(open) Divergent series of inverse primes

Indirect Proof
Thread starter fresh_42
Start date Aug 26, 2020
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Divergent Inverse Primes Series

In summary, an open divergent series of inverse primes is a series in which the terms are the reciprocals of prime numbers, and the series does not have a finite limit as the number of terms increases. It differs from a convergent series of inverse primes, which has a finite limit as the number of terms increases. While these series have been studied for their mathematical properties, they do not have any known practical applications. There is no known formula for calculating the sum of an open divergent series of inverse primes, but there are methods for estimating it. It is impossible to prove that an open divergent series of inverse primes is convergent, as it continues to grow without bound by definition.

Aug 26, 2020

fresh_42

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Show by contradiction that
$$
\sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p}
$$
diverges. Which famous result is an immediate corollary?

Skeleton (proof by Pál Erdös):

$$\underbrace{p_1<\ldots <p_k<}_{\text{small primes}}\underbrace{\underbrace{p_{k+1}<\ldots}_{\text{big primes}}}_{\displaystyle{\sum_{j>k}\dfrac{1}{p_j}<\dfrac{1}{2}}}$$
Set ##N_b =\# \{n \leq N\,|\,\exists \,j>k\, : \,p_j|n\}## and ##N_s=\#\{n\leq N\,|\,p_j|n\Longrightarrow j\leq k\}.##
##N_b < \dfrac{N}{2}##
##N_s < 2^k\sqrt{N}##
##N_b+N_s < N##

Last edited: Aug 27, 2020

1. What is a divergent series of inverse primes?

A divergent series of inverse primes is a mathematical series where the terms are the reciprocals of prime numbers, and the sum of these terms diverges or does not have a finite limit. It is also known as an infinite series, as the number of terms in the series is infinite.

2. How is a divergent series of inverse primes different from a convergent series?

A convergent series is a mathematical series where the terms approach a finite limit as the number of terms increases. In contrast, a divergent series does not have a finite limit and continues to increase as more terms are added.

3. What is the significance of studying divergent series of inverse primes?

Divergent series of inverse primes have been studied extensively in mathematics as they provide valuable insights into the behavior of infinite series. They also have applications in number theory, calculus, and other branches of mathematics.

4. Can a divergent series of inverse primes be manipulated or simplified?

Unlike convergent series, divergent series of inverse primes cannot be manipulated or simplified to obtain a finite sum. However, certain techniques, such as regularization, can be used to assign a finite value to these series.

5. Are there any real-life applications of divergent series of inverse primes?

Although divergent series of inverse primes may not have direct real-life applications, they have been used in the development of mathematical theories and concepts that have practical applications. For example, the concept of divergent series has been used in the development of quantum field theory in physics.