Understanding the Zeta function, without getting lost in the tricky paths of advanced complex analysis

Essai Scientifique , 2013 , 34 Pages

Mathématiques - Théorie des nombres

Realizing that the study of Zeta function ~(s) is dependent on the Gamma function I(s) , a function that I know well for s=x ER, I decided to search for the Zeta function in the internet ie: to get an overall satisfactory idea about the Zeta function ~(s) s=(o+i.t).

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General remarks on the Zeta function (s)

And the first 42 roots in the critical strip

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Understanding the Riemann Zeta function

And the Riemann Hypothesis

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X(s)=X(1-s) Analytic Continuation

Notice that: (1/2 + i.t) = (1/2 - i.t)

For further details see also the pdf by Theodore Yoder

An Introduction to Riemann Hypothesis

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The proof that:

(pdf by T.Yoder)

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Real s>1

s=(1/2+i.t)

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Devoted to Uncle Fotis , my Mentor in Mathematics

The easiest proof of the following:

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(1/2 + i.t) = (1/2 ­ i.t )

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Roots in the above range: 30.424876126 32.935061588

37.58617815 40.918719012 43.327073281

48.005150881 49.773832478

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Remarks about analytic continuation

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See also the pdf by Lorenzo Menici

for the complete proof of the above

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Some additional remarks on the Zeta function

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Exclusively for the members of the top club

of Prime Numbers Theorem

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Summary of the Riemann Hypothesis

(1/2 + i.t) = (1/2 ­ i.t)

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Frequently Asked Questions - General remarks on the Zeta function (s)

What is the title of this document?

The title is "General remarks on the Zeta function (s) And the first 42 roots in the critical strip"

What is this document about?

This document appears to be about the Riemann Zeta function and the Riemann Hypothesis, and it discusses aspects such as analytic continuation and remarks regarding the Zeta function and related theorems.

Who is this document devoted to?

This document is devoted to Uncle Fotis, who is referred to as the author's mentor in mathematics.

Does the document include references to external resources?

Yes, the document mentions PDFs by Theodore Yoder and Lorenzo Menici for further details and complete proofs related to the Riemann Hypothesis.

Does the document give examples of roots?

Yes, this document lists roots. They include: 30.424876126, 32.935061588, 37.58617815, 40.918719012, 43.327073281, 48.005150881, and 49.773832478.

What is the purpose of the mention of "Prime Numbers Theorem"?

It refers to the Riemann Zeta function's exclusive relevance to the members of the top club of Prime Numbers Theorem. The specifics are not elaborated upon.

• Auteur: Prof. Dr. med. John Bredakis (Auteur)
• Année de publication: 2013
• Pages: 34
• N° de catalogue: V207998
• ISBN (ebook): 9783656353768
• ISBN (Livre): 9783656354307
• Langue: anglais
• mots-clé: understanding, zeta