On the average value of π(t)-li(t)

2022/01/17 by Daniel R. Johnston, Johnston, Daniel R.
Mathematics · #11M26 #11N05 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:11M26 #msc:11N05

paper · pdf · doi:10.48550/arxiv.2201.06184

11 pages; to appear in Canad. Math. Bull

openalex publication_date 2022/01/17 · arxiv created 2022/03/07 · arxiv updated 2022/03/08 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

We prove that the Riemann hypothesis is equivalent to the condition ∫₂^x(π(t)-li(t))dt<0 for all x>2. Here, π(t) is the prime-counting function and li(t) is the logarithmic integral. This makes explicit a claim of Pintz (1991). Moreover, we prove an analogous result for the Chebyshev function θ(t) and discuss the extent to which one can make related claims unconditionally.

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