A Computerized Redux of Srinivasa Ramanujan's Tables I-III
Given in his 1916 article "On Certain Arithmetical Functions"
By Shalosh B. Ekhad
To commemorate Srinivasa Ramanujan's (1887-1920)
125-th birthday on Dec. 22, 2012, Krishna Alladi and Frank Garvan
organized an International Conference,
(see: http://www.math.ufl.edu/~fgarvan/ramanujan125.html )
where my beloved master, Doron Zeilberger gave one of the
ten plenary lectures.
His lecture was entitled: "The greatest Experimental Mathematician"
See: http://www.math.ufl.edu/~fgarvan/ramanujan125/talks/zeilberger/
He was wondering whether I can check on Srinivasa's amazing
hand calculations done in Tables I-III
of his 1916 article "On Certain Arithmetical Functions".
I am happy to report that I found no errors, and below is the
time that it took me
Table I is:
1 - 24 PHI[0, 1] = P
1 + 240 PHI[0, 3] = Q
1 - 504 PHI[0, 5] = R
2
1 + 480 PHI[0, 7] = Q
1 - 264 PHI[0, 9] = Q R
3 2
691 + 65520 PHI[0, 11] = 441 Q + 250 R
2
1 - 24 PHI[0, 13] = Q R
4 2
3617 + 16320 PHI[0, 15] = 1617 Q + 2000 Q R
3 3
43867 - 28728 PHI[0, 17] = 38367 Q R + 5500 R
5 2 2
174611 + 13200 PHI[0, 19] = 53361 Q + 121250 Q R
4 3
77683 - 552 PHI[0, 21] = 57183 Q R + 20500 Q R
6 3 2 4
236364091 + 131040 PHI[0, 23] = 49679091 Q + 176400000 Q R + 10285000 R
5 2 3
657931 - 24 PHI[0, 25] = 392931 Q R + 265000 Q R
7 4 2 4
3392780147 + 6960 PHI[0, 27] = 489693897 Q + 2507636250 Q R + 395450000 Q R
1723168255201 - 171864 PHI[0, 29] =
6 3 3 5
815806500201 Q R + 881340705000 Q R + 26021050000 R
7709321041217 + 32640 PHI[0, 31] =
8 5 2 2 4
764412173217 Q + 5323905468000 Q R + 1621003400000 Q R
Table II is:
2
288 PHI[1, 2] = -P + Q
720 PHI[1, 4] = P Q - R
2
1008 PHI[1, 6] = -P R + Q
2
720 PHI[1, 8] = P Q - Q R
3 2
1584 PHI[1, 10] = -5 P Q R + 3 Q + 2 R
3 2 2
65520 PHI[1, 12] = 441 P Q + 250 P R - 691 Q R
2 4 2
144 PHI[1, 14] = -7 P Q R + 3 Q + 4 Q R
Table III is:
3
1728 PHI[2, 3] = -P + 3 P Q - 2 R
2 2
1728 PHI[2, 5] = P Q - 2 P R + Q
2 2
1728 PHI[2, 7] = -P R + 2 P Q - Q R
2 2 3 2
8640 PHI[2, 9] = 9 P Q - 18 P Q R + 5 Q + 4 R
2 3 2 2
1728 PHI[2, 11] = -5 P Q R + 6 P Q + 4 P R - 5 Q R
4 2 2
6912 PHI[3, 4] = -P + 6 P Q - 8 P R + 3 Q
3 2 2
3456 PHI[3, 6] = P Q - 3 P R + 3 P Q - Q R
3 2 2 3 2
5184 PHI[3, 8] = -2 P R + 6 P Q - 6 P Q R + Q + R
5 3 2 2
20736 PHI[4, 5] = -P + 10 P Q - 20 P R + 15 P Q - 4 Q R
4 3 2 2 3 2
41472 PHI[4, 7] = 7 P Q - 28 P R + 42 P Q - 28 P Q R + 3 Q + 4 R
This took, 0.407, seconds .