General Dirichlet series
Introduction In the field of mathematical analysis, a '''general Dirichlet series''' is an infinite series that takes the form of: where ,  are complex numbers and  is a strictly increasing sequence of nonnegative real numbers that tends to infinity.A simple observation shows that an 'ordinary' Dirichlet series: is obtained by substituting  while a power series: is obtained when .
Fundamental theorems If a Dirichlet series is convergent at , then it is uniformly convergent in the domain: and convergent for any  where .There are now three possibilities regarding the convergence of a Dirichlet series, i.e. it may converge for all, for none or for some values of ''s''. In the latter case, there exist a  such that the series is convergent for  and divergent for . By convention,  if the series converges nowhere and  if the series converges everywhere on the complex plane.
Abscissa of convergence The '''abscissa of convergence''' of a Dirichlet series can be defined as  above. Another equivalent definition is: The line  is called the '''line of convergence'''. The '''half-plane of convergence''' is defined as: The abscissa, line and half-plane of convergence of a Dirichlet series are analogous to radius, boundary and disk of convergence of a power series.On the line of convergence, the question of convergence remains open as in the case of power series. However, if a Dirichlet series converges and diverges at different points on the same vertical line, then this line must be the line of convergence. The proof is implicit in the definition of abscissa of convergence. An example would be the series: which converges at  (alternating harmonic series) and diverges at  (harmonic series). Thus,  is the line of convergence.Suppose that a Dirichlet series does not converge at , then it is clear that  and  diverges. On the other hand, if a Dirichlet series converges at , then  and  converges. Thus, there are two formulas to compute , depending on the convergence of  which can be determined by various convergence tests. These formulas are similar to the Cauchy–Hadamard theorem for the radius of convergence of a power series.If  is divergent, i.e. , then  is given by: If  is convergent, i.e. , then  is given by:
Abscissa of absolute convergence A Dirichlet series is absolutely convergent if the series: is convergent. As usual, an absolutely convergent Dirichlet series is convergent, but the converse is not always true.If a Dirichlet series is absolutely convergent at , then it is absolutely convergent for all ''s'' where . A Dirichlet series may converge absolutely for all, for no or for some values of ''s''. In the latter case, there exist a  such that the series converges absolutely for  and converges non-absolutely for .The '''abscissa of absolute convergence''' can be defined as  above, or equivalently as: The '''line''' and '''half-plane of absolute convergence''' can be defined similarly. There are also two formulas to compute .If  is divergent, then  is given by: If  is convergent, then  is given by: In general, the abscissa of convergence does not coincide with abscissa of absolute convergence. Thus, there might be a strip between the line of convergence and absolute convergence where a Dirichlet series is conditionally convergent. The width of this strip is given by: In the case where ''L'' = 0, then: All the formulas provided so far still hold true for 'ordinary' Dirichlet series by substituting .
Other abscissas of convergence It is possible to consider other abscissas of convergence for a Dirichlet series. The '''abscissa of bounded convergence'''  is given bywhile the '''abscissa of uniform convergence'''  is given byThese abscissas are related to the abscissa of convergence  and of absolute convergence  by the formulas,and a remarkable theorem of Bohr in fact shows that for any ordinary Dirichlet series where  (i.e. Dirichlet series of the form ) ,  and  Bohnenblust and Hille subsequently showed that for every number  there are Dirichlet series  for which A formula for the abscissa of uniform convergence  for the general Dirichlet series   is given as follows: for any , let , then
Analytic functions A function represented by a Dirichlet series: is analytic on the half-plane of convergence. Moreover, for :
Further generalizations A Dirichlet series can be further generalized to the multi-variable case where , ''k'' = 2, 3, 4,..., or complex variable case where , ''m'' = 1, 2, 3,...
References * G. H. Hardy, and M. Riesz, ''The general theory of Dirichlet's series'', Cambridge University Press, first edition, 1915. * E. C. Titchmarsh, ''The theory of functions'', Oxford University Press, second edition, 1939. * Tom Apostol, ''Modular functions and Dirichlet series in number theory'', Springer, second edition, 1990. * A.F. Leont'ev, ''Entire functions and series of exponentials'' (in Russian), Nauka, first edition, 1982. * A.I. Markushevich, ''Theory of functions of a complex variables'' (translated from Russian), Chelsea Publishing Company, second edition, 1977. * J.-P. Serre, ''A Course in Arithmetic'', Springer-Verlag, fifth edition, 1973. * John E. McCarthy, '' Dirichlet Series'', 2018. * H. F. Bohnenblust and Einar Hille, ''  On the Absolute Convergence of Dirichlet Series'',  Annals of Mathematics, Second Series, Vol. 32, No. 3 (Jul., 1931), pp. 600-622.
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