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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Ueber 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 Akademie 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 = + it is used to denote a complex number while discussing the zeta function (see below) instead of the usual z = x + iy. (The notation s = + it was begun by Edmund Landau in 1903.)