In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a se...In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.展开更多
We study a class of Dirichlet functions obtained as analytic continuation across the line of convergence of Dirichlet series which can be written as Euler products. This class includes that of Dirichlet L-functions. T...We study a class of Dirichlet functions obtained as analytic continuation across the line of convergence of Dirichlet series which can be written as Euler products. This class includes that of Dirichlet L-functions. The problem of the existence of multiple zeros for this last class is outstanding. It is tacitly accepted, yet not proved that the Riemann Zeta function, which belongs to this class, does not possess multiple zeros. In a previous study we provided an example of Dirichlet function having double zeros, but that function is not an Euler product function. In this paper we deal with Euler product functions and by using the geometric properties of the mapping realized by these functions, we tackle the problem of the multiplicity of their zeros.展开更多
The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Di...The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Dirichlet series. We wanted to put to work the powerful tool of the geometry of conformal mappings of these functions, which we built in previous research, in order to study the location of their non-trivial zeros. A new approach of the concept of multiplier in Riemann type of functional equation was necessary and we have shown that with this approach the non-trivial zeros of the Dirichlet function satisfying a Reimann type of functional equation are either on the critical line, or they are two by two symmetric with respect to the critical line. The Euler product general Dirichlet series are defined, a wide class of such series is presented, and finally by using geometric and analytic arguments it is proved that for Euler product functions the symmetric zeros with respect to the critical line must coincide.展开更多
All elements in the cyclic group are generated by a generator g. The number of generators of of , namely is known to be Euler’s totient function;however, the average prob...All elements in the cyclic group are generated by a generator g. The number of generators of of , namely is known to be Euler’s totient function;however, the average probability of an element being a generator has not been discussed before. Several analytic properties of have been investigated for a long time. However, it seems that some issues still remain unresolved. In this study, we derive the average probability of an element being a generator using previous classical studies.展开更多
文摘In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.
文摘We study a class of Dirichlet functions obtained as analytic continuation across the line of convergence of Dirichlet series which can be written as Euler products. This class includes that of Dirichlet L-functions. The problem of the existence of multiple zeros for this last class is outstanding. It is tacitly accepted, yet not proved that the Riemann Zeta function, which belongs to this class, does not possess multiple zeros. In a previous study we provided an example of Dirichlet function having double zeros, but that function is not an Euler product function. In this paper we deal with Euler product functions and by using the geometric properties of the mapping realized by these functions, we tackle the problem of the multiplicity of their zeros.
文摘The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Dirichlet series. We wanted to put to work the powerful tool of the geometry of conformal mappings of these functions, which we built in previous research, in order to study the location of their non-trivial zeros. A new approach of the concept of multiplier in Riemann type of functional equation was necessary and we have shown that with this approach the non-trivial zeros of the Dirichlet function satisfying a Reimann type of functional equation are either on the critical line, or they are two by two symmetric with respect to the critical line. The Euler product general Dirichlet series are defined, a wide class of such series is presented, and finally by using geometric and analytic arguments it is proved that for Euler product functions the symmetric zeros with respect to the critical line must coincide.
文摘All elements in the cyclic group are generated by a generator g. The number of generators of of , namely is known to be Euler’s totient function;however, the average probability of an element being a generator has not been discussed before. Several analytic properties of have been investigated for a long time. However, it seems that some issues still remain unresolved. In this study, we derive the average probability of an element being a generator using previous classical studies.
