Jerzy Kaczorowski is a professor in the Faculty of Mathematics and Computer Science at Adam Mickiewicz University. Since 1996, he has been the Editor-in-Chief of Acta Arithmetica. He is a full member of the Polish Academy of Sciences and the Polish Academy of Arts and Sciences. He has authored around 120 research papers and specialises in analytic number theory. Alberto Perelli is a professor in the Department of Mathematics at the University of Genoa. He is the author of around 140 research papers and specialises in analytic number theory.

• Introduction; Notation; 1. Basic theory of the Selberg class; 2. Classical 𝐿-functions and the Selberg class; 3. Invariants; 4. The standard twist; 5. Nonlinear twists with K0 ≤ 1/; 6. Auxiliary results from complex analysis; 7. Nonlinear twists with K0 > 1/; 8. Converse theorems; 9. Further applications of nonlinear twists; Appendix A. Dirichlet series; Appendix B. Almost periodic functions; Appendix C. Complex analysis; Appendix D. Some useful identities and inequalities; Appendix E. Additive semigroups of finite type; References; Symbol index; Subject index.

Jerzy Kaczorowski is a professor in the Faculty of Mathematics and Computer Science at Adam Mickiewicz University. Since 1996, he has been the Editor-in-Chief of Acta Arithmetica. He is a full member of the Polish Academy of Sciences and the Polish Academy of Arts and Sciences. He has authored around 120 research papers and specialises in analytic number theory. Alberto Perelli is a professor in the Department of Mathematics at the University of Genoa. He is the author of around 140 research papers and specialises in analytic number theory.

• Introduction; Notation; 1. Basic theory of the Selberg class; 2. Classical 𝐿-functions and the Selberg class; 3. Invariants; 4. The standard twist; 5. Nonlinear twists with K0 ≤ 1/; 6. Auxiliary results from complex analysis; 7. Nonlinear twists with K0 > 1/; 8. Converse theorems; 9. Further applications of nonlinear twists; Appendix A. Dirichlet series; Appendix B. Almost periodic functions; Appendix C. Complex analysis; Appendix D. Some useful identities and inequalities; Appendix E. Additive semigroups of finite type; References; Symbol index; Subject index.

Jerzy Kaczorowski is a professor in the Faculty of Mathematics and Computer Science at Adam Mickiewicz University. Since 1996, he has been the Editor-in-Chief of Acta Arithmetica. He is a full member of the Polish Academy of Sciences and the Polish Academy of Arts and Sciences. He has authored around 120 research papers and specialises in analytic number theory. Alberto Perelli is a professor in the Department of Mathematics at the University of Genoa. He is the author of around 140 research papers and specialises in analytic number theory.

• Introduction; Notation; 1. Basic theory of the Selberg class; 2. Classical 𝐿-functions and the Selberg class; 3. Invariants; 4. The standard twist; 5. Nonlinear twists with K0 ≤ 1/; 6. Auxiliary results from complex analysis; 7. Nonlinear twists with K0 > 1/; 8. Converse theorems; 9. Further applications of nonlinear twists; Appendix A. Dirichlet series; Appendix B. Almost periodic functions; Appendix C. Complex analysis; Appendix D. Some useful identities and inequalities; Appendix E. Additive semigroups of finite type; References; Symbol index; Subject index.