In the first part of the talk we introduce the notion of an abstract twist of a Dirichlet series and discuss its analytic properties.

In the second part we show some classical examples of twists of L-functions. In principio we discuss a classical theorem of Hecke describing analytic properties of a twist of L-function associated to a cusp form by a Dirichlet character and the famous Weil converse theorem. Later we brieflžy review basic properties of an additive twist and explain asymmetry between multiplicative and additive twists.

In the last part we argue how the properties of a standard non-linear twist are used to obtain a new proof of a well-known Hardy Ω-theorem for the error term in the Dirichlet divisor problem and how this new method clarifies the presence of the exponent 1/4.