欧拉-高斯-狄利克雷-黎曼间和谐的灵感(英文)

L'estro Armonico del Circolo di Euler-Gauss-Dirichlet-Riemann

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.

This paper contains material presented by the first authors in CIMPA School at Kathmandu University. , July 26, 27, 28, 2010, to be included in [ 13 ], 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 [ 16], 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 treat- ment 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 uni- form 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.