What is Bartlett's theory? This neuroimaging paper has been cited thousands of times.  In it, a method is proposed for computing the correlations among several seed regions and all other brain voxels.  Part of this method involves z-scoring Fisher z values:

To combine results across subjects and compute statistical significance, correlation coefficients were converted to a normal distribution by Fischer's [sic] z transform (25).  These values were converted to z scores (i.e., zero mean, unit variance, Gaussian distributions) by dividing by the square root of the variance, computed as $1 / \sqrt{(n - 3)}$, where $n$ is the degrees of freedom in the measurement.  Because individual time points in the BOLD signal are not statistically independent, the degrees of freedom must be corrected according to Bartlett's theory (25).  The correction factor for independent frames was calculated to be 2.34, resulting in 318 / 2.34 = 135.9 df.

No other mention is made of Bartlett's theory -- that is, they don't explain how they came to the value of 2.34.
Reference 25 is this textbook.  Although the textbook seems excellent and reasonably priced, I would like to acquire a basic understanding of Bartlett's theory without having to buy it, so that I might replicate this method.  Of course my first course of action was to perform several Google searches; they, however, turned up nothing.
What is Bartlett's theory, in this context?
 A: In the supporting information for another paper from the same group, I found this elaboration:

Because individual time points in the BOLD signal are not statistically independent, the degrees of freedom must be computed according to Bartlett’s theory, i.e., computing the integral across all time of the square of the autocorrelation function (11).

Reference 11 is the same Jenkins and Watts textbook.
This raises the question, Which autocorrelation function is used: the autocorrelation for the seed region or the autocorrelation for the voxel with which it has been correlated?  And what is meant by "across all time" -- across all time lags?
Michael Fox, in a personal communication, responds that the autocorrelation function for the seed region is used, and that "across all time" indeed means across all time lags.
(But why use the autocorrelation for the seed region and ignore the other timeseries?)
I can't find any independent support for this approach.
A handout I found from a course in Atmospheric Sciences at U Dub suggests a vaguely similar approach and attributes it to Bartlett:

Indeed, Bretherton et al, (1999) show that, assuming that one is looking at quadratic
  statistics, such as variance and covariance analysis between two variables $x_1$ and $x_2$, and
  using Gaussian red noise as a model then a good approximation to use is:
$\frac{N^*}{N} = \frac{1− r_1(\Delta t)r_2 (\Delta t)}{
1+ r_1(\Delta t)r_2 (\Delta t)}$
where, of course, if we are covarying a variable with itself, $r_1(\Delta t)r_2(\Delta t) = r(\Delta t)^2$. This
  goes back as far as Bartlett (1935). Of course, if the time or space series is not Gaussian red noise, then the formula is not accurate. But it is still good practice to use it.

In this, $N^*$ is the degrees of freedom, $N$ is the number of observations, and $r_x$ is the autocorrelation function for signal $x$.
The operative words here are "vaguely similar."
I have now read a good portion of Jenkins and Watts, and skimmed through even more, but have found no description in the text of the correction described by Fox et al. or of the Fisher z-transform.
All that said, I have decided to compute the correction factor the same way Michael says I should . . . but wait: How do I do this?
A: Bartlett's theory here refers to results from this paper 
On the Theoretical Specification and Sampling Properties of Autocorrelated Time-Series.
The main idea here is that if you have $n$ discrete time observations, the effective number of degrees of freedom needs to be less than n since the observations are not independent, given by $n \cdot\text{correction factor}$
For a linear autoregressive model with $s$ lags
$$ x_{t+1} = \sum_{l=1}^{s}x_{t-l}\rho_{l} + e_t $$
Suppose $s=1$. The asymptotic correction factor for AR-(1) model is given by $\frac{1-\rho^2}{1+\rho^2}$. Thus instead of an asymptotic variance of $\frac{1}{n}$, we have a larger asymptotic variance of $\frac{1}{n}\frac{1+\rho^2}{1-\rho^2}$
You can similarly find a derivation of Bartlett's formula for asymptotic variance vector of $(\rho_1,\ldots, \rho_s)$ autocorrelation coefficients case in Theorem 7.2.1 in [Brockwell and Davis][2] and for a number of other time-series models as well. 
For the case of asymptotic variance between two different time-series, $x(t)$ and $y(t)$, a similar formula applies and is nicely discussed in this paper by [Haugh][3]. Even if you do whiten each individual time-series first, the test statistic for the sample cross-correlation still needs to adjust the asymptotic variance estimate. 
[2]: Time Series: Theory and Methods by Brockwell and Davis. Available for free at http://link.springer.com/book/10.1007%2F978-1-4419-0320-4
[3]: Haugh, Larry D. "Checking the independence of two covariance-stationary time series: a univariate residual cross-correlation approach." Journal of the American Statistical Association 71.354 (1976): 378-385.
APA 
