The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has… 1,351 more words

## Tags » Math.CA

#### 254A, Notes 5: Bounding exponential sums and the zeta function

We return to the study of the Riemann zeta function , focusing now on the task of upper bounding the size of this function within the critical strip; as seen in Exercise 43 of… 4,209 more words

#### 254A, Supplement 3: The Gamma function and the functional equation (optional)

In Notes 2, the Riemann zeta function (and more generally, the Dirichlet -functions ) were extended meromorphically into the region in and to the right of the critical strip. 8,413 more words

#### 254A, Supplement 2: A little bit of complex and Fourier analysis

We will shortly turn to the complex-analytic approach to multiplicative number theory, which relies on the basic properties of complex analytic functions. In this supplement to the main notes, we quickly review the portions of complex analysis that we will be using in this course. 5,281 more words

#### Real analysis relative to a finite measure space

In the traditional foundations of probability theory, one selects a probability space , and makes a distinction between *deterministic* mathematical objects, which do not depend on the sampled state , and… 9,931 more words

#### Hilbert's fifth problem and approximate groups

Due to some requests, I’m uploading to my blog the slides for my recent talk in Segovia (for the birthday conference of Michael Cowling) on “ 47 more words

#### Algebraic probability spaces

As laid out in the foundational work of Kolmogorov, a *classical probability space* (or probability space for short) is a triplet , where is a set, is a -algebra of subsets of , and is a countably additive probability measure on . 2,721 more words