Dedekind Sums

02262024, 02:28 PM
Post: #1




Dedekind Sums
Definition
The Dedekind Sum is defined as follows: Let P and Q be relatively prime integers, that is GCD(P, Q) = 1. Then S is the Dedekind sum as: S = Σ( ((I ÷ Q)) × ((P × I ÷ Q)), for I=1 to Q) The double parenthesis around the terms I ÷ Q and P × I ÷ Q signify a custom function: (( X )) = 0, if X is an integer X – FLOOR(X) – 1/2, if X is not an integer If X is positive, X – INTG(X) – 1/2 HP Prime: DEDEKIND Code: EXPORT DEDEKIND(p,q) P = 2, Q = 17: 0.4705882353 P =14, Q = 57: 0.8187134503 Sources Shipp, R. Dale. “Table of Dedekind Sums” Journal of Research of the National Bureau of StandardsB. Mathematics and Mathematical Physics Vol. 69B, No 4, OctoberDecember 1965 https://nvlpubs.nist.gov/nistpubs/jres/6...59_A1b.pdf Retrieved February 21, 2024 Weisstein, Eric W. "Dedekind Sum." From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/DedekindSum.html Retrieved February 18, 2024 

02262024, 09:47 PM
(This post was last modified: 02262024 09:49 PM by johnb.)
Post: #2




RE: Dedekind Sums
(02262024 02:28 PM)Eddie W. Shore Wrote: I don't have a prime, so I don't recall whether it uses a IEEE754 binary floating point, or a BCD representation, so I may be offbase here. If it's any "binary, mantissa+exponent" representation, then 0.5 is not exactly the same thing as 1/2. You might squeeze an extra ULP or two out of your answer by computing 'a' and 'b' as twice their values and subtract 1, then return (s/2). Oops. LOL! Let this be a lesson to my fellow amateur mathematicians: do not attempt what you think are reversible transformations in the real domain when discontinuous functions are involved! FLOOR(2x) / 2 ≠ FLOOR(x) [crawling back into my hole in the ground...] Daily drivers: 15c, 32sII, 35s, 41cx, 48g, WP 34s/31s. Favorite: 16c. Latest: 15ce, 48s, 50g. Gateway drug: 28s found in yard sale ~2009. 

« Next Oldest  Next Newest »

User(s) browsing this thread: 1 Guest(s)