Can Trigonometric Series be Evaluated Using Euler-Maclaurin Sum Formula?

[eljose]

Let be the series:

$$\sum_{n} e^{if(n)}$$ where f is a function perhaps a Polynomial ..then my question is..how can this series to be evaluated (at least approximately) ?..perhaps using Euler-Bernoulli sum formula, and another question what are they used for?, i heard in a book that Goldbach conjecture could be proved using them.

Another question if we have..$$\sum_{n} e^{if(n)}$$ summed over the integers or a subset of integers..could the numbers f(n) be considered "frecuencies of vibration" or eigenvalues of a certain operator?..in fact there is an interesting connection with Physics ..if we define the partition function:

$$Z(u)= \sum_{n>0}e^{-uE(n)}$$ under "complex rotation" (u-->ix ) the partition function becomes a trigonometric sum..where in this case E(n) are the "energies" (eigenvalues) of a certain Hamiltonian.

 

[Office_Shredder]

If f(n) is a polynomial (just for the top part specifically), then you would have trouble finding one that has a convergent series (I would imagine you get a random walk type effect

 

[shmoe]

If f(n) is a polynomial with real coefficients, the sum can be estimated decently (this sum over a finite range of course). If it's linear you are looking at a geometric series, so no problem. If it's higher degree, you can look at the square of the modulus of the sum (i.e. multiply by the complex conjugate), you'll get a double sum with terms like exp(i*(f(m+n)-f(m)). f(m+n)-f(m) is a polynomial in m of smaller degree, so you can use this recursively to get down to the linear case.

An excellent place to start for this stuff is Graham and Kolesnik's "van der Corput's method of exponential sums". The Iwaniec and Kowalski Analytic Number theory text has some great info as well (no suprise).

Applications are all over, equidistribution of sequences modulo 1 (see "Weyl's criteria"), estimating the zeta function, the circle problem (counting lattice points inside a circle), the divisor problem (counting lattice points under a hyperbola), etc. Vinogradov's proof of the ternary goldbach required (among other things) estimates for certain exponential sums over primes.

 

[shmoe]

If f(n) is a polynomial (just for the top part specifically), then you would have trouble finding one that has a convergent series (I would imagine you get a random walk type effect

Ran across this page in the "free math textbook thread":

http://www.math.harvard.edu/~elkies/M259.98/index.html

the background is what's relevant here. This is a plot of the partial sums of the series exp(2*Pi*i*n^(3/2)/5), n ranging from 1 to "something". In going from one swirl to the next, the derivative of the n^(3/2)/5 part needs to be close to an integer, then (n+1)^(3/2)/5 - n^(3/2)/5 will be close to an integer so the corresponding exponential terms in the sum will be in the same direction and the can conspire to get the sum going in that direction. The swirly bits correspond to stretches where n^(3/2)/5 mod 1 is hopping around the interval [0,1) kinda randomly. You can work out the ranges

that's a vague description of what's going on, it can be explained much more precisely via van der corput's stuff, see the references I gave above.

For polynomials, if you are summing exp(2*Pi*i*f(n)) where f(n) is a polynomial of degree d>0 with lead coefficient y, then the sum from 1 to N will be o(N) if y is not rational. This is equivalent to saying the values f(n) mod 1 are equidistributed in the unit interval, essentially there will be enough cancellation in the exponential sum from things pointing in different directions to keep the sum "small". How small will depend on how well y can be approximated by rationals (meaning what can you say about the denominator of the approximation relative to how close this approximation is).

 

[eljose]

And HOw about using the method of the "Euler-Bernoulli" sum formula:

$$\int_{0}^{N}dx e^{if(x)}= C+ \sum_{n=1}^{N-1}e^{if(n)}-\sum_{r=1}^{\infty}D^{2r-1}e^{if(x)}|_{0}^{N}$$ ?

This simply would give an approximate evaluation of the Trigonometric sum, supposing f(x) is "smooth" enough so it can be applied.

 

[shmoe]

You mean Euler-Maclaurin* and it really depends on what you are summing. Applying it directly to something like the dirichlet series for the zeta function for example turns out not so hot (depending also on what you are trying to do), but you can handle the 'tail' of the Dirichlet series fairly well this way to approximate zeta with a finite sum of not "too many" terms.

It really depends on f on how well the sum is approximated by the corresponding integral. One of the key parts of van der Corput's approach was to replace the sum with integrals (then evaluate these via stationary phase or the like). You can do this by the first order E-M summation formula and replacing the periodic bernoulli function by it's corresponding Fourier series and integrating term by term. How many of these that end up providing a major contribution depends on how f' varies throughout the interval (essentially you'll get a term for each point where f'/2/Pi takes on an integer value, this expalins the 'swirls' in that background picture. By the way, the exponent is usually normalized to be exp(2*Pi*i*f(n)), a standard function in number theory texts is e(x)=exp^(2*Pi*i*x)).

Pick up either of the references I suggested. Basics can be had in books on the Zeta function as well, like Titchmarsh or Ivic.



*it's somewhat ironic that the part of it that makes you want to add "Bernoulli" to the name is, in fact, missing from what you have written.