A278715 Table T read by rows. T(k, h) gives the numerators of the Dedekind sums s(h, k) with gcd(h, k) = 1 or 0 otherwise.

0, 1, -1, 1, 0, -1, 1, 0, 0, -1, 5, 0, 0, 0, -5, 5, 1, -1, 1, -1, -5, 7, 0, 1, 0, -1, 0, -7, 14, 4, 0, -4, 4, 0, -4, -14, 3, 0, 0, 0, 0, 0, 0, 0, -3, 15, 5, 3, 3, -5, 5, -3, -3, -5, -15, 55, 0, 0, 0, -1, 0, 1, 0, 0, 0, -55, 11, 4, 1, -1, 0, -4, 4, 0, 1, -1, -4, -11, 13, 0, 3, 0, 3, 0, 0, 0, -3, 0, -3, 0, -13, 91, 7, 0, 19, 0, 0, -7, 7, 0, 0, -19, 0, -7, -91

Offset: 2

Author

Wolfdieter Lang, Nov 28 2016

Keywords

• sign
• tabl
• frac
• easy

Comments

For the denominators see A278716.

The Dedekind sums are s(h,k) = Sum_{r=1..k-1} (r/k)*(h*r/k - floor(h*r/k) - 1/2) = Sum_{r=1..k-1} ((r/k))*((h*r)/k) with the period 1 sawtooth function ((x)) = x - floor(x) - 1/2 if x is not an integer and 0 otherwise. One assumes gcd(h,k) = 1, and for other h, k values the table has 0's. See the references Apostol pp. 52, 61-69, 72-73 and Ayoub pp. 168, 191. See also the Weisstein link.

In order to have a regular triangle T(k, h) one starts with k >= 2 and h = 1, 2, ..., k-1. s(1,1) = 0.

Examples

			The triangle T(k,h) begins (if gcd(k,h) is not 1 we use o instead of 0):
k\h  1  2  3  4  5  6  7   8  9  10  11  12
2:   0
3:   1 -1
4:   1  o -1
5:   1  0  0 -1
6:   5  o  o  o -5
7:   5  1 -1  1 -1 -5
8:   7  o  1  o -1  o -7
9:  14  4  o -4  4  o -4 -14
10:  3  o  0  o  o  o  0   o -3
11: 15  5  3  3 -5  5 -3  -3 -5 -15
12: 55  o  o  o -1  o  1   o  o   o -55
13: 11  4  1 -1  0 -4  4   0  1  -1  -4 -11
...
n = 14: 13 o 3 o 3 o o o -3 o -3 o -13,
n = 15: 91 7 0 19 0 0 -7 7 0 0 -19 0 -7 -91.
...
---------------------------------------------
The rational triangle s(h,k) begins (here o is used if gcd(h,k) is not 1):
k\h  1      2     3    4     5     6     7
2:   0
3:  1/18 -1/18
4:  1/8     o  -1/8
5:  1/5     0    0   -1/5
6:  5/18    o    o     o   -5/18
7:  5/14  1/14 -1/14  1/14 -1/14 -5/14
8:  7/16    o   1/16   o   -1/16   o   -7/16
...
n = 9: 14/27 4/27 o -4/27 4/27 o -4/27 -14/27,
n = 10: 3/5 o 0 o o o 0 o -3/5,
n = 11: 15/22 5/22 3/22 3/22 -5/22 5/22 -3/22 -3/22 -5/22 -15/22,
n = 12:  55/72 o o o -1/72 o 1/72 o o o  -55/72,
n = 13: 11/13 4/13 1/13 -1/13 0 -4/13 4/13 0 1/13 -1/13 -4/13 -11/13,
n = 14: 13/14 o 3/14 o 3/14 o o o -3/14 o -3/14 o -13/14,
n = 15: 1/90 7/18 o 19/90 o o -7/18 7/18 o o -19/90 o -7/18 -91/90.
...
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References

• Apostol, Tom, M., Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990.
• Ayoub, R., An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963.

Links

• G. C. Greubel, Rows n=2..100 of triangle, flattened
• Eric Weisstein's World of Mathematics, Dedekind Sum.

Crossrefs

Cf. A278716, A264388/A264389 (h=1), A278713/A278714 (h=2 odd k).

Programs

Magma

[[GCD(n,k) eq 1 select Numerator((&+[(j/n)*(k*j/n - Floor(k*j/n) - 1/2): j in [1..(n-1)]])) else 0: k in [1..(n-1)]]: n in [2..15]]; // G. C. Greubel, Nov 22 2018

Mathematica

T[n_, k_]:= If[GCD[n, k] == 1, Sum[(j/n)*(k*j/n - Floor[k*j/n] - 1/2), {j, 1, n - 1}], 0]; Numerator[Table[T[n, k], {n,2,15}, {k,1,n-1}]] //Flatten (* G. C. Greubel, Nov 22 2018 *)

PARI

{T(n,k) = if(gcd(n,k)==1, sum(j=1,n-1, (j/n)*(k*j/n - floor(k*j/n) - 1/2)), 0)};
for(n=2,15, for(k=1,n-1, print1(numerator(T(n,k)), ", "))) \\ G. C. Greubel, Nov 22 2018

Sage

def T(n,k):
    if gcd(n,k)==1:
       return numerator(sum((j/n)*(k*j/n - floor(k*j/n) - 1/2) for j in (1..(n-1))))
    elif gcd(n,k)!=1:
        return 0
    else:
        0
[[T(n,k) for k in (1..n-1)] for n in (2..15)] # G. C. Greubel, Nov 22 2018

Formula

T(k ,h) = numerator(s(h,k)) with the Dedekind sums s(h,k) given in a comment above and gcd(h,k) = 1. k >=2, h = 1, 2, ..., k-1. If gcd(h,k) is not 1 then T(k,h) is put to 0 (in the example o is used). Note that T(k,h) can vanish also for gcd(h,k) = 1.