Dirichlet series

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In summary, the conversation discusses the series g(s) in the form of a Dirichlet series and the attempt to obtain an exact or approximate expression for it. Other techniques, such as partial summation and Mellin transform, are also mentioned in the discussion. The conversation ends with a question about the number of accounts the person has on the platform.
  • #1
Let be the series in the form [tex] g(s)= \sum_{1 \le n } |\Lambda (n) |^{2} n^{-s} [/tex] where lambda is Von Mangoldt function, my question is how could i get an exact or at least almost exact expresion for g(s) . My other question is how could i obtainthe Mellin transform of the function [tex] \Lambda (n+2) \Lambda (n+1) [/tex] i have tried sum by parts but got no results.
 
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  • #2
of course i tried (wrong ??) with the property, given a Dirchlet series [tex] g(s)= \sum_{1 \le n} f(n)n^{-s} [/tex] then [tex] -g'(s)/g(s)= \sum_{1 \le n} f(n) \Lambda (n) n^{-s} [/tex] but i think this does not work unless f(n) is multiplicative, however could it be used at least as a good approximation or modifying it a bit coul work??
 
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  • #3
Welcome back, Jose.
 
  • #4
readng Tom Apostol's 'Introduction to Analytic number theory' the desired series is just [tex] AT^{-1} \int_{-T}^{T}dt | \frac{ \zeta ' (a+it)}{\zeta (a+it)} |^{2} = \sum_{1 \le n} | \Lambda (n) |^{2} n^{2a} [/tex] , with a>1 and T tending to infinity. but i don't know the value of the integral, could someone provide an approximate value of the integral over t (-T,T) above ??, thank you.
 
  • #5
How many accounts has this guy made on here?
 
  • #6
I have also tried using partial summation so:

[tex] \sum_{n \le x } \Lambda (n+2) \Lambda (n) = B(x)= \Lambda (x+2) \Psi (x) - \sum_{n \le x } \Psi (x) ( \Lambda (x+2) - \Lambda (x+1) ) [/tex]

so [tex] g(s)= \sum_{1 \le n} \Lambda (n+2) \Lambda (n) n^{-s}= s \int_{1}^{\infty}B(x) x^{- (s+1)} [/tex]

to obtain 'g(s)' any hint please?? thanx.
 

1. What is a Dirichlet series?

A Dirichlet series is an infinite series of the form $\sum_{n=1}^\infty \frac{a_n}{n^s}$, where $a_n$ are complex coefficients and $s$ is a complex variable. These series have applications in number theory, complex analysis, and other areas of mathematics.

2. What is the Dirichlet series convergence theorem?

The Dirichlet series convergence theorem states that if a Dirichlet series converges at a point $s_0$ in the complex plane, then it converges absolutely at all points $s$ with $\mathrm{Re}(s) > \mathrm{Re}(s_0)$. This theorem is useful for determining the convergence of Dirichlet series.

3. How are Dirichlet series related to the Riemann zeta function?

The Riemann zeta function, denoted by $\zeta(s)$, is closely related to Dirichlet series. In fact, the Riemann zeta function can be expressed as a Dirichlet series: $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. This connection allows for the study of the Riemann zeta function using techniques from Dirichlet series theory.

4. What is the Dirichlet series analytic continuation?

The analytic continuation of a Dirichlet series is the process of extending the definition of the series to a larger region in the complex plane. This allows for the evaluation of the series at points where it may not have initially converged. Analytic continuation is a powerful tool for studying the behavior of Dirichlet series.

5. How are Dirichlet series used in number theory?

Dirichlet series have many applications in number theory. For example, the Riemann zeta function, which can be expressed as a Dirichlet series, is closely related to the distribution of prime numbers. Other Dirichlet series, such as the Dirichlet L-series, are also important in number theory, particularly in the study of quadratic forms and quadratic characters.

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