Abstract
In this chapter I would like to interrupt the historic line in order to put into evidence what I just said, namely that the proof of RHp could have been found already in 1937, in the framework of the theory of function fields. I will present here such a proof. In principle it can be regarded as a translation of Severi’s proof from the language of algebraic geometry into the language of algebra. But I will not use any knowledge of the terminology and results of algebraic geometry. I shall use those notions and facts from the theory of function fields which were available to and preferred by Hasse at the time of the Göttingen workshop which I have discussed above.
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There is some ambiguity in the use of the words “integral” and “integer”. In analysis an “integral” means the result of “integrating” a differential f(x)dx, the result being denoted by \(\int f(x)dx\) in the notation of Leibniz. In number theory the attribute “integral” is sometimes used in the meaning of “being an integer”. In order to avoid misunderstandings I do not use here “integral” in this meaning. Instead, I use “integer” also as an attribute. In the theory of function fields this is interpreted as having no denominator or pole; this can happen not only for divisors but also for differentials. Classically one says “differential of the first kind” instead of “integer differential.”
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Roquette, P. (2018). A Virtual Proof. In: The Riemann Hypothesis in Characteristic p in Historical Perspective. Lecture Notes in Mathematics(), vol 2222. Springer, Cham. https://doi.org/10.1007/978-3-319-99067-5_10
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