Bernhard Riemann

The properties of a function are completely determined by its singularities and its behavior at infinity.

The distinction between algebraic functions and transcendental functions becomes clear when one studies their Riemann surfaces.

The notion of a manifold of constant curvature is fundamental for non-Euclidean geometry.

The axioms of geometry are not self-evident truths, but hypotheses whose validity must be tested against experience.

The infinitely small is the true domain of geometry.

The representation of a function by a series is only possible in a region where the function is analytic.

The theory of functions of a complex variable is the theory of harmonic functions in two dimensions.

The prime number theorem is the most important result in the analytic theory of numbers.

The zeros of the zeta function all seem to lie on the line with real part 1/2; this is a property whose proof I consider to be of the greatest importance.

The concept of a function is extended by considering surfaces on which the function is single-valued.

The integration of differential equations often depends on the properties of the functions defined by these equations.

The curvature of space is a quantity that can be determined by physical measurements.

The study of mathematics is the study of the relationships between quantities and forms.

The theory of Abelian integrals is simplified by the introduction of the Riemann surface.

The representation of a function by a power series is a local property.

The problem of the distribution of prime numbers is intimately connected with the properties of the zeta function.

The notion of a topological space is implicit in the study of manifolds.

The conformal mapping of a surface upon a plane is the geometric interpretation of an analytic function.

The properties of a function are reflected in the geometry of its Riemann surface.

The integral of a function around a singularity gives a residue which is crucial for evaluation.