Dedekind Sums

[Eddie W. Shore]

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)
BEGIN
// 2024-02-21 EWS
LOCAL s,i,a,b;

// Calculation
IF CAS.gcd(p,q)==1 THEN
s:=0;

FOR i FROM 1 TO q DO

a:=i/q;
IF FP(a)==0 THEN
a:=0;
ELSE
a:=a-FLOOR(a)-0.5;
END;

b:=p*i/q;
IF FP(b)==0 THEN
b:=0;
ELSE
b:=b-FLOOR(b)-0.5;
END;
s:=s+a*b; 
END;
RETURN s; 
ELSE
RETURN "p and q are not relatively prime.";
END; 
END;

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 Standards-B. Mathematics and Mathematical Physics Vol. 69B, No 4, October-December 1965 https://nvlpubs.nist.gov/nistpubs/jres/6...59_A1b.pdf
Retrieved February 21, 2024

Weisstein, Eric W. "Dedekind Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DedekindSum.html
Retrieved February 18, 2024

[johnb]

(02-26-2024 02:28 PM)Eddie W. Shore Wrote:  

Code:

...
a:=a-FLOOR(a)-0.5;
...
b:=b-FLOOR(b)-0.5;
...
s:=s+a*b; 
...
RETURN s; 
...

I don't have a prime, so I don't recall whether it uses a IEEE-754 binary floating point, or a BCD representation, so I may be off-base 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...]