Average orders of multiplicative arithmetical functions of integer matrices
Average r-rank Artin conjecture
Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups , with and , with a range of uniformity for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar...
Average size of 2-Selmer groups of elliptic curves, II
Average Value of the Euler Function on Binary Palindromes
We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on .
Average values of L-series in function fields.
Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution
Averages of short exponential sums
Averages of short exponential sums, II
Averages of twisted elliptic L-functions
Aviezri Fraenkel's work in number theory.