Thomas R. Nicely
Last updated 1000 GMT 18 January 2010.
Li(x) = integral(1/ln(t), t, 0, x) = Li(2) + integral(1/ln(t), t, 2, x) .
where
Li(2) = 1.0451637801174927848445888891946131365226155781512...
and where the singularity at x=1 is treated using the Cauchy principal value.
Li(x) = Li(0, x) = Li(2, x) + Li(2) .
Li(x) is now usually defined as Li(0, x), with a lower limit of zero; the Cauchy principal value is employed near the singularity at t=1, and the singularity at t=0+ is integrable. In some references, Li(2, x) is still used, with the lower limit of 2. Adoption of the lower limit zero eliminates the trivial exceptions (n=2,3,4,5,7) to the conditional inequality Li(n) > pi(n) (n an integer > 1). The fact that Li(0, x) < 0 for 0 < x < 1 and for 1 < x < 1.45136923488338105... is not considered to constitute an exception, the domain of the inequality being commonly taken as those integers n > 1.
R(x) = sum(µ(n)*Li(x^(1/n)/n, n, 1, +infinity) ,where µ(n) is Möbius' function,
µ(n) = 1 , if n=1; µ(n) = 0 , if n contains a repeated prime factor; µ(n) = (-1)^k , if n is the product of k distinct primes.
R(x) = 1 + sum((ln x)^m/(m!*m*Zeta(m+1)), m, 1, +infinity) ,where Zeta is the Riemann zeta function.
====================================================================== x pi(x) Li(x)-pi(x) R(x)-pi(x) ====================================================================== 1.000000e+01 4 2.1656 0.5646 2.000000e+01 8 1.9053 -0.4728 3.000000e+01 10 3.0226 0.1691 4.000000e+01 12 3.8395 0.6325 5.000000e+01 15 3.4687 -0.0243 6.000000e+01 17 3.9654 0.2300 7.000000e+01 19 4.3618 0.4146 8.000000e+01 22 3.6786 -0.4576 9.000000e+01 24 3.9299 -0.3776 1.000000e+02 25 5.1261 0.6616 2.000000e+02 46 4.1922 -1.4100 3.000000e+02 62 6.3336 -0.0346 4.000000e+02 78 7.4179 0.4515 5.000000e+02 95 6.7939 -0.6720 6.000000e+02 109 8.6465 0.7474 7.000000e+02 125 8.0889 -0.1957 8.000000e+02 139 9.1967 0.5631 9.000000e+02 154 9.0236 0.0699 1.000000e+03 168 9.6097 0.3594 2.000000e+03 303 11.8092 0.3353 3.000000e+03 430 12.7592 -0.2741 4.000000e+03 550 15.3645 1.0857 5.000000e+03 669 15.2808 -0.0539 6.000000e+03 783 17.4141 1.1525 7.000000e+03 900 14.3308 -2.7629 8.000000e+03 1007 19.4164 1.5637 9.000000e+03 1117 19.9488 1.3955 1.000000e+04 1229 17.1372 -2.0688 2.000000e+04 2262 26.6140 2.4194 3.000000e+04 3245 31.8991 4.1257 4.000000e+04 4203 30.0113 -0.6586 5.000000e+04 5133 33.5468 0.3986 6.000000e+04 6057 25.8469 -9.4916 7.000000e+04 6935 50.2945 12.9786 8.000000e+04 7837 39.2132 0.0851 9.000000e+04 8713 44.2916 3.4842 1.000000e+05 9592 37.8090 -4.5683 2.000000e+05 17984 52.0521 -2.4680 3.000000e+05 25997 89.6922 26.3336 4.000000e+05 33860 62.6219 -7.9510 5.000000e+05 41538 68.2888 -8.4941 6.000000e+05 49098 74.8270 -7.4695 7.000000e+05 56543 101.6627 14.3704 8.000000e+05 63951 86.2974 -5.5873 9.000000e+05 71274 88.0546 -8.0972 1.000000e+06 78498 129.5492 