Change search

Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf
An introduction to the Riemann hypothesis
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2014 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
##### Abstract [en]

This paper exhibits the intertwinement between the prime numbers and the zeros of the Riemann zeta function, drawing upon existing literature by Davenport, Ahlfors, et al.

We begin with the meromorphic continuation of the Riemann zeta function $\zeta$ and the gamma function $\Gamma$ . We then derive a functional equation that relates these functions and formulate the Riemann hypothesis.

We move on to the topic of nite-ordered functions and their Hadamard products. We show that the xi function $\xi$ is of finite order, whence we obtain many useful properties. We then use these properties to and a zero-free region for $\zeta$ in the critical strip. We also determine the vertical distribution of the non-trivial zeros.

We finally use Perron's formula to derive von Mangoldt's explicit formula, which is an approximation of the Chebyshevfunction $\psi$ . Using this approximation, we prove the prime number theorem and conclude with an implication ofthe Riemann hypothesis.

2014. , 41 p.
Mathematics
##### Identifiers
OAI: oai:DiVA.org:kth-153636DiVA: diva2:752775
##### Educational program
Master of Science in Engineering -Engineering Physics
##### Supervisors
Available from: 2014-10-06 Created: 2014-10-06 Last updated: 2014-10-09Bibliographically approved

#### Open Access in DiVA

##### File information
File name FULLTEXT01.pdfFile size 1843 kBChecksum SHA-512
7b1fb08c2e36499bdf1fe82c21f6bae06302264567164781215f3cf64f343d2b40200dbe516091933f6ee5e08cdbb9cb0814f170eedf902a64960eecddacc29b
Type fulltextMimetype application/pdf
##### By organisation
Mathematics (Dept.)
Mathematics

#### Search outside of DiVA

The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available
urn-nbn

#### Altmetric score

urn-nbn
Total: 236 hits

Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf
v. 2.29.1
|