Bernhard Riemann

The theory of functions of a complex variable is the theory of the conformal mapping of a surface upon a plane.

The zeta function converges for all real values of s greater than 1.

One finds in fact approximately this number of prime numbers within these limits; however, it is found that the density of prime numbers is on average inversely proportional to the logarithm.

It is very probable that all roots [of the zeta function] are real. One would of course like to have a rigorous proof of this; however, I have left this research aside for the time being after some fleeting vain attempts.

The integral of a function along a path in the complex plane depends not merely on the endpoints but on the topology of the region.

The concept of a multiply extended magnitude is one which can be developed in different ways.

I have in this memoir treated the problem of the distribution of prime numbers, a subject which may seem to belong to pure mathematics alone, but whose investigation, as I hope to show, is profitable for the knowledge of the construction of the series of whole numbers.

The principles of geometry are not derivable from general notions of magnitude, but the properties which distinguish space from other thinkable triply extended magnitudes can only be deduced from experience.

The measure of curvature at a point on a surface is an intrinsic property, independent of the surrounding space.

The possibility of representing a surface upon a plane in such a way that the representation is similar in its smallest parts to the surface represented, is the foundation of the theory of functions.

The notion of a manifold is fundamental; it is the concept of a collection of elements which can be determined by a certain number of continuous or discrete variables.

The analytical treatment of geometry is based upon the notion of coordinates, and the idea of a manifold is a generalization of this notion.

The theory of functions of a complex variable derives its immense importance from the fact that it is the theory of the conformal mapping of surfaces.

The representation of a given surface upon another in such a way that the smallest parts are similar is possible only when the two surfaces have the same constant measure of curvature.

The number of prime numbers less than a given quantity x is approximately equal to the integral from 2 to x of dt/log t.

The connection between the theory of numbers and the theory of functions is one of the most profound and fruitful in all of mathematics.

The concept of a Riemann surface is essential for the complete understanding of many-valued functions.

The integral of a function along a closed path is zero if the function is single-valued and analytic within the region enclosed by the path.

The problem of the conformal representation of a simply connected plane surface upon another is always solvable.

The theory of elliptic functions is transformed and clarified when one considers the corresponding Riemann surface, the torus.