Sine sum

Sam Walters posted something interesting on Twitter yesterday I hadn't seem before:

The sines of the positive integers have just the right balance of pluses and minuses to keep their sum in a fixed interval. (Not hard to show.) #math pic.twitter.com/RxeoWg6bhn

- Sam Walters (@SamuelGWalters) November 29, 2018

If for some reason your browser doesn't render the embedded tweet, he points out that

for all positive integers N.

Here's a little Python script to illustrate the sum in his tweet.

 from scipy import sin, arange import matplotlib.pyplot as plt x = arange(1,100) y = sin(x).cumsum() plt.plot(x, y) plt.plot((1, 100), (-1/7, -1/7), "g--") plt.plot((1, 100), (2, 2), "g--") plt.savefig("sinsum.svg")

Exponential sums are interesting because crude application of the triangle inequality won't get you anywhere. All it will let you conclude is that the sum is between -N and N.

(Why is this an exponential sum? Because it's the imaginary part of the sum over eⁱⁿ.)

For more on exponential sums, you might like the book Uniform Distribution of Sequences.

Also, I have a page that displays the plot of a different exponential sum each day, the coefficients in the sum

being taking from the day's date. Because consecutive numbers have very different number theoretic properties, the images vary quite a bit from day to day.

Here's a sneak peak at what the exponential sum for Christmas will be this year.