On the average value of $\pi(t)-\text{li}(t)$

Publication date: 16 January 2022

Abstract: We prove that the Riemann hypothesis is equivalent to the condition

i n t_{2}^{x} l e f t (p i (t) - e x t l i (t) i g h t) m a t h r m d t < 0

for all

x > 2

. Here,

p i (t)

is the prime-counting function and

e x t l i (t)

is the logarithmic integral. This makes explicit a claim of Pintz (1991). Moreover, we prove an analogous result for the Chebyshev function

h e t a (t)

and discuss the extent to which one can make related claims unconditionally.

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