2008/10/12 by Yuanyou Furui Cheng, Cheng, Yuanyou
Mathematics · #11M26 #30D99 #32A60 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.0810.2103
openalex publication_date 2008/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that the non-trivial zeros of ζ(s) lie on the line \Re(s) =1/2. The density hypothesis is a conjectured estimate N(λ, T) =O(T\sp2(1-λ) +ε ) for any ε>0, where N(λ, T) is the number of zeros of ζ(s) when \Re(s) ≥λ and 0 0. The Riemann-von Mangoldt Theorem confirms this estimate when λ=1/2, with T\spε being replaced by log T. In an attempt to transform Backlund's proof of the Riemann-von Mangoldt Theorem to a proof of the density hypothesis by convexity, we discovered a different approach utilizing an auxiliary function. The crucial point is that this function should be devised to be symmetric with respect to \Re(s) =1/2 and about the size of the Euler Gamma function on the right hand side of the line \Re(s) =1/2. Moreover, it should be analytic and without any zeros in the concerned region. We indeed found such a function, which we call pseudo-Gamma function. With its help, we are able to establish a proof of the density hypothesis. Actually, we give the result explicitly and our result is even stronger than the original density hypothesis, namely it yields N(λ, T) ≤ 8.734 log T for any 1/2 < λ< 1 and T≥ 2445999554999.