6 papers
On a weighted sum over multiplicative functions and its applications to the GPY sieve
Zihao Liu
In this paper, we investigate the asymptotics of a class of weighted sums over multiplicative functions and apply our results to deduce a stronger asymptotic form of Yitang Zhang's…
A Corrected Simplified Proof of Chen's Theorem
Zihao Liu
In 1973, J.-R. Chen showed that every large even integer is a sum of a prime and a product of at most two primes. In this paper, the author indicates and fixes the issues in a simp…
A Simple Proof of Siegel's Theorem Using Mellin Transform
Zihao Liu
In this paper, we present a simple analytic proof of Siegel's theorem that concerns the lower bound of for primitive quadratic . Our new method compares an elementary l…
On Goldfeld's Proof of Siegel's Theorem
Zihao Liu
In this paper, we give a detailed account of Goldfeld's proof of Siegel's theorem. Particularly, we present complete proofs of the nontrivial assumptions made in his paper.
Scaled Homology and Topological Entropy
Bingzhe Hou, Kiyoshi Igusa, Zihao Liu
In this paper, we build up a scaled homology theory, -homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newl…
A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function
Zihao Liu
In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin's form: …