n ≤ x+w provided the interval is suffciently long with respect to x.