Riemann 1859 — Über die Anzahl der Primzahlen unter einer gegebenen Grösse

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• builds on → Euler 1737 — Variae observationes circa series infinitas (E72) — zeta continued to the plane, the functional equation, the explicit formula through log zet
• enables → Mangoldt 1905 — Zur Verteilung der Nullstellen der Riemannschen Funktion xi(t) — N(T) = (T/2pi) log(T/2pi) - T/2pi + O(log T): Riemann's count of the zeros proven
• connects at → The Riemann hypothesis — where the logarithm enters: the explicit formula runs through log zeta - the product over
• builds on → Dirichlet 1837 — Beweis des Satzes, dass jede unbegrenzte arithmetische Progression ... unendlich viele Primzahlen enthält — zeta continued to the plane, the functional equation, the explicit formula through log zet
• builds on → Gauss 1849 — Letter to Encke, 24 December 1849 (the count made in 1792–93) — zeta continued to the plane, the functional equation, the explicit formula through log zet
• builds on → Chebyshev 1852 — Mémoire sur les nombres premiers — zeta continued to the plane, the functional equation, the explicit formula through log zet
• enables → Hadamard 1896 — Sur la distribution des zéros de la fonction zeta(s) et ses conséquences arithmétiques — the prime number theorem: no zero on the line Re s = 1
• enables → Poussin 1896 — Recherches analytiques sur la théorie des nombres premiers — the prime number theorem, independently, with the first zero-free region