Conjecture about Dirichlet series.

In summary, the conversation discusses the idea of using Ramanujan resummation to obtain a regularized sum for a Dirichlet series that converges for Re(s) > a. It is suggested that this can be achieved in the form of a tail integral with constants c, a, and b, and the result can be recovered for a series with all terms equal to 1. The question is raised whether this integral will always evaluate in the given form.
  • #1
mhill
189
1
Hi, i hope it is not a crack theory

i had the idea when reading Ramanujan resummation, i believe that for a Dirichlet series which converges for Re (s) > a , with a a positive real number then we can obtain a regularized sum of the divergent series in the form:

[tex] \sum_{n >1}a_{n}n^{-s}- C(s-a)^{b} [/tex]

with C, a and b real numbers , in case a_n are all 1 we recover the result

[tex] \sum_{n >1}n^{-s}- (s-1)^{-1} [/tex]

however i believe that 'Ramanujan resummation of series' given by

[tex] \sum_{n >1}a_{n}n^{-s}-\int_{1}^{\infty}a(x)x^{-s} [/tex]

must be satisfied no matter what kind of numerical series is.
 
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  • #2
I think you're saying that the tail integral which is characteristic of Ramanujan resummation can be put in the form c(s-a)^b always for a, b, c constants. However can it be shown that the integral will evaluate in the given form always?
 

FAQ: Conjecture about Dirichlet series.

What is a Dirichlet series?

A Dirichlet series is an infinite series of the form $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^s}$, where $s$ is a complex variable and $a_n$ are complex coefficients. It is named after the mathematician Peter Gustav Lejeune Dirichlet.

What is the conjecture about Dirichlet series?

The conjecture about Dirichlet series, also known as the Dirichlet series conjecture, is a mathematical conjecture proposed by the mathematician Edmund Landau in 1911. It states that for any Dirichlet series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ with non-negative coefficients $a_n$, if the series converges at a point $s_0$, then it converges uniformly in the half-plane $\text{Re}(s) > s_0$.

What is the significance of the conjecture about Dirichlet series?

If the conjecture about Dirichlet series is true, it would have important implications in number theory and complex analysis. It would provide a powerful tool for studying the behavior of Dirichlet series and their associated functions.

Has the conjecture about Dirichlet series been proven?

No, the conjecture about Dirichlet series has not been proven. It remains an open problem in mathematics and continues to be an active area of research.

Are there any known counterexamples to the conjecture about Dirichlet series?

No, there are no known counterexamples to the conjecture about Dirichlet series. However, there are some known cases where the conjecture has been proven to be true, such as when the coefficients $a_n$ are all equal to 1.

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