1935/09/02 by E. C. Titchmarsh · 79 citations
Mathematics · #Analytic Number Theory Research #Meromorphic and Entire Functions #Riemann zeta function #Riemann hypothesis #Mathematics #Critical line #Riemann Xi function #Pure mathematics #Number theory #Prime (order theory) #Function (biology) #Combinatorics #Mathematical analysis #Mathematical physics #Physics
paper · doi:10.1098/rspa.1935.0146
published in Proceedings of the Royal Society of London A Mathematical and Physical Sciences 151(873), 234-255 (Royal Society)
openalex publication_date 1935/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/27
Abstract 1―It is well known that the distribution of the zeroes of the Riemann zeta-function ζ(s) = ∞Σ n=1 1/n8 (s= σ + it) plays a fundamental part in the theory of prime numbers. It was conjectured by Riemann that all the complex zeroes of ζ(s) lie on the line σ = 1/2, but this hypothesis has never been proved or disproved. It is therefore natural to enquiry how far the hypothesis is supported by numerical calculations. The most extensive calculations of this kind have been undertaken by Gram, Backlund, and Hutchinson. The final result obtained by Hutchinson is that ζ(s) has 138 zeroes on σ = 1/2 between t = 0 and t = 300, and no other zeroes between these values of t.