Convergence of a sum over primes

In summary, the claim is that, given a nonincreasing sequence of positive numbers, then the sum of its prime indices converges. If the intuitive step of proving that the slope of the linear interpolation grows with order 1/log(x) is impossible to rigorously do, then the claim is unproven.
  • #1
Boorglar
210
10
I am trying to understand a condition for a nonincreasing sequence to converge when summed over its prime indices. The claim is that, given [itex]a_n[/itex] a nonincreasing sequence of positive numbers,
then [itex]\sum_{p}a_p[/itex] converges if and only if [itex]\sum_{n=2}^{\infty}\frac{a_n}{\log(n)}[/itex] converges.

I have tried various methods to prove this but my error estimates are always too large.
The closest I came to a proof is this: first, extend the sequence [itex]a_n = a(n)[/itex] to positive reals by "connecting the dots" (interpolating by some nondecreasing function that takes on the same values as [itex]a_n[/itex] on the integers. Then, do the same for [itex]\pi(x)[/itex] (prime counting function) and [itex]p(x)[/itex] (the n-th prime). The goal is to use the integral test to relate the two sums.

So [itex]\sum_{p}a_p[/itex] converges if and only if [itex]\int_{1}^{\infty}a(p(x))dx[/itex] converges.
Using the substitution [itex] t = p(x) [/itex] (so [itex]x = \pi(t), dx = \pi'(t)dt[/itex]), the second integral equals [itex]\int_{2}^{\infty}a(t)\pi'(t)dt[/itex]. By the Prime Number Theorem, [itex]\pi(t) = \frac{t}{\log(t)} + O\left(\frac{t}{\log^2(t)}\right)[/itex], so the derivative is (intuitively) [itex]\pi'(t) = \frac{1}{\log(t)} + O\left(\frac{1}{\log^2(t)}\right)[/itex]. So, assuming this "intuition" is correct, the integral is [itex]\int_{2}^{\infty}\frac{a(t)}{\log(t)}dt + ...[/itex] where the ellipsis are terms of lower order than the main term. This integral converges if and only if [itex]\sum_{n=2}^{\infty}\frac{a_n}{\log(n)}[/itex] converges.

That would be good, but I am unable to prove the "intuitive" step. I need some estimate on the order of the derivative of [itex]\pi(x)[/itex], but the only information I have is the big-Oh of the function, not its derivative.
 
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  • #2
can pi(x) be defined for non integer x as the linear interpolation between neighboring integer values? then the derivative has some meaning and the proof looks pretty convincing
 
  • #3
Well, that's what I want to do. My problem is how to rigorously prove that the slopes of the linear interpolation grow with order 1/log(x). The main issue is that pi(x) will be constant most of the time, except at the primes where it jumps by 1, so there will be infinitely many points where the slope will be 1, so it can't be O(1/log(x)).
 
  • #4
pi(x) , meaning the linearly interpolated version, has a slope of 1/distance between neighboring primes, doesn't it?
 
  • #5
Oh oops. Yeah it would be 1 / (p(n+1) - p(n)). But still, assuming the twin prime conjecture is true, there would be infinitely many cases where the slope is 1/2... Clearly this happens rarely, but does it compensate for the desired O(1/log x) order?
 
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Related to Convergence of a sum over primes

1. What is the "Convergence of a sum over primes"?

The "Convergence of a sum over primes" is a mathematical concept that refers to the behavior of a series where the terms are sums of prime numbers. It is a fundamental topic in number theory and has important applications in various fields such as cryptography and computer science.

2. How is the convergence of a sum over primes determined?

The convergence of a sum over primes is determined by analyzing the behavior of the partial sums of the series. If the partial sums approach a finite limit as the number of terms increases, then the series is said to converge. Otherwise, if the partial sums diverge or oscillate, the series is said to diverge.

3. What is the significance of the convergence of a sum over primes?

The convergence of a sum over primes is significant because it helps to understand the distribution of prime numbers and their properties. It also has important applications in various areas of mathematics, including number theory, analysis, and algebra.

4. Are there any known results or theorems related to the convergence of a sum over primes?

Yes, there are several known results and theorems related to the convergence of a sum over primes. Some of the most well-known ones include the Prime Number Theorem, the Goldbach Conjecture, and the Riemann Hypothesis. These results provide insights into the behavior of prime numbers and their relationship to other mathematical concepts.

5. Is the convergence of a sum over primes a solved problem?

No, the convergence of a sum over primes is still an open problem in mathematics. While there have been many significant results and conjectures related to this topic, there is still much to be discovered and understood about the convergence of sums over primes. It remains an active area of research for mathematicians.

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