This paper contains material presented by the first authors in CIMPA School at Kathmandu University.,July 26,27,28,2010,to be included in ,and is intended for a rambling introduction to number-theoretic concepts throu...This paper contains material presented by the first authors in CIMPA School at Kathmandu University.,July 26,27,28,2010,to be included in ,and is intended for a rambling introduction to number-theoretic concepts through built-in properties of(number-theoretic) special functions.We follow roughly the historical order of events from somewhat more modern point of view.§1 deals with Euler's fundamental ideas as expounded in [6] and ,from a more advanced standpoint.§2 gives some rudiments of Bernoulli numbers and polynomials as consequences of the partial fraction expansion.§3 states sieve-theoretic treatment of the Euler product.Thus,the events in §1-§3 more or less belong to Euler's era.§4 deals with RSA cryptography as motivated by Euler's function,with its several descriptions being given.§5 contains a slight generalization of Dirichlet's test on uniform convergence of series,which is more effectively used in §6 to elucidate Riemann's posthumous Fragment II than in [1].Thus §5-§6 belong to the Dirichlet-Riemann era.§7 gives the most general modular relation which is the culmination of the Riemann-Hecke-Bochner correspondence between modular forms and zeta-functions.Appendix gives a penetrating principle of the least period that appears in various contexts.展开更多
文摘This paper contains material presented by the first authors in CIMPA School at Kathmandu University.,July 26,27,28,2010,to be included in ,and is intended for a rambling introduction to number-theoretic concepts through built-in properties of(number-theoretic) special functions.We follow roughly the historical order of events from somewhat more modern point of view.§1 deals with Euler's fundamental ideas as expounded in [6] and ,from a more advanced standpoint.§2 gives some rudiments of Bernoulli numbers and polynomials as consequences of the partial fraction expansion.§3 states sieve-theoretic treatment of the Euler product.Thus,the events in §1-§3 more or less belong to Euler's era.§4 deals with RSA cryptography as motivated by Euler's function,with its several descriptions being given.§5 contains a slight generalization of Dirichlet's test on uniform convergence of series,which is more effectively used in §6 to elucidate Riemann's posthumous Fragment II than in [1].Thus §5-§6 belong to the Dirichlet-Riemann era.§7 gives the most general modular relation which is the culmination of the Riemann-Hecke-Bochner correspondence between modular forms and zeta-functions.Appendix gives a penetrating principle of the least period that appears in various contexts.
