# The Riemann Hypothesis: A Resource for the Afficionado and by Peter Borwein, Stephen Choi, Brendan Rooney, Andrea

By Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller

This ebook offers the Riemann speculation, hooked up difficulties, and a flavor of the physique of concept built in the direction of its resolution. it's special on the expert non-expert. just about all the fabric is offered to any senior arithmetic scholar, and masses is offered to somebody with a few college arithmetic. The appendices comprise a variety of unique papers that surround crucial milestones within the evolution of thought hooked up to the Riemann speculation. The appendices additionally comprise a few authoritative expository papers. those are the “expert witnesses” whose perception into this box is either important and irreplaceable.

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For a proof of this theorem, see Section 63 of [32]. Now let s = 12 + it for t ∈ R, and let N1 (T ) be the number of zeros of ζ(s) in the rectangle R with vertices at −1 − iT, 2 − iT, 2 + iT, −1 + iT (note that N1 (T ) = 2N (T ), where N (T ) is as defined in the first section of Chapter 2). 1 we have N1 (T ) − 1 = 1 2πi − ∂R ζ (s) ds, ζ(s) as long as T is not the imaginary part of a zero of ζ(s). The −1 term, which appears on the left-hand side of the equation, corresponds to the pole of ζ(s) located at s = 1.

1 Number-Theoretic Equivalences Number-theoretic equivalences of the Riemann hypothesis provide a natural method of explaining the hypothesis to nonmathematicians without appealing to complex analysis. While it is unlikely that any of these equivalences will lead directly to a solution, they provide a sense of how intricately the Riemann 46 5 Equivalent Statements zeta function is tied to the primes. We begin by repeating our definition of the Liouville function. 1. The Liouville function is defined by λ(n) := (−1)ω(n) , where ω(n) is the number of, not necessarily distinct, prime factors of n, counted with multiplicity.

The assertion that √ π(x) = Li(x) + O( x log x) is equivalent to the Riemann hypothesis [19]. The next equivalence involves Mertens’ function, for which we will need the M¨obius function. 6. The M¨ obius function, µ(n), is defined in the following way: if n has a square factor, 0 µ(n) := 1 if n = 1, k (−1) if n is a product of k distinct primes. We now define the Mertens function as follows. 7. 370]. In terms of the Mertens function we have the following equivalence. 8. The Riemann hypothesis is equivalent to 1 M (x) = O(x 2 +ε ) for every ε > 0 [155].