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Über die Anzahl der Primzahlen unter einer gegebenen Größe (Usual English translation: On the Number of Primes Less Than a Given Magnitude) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akadademie der Wissenschaften zu Berlin. Although it is the only paper he ever published on number theory, it contains ideas which influenced thousands 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. The paper was so influential that the notation s = σ + it is used to denote a complex number while discussing the zeta function (see below) instead of the usual z=x+iy.

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 (nontrivial) 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.

• English translation of Riemann's paper
• Number theory and physics website

• Riemann's Zeta Function, H. M. Edwards, Dover, 1974, ISBN 0-486-41740-9