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The maximal order of the shifted-prime divisor function
Steve Fan, Paul Pollack
For each positive integer , we denote by the number of shifted-prime divisors of , i.e., \[ω^*(n):=\sum_{p-1\mid n}1.\] First introduced by Prachar in 1955, th…
Counting primes with a given primitive root, uniformly
Steve Fan, Paul Pollack
The celebrated Artin conjecture on primitive roots asserts that given any integer which is neither nor a perfect square, there is an explicit constant such that t…
Extremal elasticity of quadratic orders
Steve Fan, Paul Pollack
We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if is an ima…
The typical elasticity of a quadratic order
Steve Fan, Paul Pollack
For an atomic domain , the of is defined as $\sup\{r/s: π_1\cdots π_r = ρ_1 \cdots ρ_s,~ \text{where each $π_i, ρ_j$ is irreducible}\}$; the elasticity p…
Mean values of multiplicative functions and applications to residue-class distribution
Paul Pollack, Akash Singha Roy
We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \…