On the Number of Primes Less Than a Given Magnitude

Overview

• The use of the Greek letter zeta for a function previously mentioned by Euler
• The analytic continuation of this zeta function ζ to all complex s ≠ 1
• The entire function ξ, related to the zeta function through the gamma function
• The discrete function J defined for x ≥ 0, which is defined by J = 0 and J jumps by 1/n at each prime power pⁿ.

• Two proofs of the functional equation of ζ
• Proof sketch of the product representation of ξ
• Proof sketch of the approximation of the number of roots of ξ whose imaginary parts lie between 0 and T.

• The Riemann hypothesis, that all zeros of ζ have real part 1/2. Riemann states this in terms of the roots of the related ξ function, "... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien." That is, "it is very probable that all roots are real. One would, however, wish for a strict proof of this; I have, though, after some fleeting futile attempts, provisionally put aside the search for such, as it appears unnecessary for the next objective of my investigation."

• Functional equations arising from automorphic forms
• Analytic continuation
• Contour integration
• Fourier inversion.