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Q: Is there a reference for a detailed proof of Riemann's explicit formula ?

I am reading the TeXified version of Riemann's manuscript, but sometimes I can't follow (I think the author has kept typos from the orignal paper).

Here are some points I have trouble with (but there are others) :

  • How does he calculate $$\int_{a+i\mathbb{R}} \frac{d}{ds} \left( \frac{1}{s} \log(1-s/\beta)\right) x^s ds$$ on page 4 ?

  • What do I need to know about $Li(x)$ to see how the terms $Li(x^{1/2+i\alpha})$ appear ?

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2 Answers 2

up vote 3 down vote accepted

I would recommend reading Edwards' excellent 'Riemann's Zeta Function' (it's cheap !) since it details every part of Riemann's article (with more rigorous proofs of course !).

Concerning your second question see page 27 of Edwards' book.
About the $\mathrm{Li}$ function you probably won't need more than Wikipedia's informations.
What is really required to understand Riemann's proof is a good knowledge of complex integration (the venerable Whittaker and Watson may be useful for this).

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Answering the question in the title, not the body, Riemann's explicit formula is stated on page 244 of Stopple, A Primer of Analytic Number Theory, and discussed over the next several pages. By the way, it's considered that Riemann only gave a "heuristic proof," the first rigorous proof being given by von Mangoldt in 1895.

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