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eljose79
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given the Dirichlet series Sum(0,infinite)a(n)n-^s=f(s) would be an analityc formula to invert the Dirichlet series to obtain a(n)?..i have searched it at google but found no results.
eljose79 said:Another question ..how do you obtain the convergence radius for a Dirichlet series? in fact is this the criterion? if the series is given by a(n)n^-s then s must satisfy taking logarithms s>nlog[a(n)/a(n+1)] with [x] the modulus of x,is that true?...(i have taken the criterion that [b(n+1)/b(n)]<1 with b(n)=a(n)n-^s it gives the result [a(n+1)/a(n)].[1/1+1/n]^s taking logarithm and knowing [Log(1/1+1/n)]=1/n
eljose79 said:Thanks for your explanation fo Perron,s formula..accoridng to that it can not invert the Fourier series but only to give a finite sum of the coefficients a(n) but not any a(n) itself i was looking for a formula that given g(s) sum of idirichlet series would give us an expresion for a(n)
eljose79 said:But for evaluating integrals in the form Int(c-i8,c+i8) involving Riemann,s function R(s) and its derivatives (8 here means infinity) will be any numerical method won,t it?
eljose79 said:Another question, how do you obtain the half plane of convergence for the Dirichlet series?...
eljose79 said:Note: where i could find the prove that for mu(n)=O(n^e) e>0 or that the series with general term mu(n)/n^-s converges for s>1?..
A Dirichlet series inversion is a mathematical operation that allows one to find the coefficients of a function given its Dirichlet series representation. This is useful for studying the behavior and properties of a function, as well as for solving problems in number theory, analysis, and other fields.
Dirichlet series inversion is based on the fact that a Dirichlet series can be written as an infinite sum of coefficients multiplied by powers of the variable s. By manipulating this series, one can find a closed-form expression for the coefficients, which can then be used to reconstruct the original function.
Dirichlet series inversion has various applications in mathematics, including number theory, complex analysis, and probability theory. It can be used to study the distribution of prime numbers, evaluate infinite sums and integrals, and solve problems related to modular forms and special functions.
Like any mathematical technique, Dirichlet series inversion has its limitations. It may not always be possible to find a closed-form expression for the coefficients of a function, or the series may not converge for certain values of the variable s. Additionally, the inversion process can be time-consuming and may require advanced mathematical skills.
While Dirichlet series inversion is primarily a theoretical tool, it has been applied in various real-world problems, such as in signal processing and data analysis. It can also be used to approximate functions and solve differential equations, making it a valuable tool in engineering and scientific research.