Saturday, March 30, 2024

HP Prime and Casio fx-CG 50: Dedekind Sums

 HP Prime and Casio fx-CG 50: 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

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;


Casio fx-CG 50: DEDEKIND

244 bytes


Code:

 “DEDEKIND SUM: S(P,Q)”

P”? → P

Q”? → Q


If GCD(P,Q)≠1

Then

P AND Q ARE NOT RELATIVELY PRIME”

Stop


For 1→ I To Q

I÷Q → A

Frac A=0 ⇒ 0 → A

Frac A≠0 ⇒ A – Intg A – 0.5 → A

P × I ÷ Q → B

Frac B=0 ⇒ 0 → B

Frac B≠0 ⇒ B – Intg B – 0.5 → B
S + A × B → S

Next

S


Note: The are 6 spaces between NOT and RELATIVELY to align the text.


Examples


P

Q

Results (fraction)

Results (decimal)

2

17

8/17

0.4705882353

1

21

95/63

1.50793650794

9

43

27/86

0.3139534884

8

67

53/134

0.3955223881

4

75

649/450

1.442222222

14

57

-140/171

-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/69B/jresv69Bn4p259_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


Eddie


All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.