On the Number of Primes Less Than a Given Magnitude

Helping orphans the way you would do it

On the Number of Primes Less Than a Given Magnitude (or Ueber die Anzahl der Primzahlen unter einer gegebenen Grősse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory.

Among the new definitions introduced:

The analytic continuation of the Riemann zeta function ζ(s) to all complex s ≠ 1
The entire function ξ(s)
The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power pⁿ

Among the proofs and sketches of proofs:

Two proofs of the functional equation of ζ(s)
"Proof" of the product representation of ξ(s)
"Proof" of the approximation of the number of roots of ξ(s) whose imaginary part lies between 0 and T

Among conjectures made:

The Riemann hypothesis, that all zeros of ζ(s) have real part 1/2

New methods and techniques used in number theory:

Analytic continuation (although not in the spirit of Weierstrass)
Contour integration
Fourier inversion

Riemann also discussed the relationship between ζ(s) and the distribution of the prime numbers, using the function J(x) essentially as a measure for Stieltjes integration. He then obtained the main result of the paper, a formula for J(x), by comparing with ln(ζ(s)). Riemann then attempted to make an approximate formula for the prime-counting function π(x), although he himself admits he is aware of the defects of his arguments.

External links:

References:

Riemann's Zeta Function, H. M. Edwards, Dover, 1974, ISBN 0486417409