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A RidiculouslySimple Direct Proof of FLT
Analytic number theorists are often interested in the error of approximations such as the prime number theorem. In this case, the error is smaller than /log . Riemann's formula for π() shows that the error term in this approximation can be expressed in terms of the zeros of the zeta function. In , Riemann conjectured that all the "nontrivial" zeros of ζ lie on the line but never provided a proof of this statement. This famous and longstanding conjecture is known as the and has many deep implications in number theory; in fact, many important theorems have been proved under the assumption that the hypothesis is true. For example, under the assumption of the Riemann Hypothesis, the error term in the prime number theorem is .
However, one could get numerical evidence against $NP \subseteq P/poly$ of a similar sort of that found for the Riemann Hypothesis or Goldbach conjecture, by using explicit computations to show that, say, SAT up to length 10 does not have circuits of size 20 (where you let the values of 10 and 20 vary).
Human Knowledge: Foundations and Limits
The Riemann Hypothesis does not play a role in the line of argument presented in this paper, the aim of which is to show that the prime sequence is highly organized while fully opaque.
The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an Lfunction. The most famous Lfunction is the Riemann zeta function, and as well as being ubiquitous in number theory itself, Lfunctions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and SwinnertonDyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to Lfunctions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the Lfunctions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of Lfunctions and modular forms in a systematic, comprehensive, and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to Lfunctions for nonexperts and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website . I also showed how this has been created by a worldwide opensource collaboration, which we hope may become a model for others.
ShtetlOptimized » Blog Archive » Eight Signs A Claimed …
,,. Such is the logical conclusion from the preceding sections. The line of argument of the present paper did not set out to make this statement about the Riemann Hypothesis. But the statement in question is just what logically follows from the paper’s line of argument. If expression (16) is demonstrably opaque, then how could the Riemann hypothesis—proven or not—say anything comprehensible about it?
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