Approximate evaluation of this series (exponential sum)

In summary, the series is oscillating for big primes, and can be approximated using partial/Abel summation.
  • #1
Kevin_spencer2
29
0
Let be the series

[tex] \sum_{p<N}e^{2\pi p ix}=f(x) [/tex] where the sum is intended to be

over all primes less or equal than a given N.

My question is if there are approximate methods to evaluate this series for N big , since for a big prime the exponential sum is very oscillating would it be an 'intelligent' form to evaluate it for big N?, of course we know the trivial bound [tex] f(x)<\pi(N) [/tex] however i think this is rather useless.
 
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  • #2
Maybe this is not in the ballpark of what you're looking for, but I believe you can approximate this using partial/Abel summation. We can approximate [tex]\pi(N)=\sum_{p<N} 1[/tex] and use this to approximate [tex]f(x)[/tex]:

In particular, [tex]f(x)[/tex] can be written as the Riemann-Stieltjes Integral
[tex]f(x)=\int_{1}^{N} e^{2\pi i t x} d\pi(t)[/tex]
which then can be evaluated using integration by parts to get
[tex]f(x)=\pi(t) e^{2\pi i t x} |^{N}_{1} -2\pi i x \int_{1}^{N} e^{2\pi i t x} \pi(t) dt[/tex]
Now you can use some approximations of [tex]\pi(t)[/tex] to approximate the integral, and maybe that would give a decent answer. I don't know, I haven't worked it out.
 
  • #3
Since [tex]e^{2\pi \i}[/tex] is equal to 1, and one 1 the power of anything is equal to one, function is the addition of 1 [tex]\pi(N)[/tex] times. This basically means f(x) is a constant function, but dependant on N. Not sure about my answer though...
 
  • #4
x doesn't have to be an integer.
 
  • #5
Gib Z said:
Since [tex]e^{2\pi \i}[/tex] is equal to 1, and one 1 the power of anything is equal to one, function is the addition of 1 [tex]\pi(N)[/tex] times. This basically means f(x) is a constant function, but dependant on N. Not sure about my answer though...

haha, I should have noticed that :redface:. Perhaps the original poster meant [tex]e^{2\pi i/p}[/tex], which would make the question slightly more interesting.

edit: or even better, what Hurkyl said.
 
Last edited:
  • #6
Gib Z said:
Since [tex]e^{2\pi \i}[/tex] is equal to 1, and one 1 the power of anything is equal to one, function is the addition of 1 [tex]\pi(N)[/tex] times. This basically means f(x) is a constant function, but dependant on N. Not sure about my answer though...

Just so you're clear on what was meant above: exp{2pi i x} is 1 if and only if x is an integer. It should not be thought of as exp(2 pi i) to the power x. Raising things to powers creates issues anyway with branches.
 
  • #7
Isn't each term looking for points mod p on the unit circle (you can think of a p lattice on the unit circle, and x maps to some point in the one of the the domains). You are in adding a bunch of number mod different primes in essence, which being all coprime might make it easier.

Anyway, it seemed like going down that path might produce something useful. You could even "unroll" the unit circle into a full axis and put a lattice there if it were easier (not sure it is).

Just some random ideas.
 
  • #8
Just realized that it was [itex]2 \pi i p[/itex] and not [itex] {2 \pi i \over p} [/itex]. Not sure anything I said still applies.
 

1. What is the purpose of approximate evaluation of this series?

The purpose of approximate evaluation of this series is to estimate the value of the series without having to calculate each term individually. This can save time and resources, especially for series with a large number of terms.

2. How is the approximate value of a series calculated?

The approximate value of a series is calculated using various mathematical techniques such as truncation, interpolation, and extrapolation. These methods involve approximating the series with simpler functions or using partial sums to estimate the overall value.

3. What are the limitations of approximate evaluation?

One limitation of approximate evaluation is that it may not provide an exact value for the series, as it relies on estimation techniques. It is also important to note that the accuracy of the approximation depends on the complexity of the series and the chosen method of approximation.

4. Can approximate evaluation be used for all types of series?

No, approximate evaluation may not be suitable for all types of series. It is most commonly used for infinite series with known patterns or for series with a large number of terms. For more complex or irregular series, other methods of evaluation may be more appropriate.

5. How can the accuracy of approximate evaluation be improved?

The accuracy of approximate evaluation can be improved by using more sophisticated techniques, such as higher-order truncations or more precise interpolation methods. Additionally, using a larger number of terms in the approximation can also improve the accuracy, but this may come at the cost of increased computation time.

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