Über die Anzahl der Primzahlen unter einer gegebenen Grösse
E47355
Über die Anzahl der Primzahlen unter einer gegebenen Grösse is Bernhard Riemann’s seminal 1859 paper that introduced the Riemann zeta function and laid the foundations of analytic number theory, including the famous Riemann Hypothesis.
Aliases (4)
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical paper
→
scientific article → work on number theory → |
| alsoKnownAs |
On the Number of Primes Less Than a Given Magnitude
→
Riemann’s 1859 memoir on primes → |
| associatedConjecture |
Riemann Hypothesis
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|
| author |
Bernhard Riemann
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|
| authorBirthYear |
1826
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|
| authorDeathYear |
1866
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|
| centuryOfPublication |
19th century
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|
| concerns |
approximation of π(x) by the logarithmic integral li(x)
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asymptotic behavior of the prime-counting function π(x) → error term in the prime number theorem → |
| countryOfPublication |
Kingdom of Prussia
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| defines |
Riemann zeta function
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|
| field |
analytic number theory
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complex analysis → number theory → |
| historicalSignificance |
foundational work of analytic number theory
→
introduced complex-analytic methods into the study of primes → |
| influenceOn |
Hilbert–Pólya conjecture
→
development of analytic number theory → modern research on L-functions → prime number theorem → spectral interpretations of zeros of ζ(s) → |
| institutionalContext |
Königlich Preußische Akademie der Wissenschaften zu Berlin
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|
| introduces |
Riemann Hypothesis
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|
| length |
approximately 8 pages
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|
| originalLanguage |
German
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|
| proposes |
all non-trivial zeros of the Riemann zeta function have real part one-half
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|
| publicationDate |
1859-11-19
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|
| publicationYear |
1859
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|
| publishedIn |
Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin
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|
| states |
explicit formula relating π(x) to zeros of ζ(s)
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|
| topic |
Euler product
→
Riemann Hypothesis → Riemann zeta function → analytic continuation → distribution of prime numbers → explicit formulas in prime number theory → functional equation of the Riemann zeta function → non-trivial zeros of the zeta function → prime-counting function → |
| usesMethod |
analytic continuation of ζ(s)
→
complex analysis of Dirichlet series → contour integration → functional equation of ζ(s) → |
Referenced by (5)
| Subject (surface form when different) | Predicate |
|---|---|
|
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
("On the Number of Primes Less Than a Given Magnitude")
→
Über die Anzahl der Primzahlen unter einer gegebenen Grösse ("Riemann’s 1859 memoir on primes") → |
alsoKnownAs |
|
Riemann zeta function
("Riemann 1859 memoir "On the Number of Primes Less Than a Given Magnitude"")
→
|
historicalPaper |
|
Bernhard Riemann
→
|
notableWork |
|
Riemann hypothesis
("Riemann's 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse"")
→
|
statedIn |