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
The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mist...The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.展开更多
We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating t...We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser’s theorem. We have also proved that those functions do not possess multiple zeros of order higher than 2 and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof.展开更多
This work represents the development and detailing of works in [1] [2], and work is dedicated to the promotion of the results Abels obtained modifying zeta functions. The properties of zeta functions are studied;these...This work represents the development and detailing of works in [1] [2], and work is dedicated to the promotion of the results Abels obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem.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.
文摘The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.
文摘We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser’s theorem. We have also proved that those functions do not possess multiple zeros of order higher than 2 and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof.
文摘This work represents the development and detailing of works in [1] [2], and work is dedicated to the promotion of the results Abels obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem.
