2022/01/20 by Olivier Ramaré, Ramare, Olivier, Alisa Sedunova +3
Mathematics · Computer Science · #Mathematical and Theoretical Analysis #Numerical Methods and Algorithms #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.2201.08076
When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for ∑n ≤ Xf(n)/n with error term O((log X)κ-h-1+ε) (for any positive ε>0) as soon as we have ∑p≤ Qf(p)(log p)/p=κlog Q+η+O(1/(log2Q)h) for a non-negative κ and some non-negative integer h. The method generalizes the 1967-approach of Levin and Fainleib and uses a differential equation.