29.3994 2.000000e+06 148933 121.8332 -9.4921 3.000000e+06 216816 154.5644 0.3228 4.000000e+06 283146 206.2546 33.1979 5.000000e+06 348513 125.1150 -64.2044 6.000000e+06 412849 227.5041 23.6997 7.000000e+06 476648 178.8123 -38.1506 8.000000e+06 539777 222.6901 -6.3947 9.000000e+06 602489 187.2538 -53.1147 1.000000e+07 664579 339.4050 88.4476 2.000000e+07 1270607 298.0363 -35.9674 3.000000e+07 1857859 354.0973 -41.3796 4.000000e+07 2433654 362.4394 -83.7209 5.000000e+07 3001134 423.4267 -66.6722 6.000000e+07 3562115 568.4771 39.1531 7.000000e+07 4118064 520.9295 -44.0939 8.000000e+07 4669382 708.7454 110.7840 9.000000e+07 5216954 856.2234 227.5602 1.000000e+08 5761455 754.3754 96.8673 2.000000e+08 11078937 1037.8525 153.1722 3.000000e+08 16252325 1084.1472 30.4409 4.000000e+08 21336326 1052.0119 -141.4858 5.000000e+08 26355867 965.1489 -349.8050 6.000000e+08 31324703 1342.2521 -81.3140 7.000000e+08 36252931 1311.0438 -211.5094 8.000000e+08 41146179 1683.2333 69.2433 9.000000e+08 46009215 2433.6333 734.3285 1.000000e+09 50847534 1700.9570 -78.5723 2.000000e+09 98222287 3015.1378 601.7873 3.000000e+09 144449537 3191.7309 304.9513 4.000000e+09 189961812 2778.7703 -500.4800 5.000000e+09 234954223 4557.8739 937.0539 6.000000e+09 279545368 3498.9274 -427.7411 7.000000e+09 323804352 4067.5164 -138.1936 8.000000e+09 367783654 5241.6802 777.9819 9.000000e+09 411523195 5058.5772 353.9783 1.000000e+10 455052511 3103.5866 -1827.6932 2.000000e+10 882206716 8163.2048 1436.6258 3.000000e+10 1300005926 10205.6671 2134.0410 4.000000e+10 1711955433 8698.8221 -489.8988 5.000000e+10 2119654578 11961.2941 1799.0736 6.000000e+10 2524038155 10162.7505 -872.0577 7.000000e+10 2925699539 10280.3473 -1551.2344 8.000000e+10 3325059246 12346.4597 -222.3001 9.000000e+10 3722428991 15270.6039 2013.0737 1.000000e+11 4118054813 11587.6216 -2318.3686 2.000000e+11 8007105059 13344.6556 -5707.0635 3.000000e+11 11818439135 23766.7958 850.6010 4.000000e+11 15581005657 35813.5524 9683.0924 5.000000e+11 19308136142 35586.1714 6651.6536 6.000000e+11 23007501786 30191.6689 -1258.3116 7.000000e+11 26684074310 39428.2630 5679.8281 8.000000e+11 30341383527 34506.1467 -1370.0344 9.000000e+11 33981987586 45948.4896 8083.3095 1.000000e+12 37607912018 38262.8049 -1475.7741 2.000000e+12 73301896139 48195.4975 -6432.3993 3.000000e+12 108340298703 80927.2802 15095.6042 4.000000e+12 142966208126 61847.6795 -13313.8699 5.000000e+12 177291661649 72125.9698 -11181.7988 6.000000e+12 211381427039 99289.4065 8668.9067 7.000000e+12 245277688804 110789.8197 13483.7731 8.000000e+12 279010070811 79478.0329 -24019.8995 9.000000e+12 312600354108 127830.8386 18542.4553 1.000000e+13 346065536839 108971.0503 -5773.1740 1.100000e+13 379418886337 112657.5542 -7258.1337 1.200000e+13 412671294782 88008.6253 -36833.0266 1.300000e+13 445831610611 157230.3263 27677.3989 1.400000e+13 478907544860 151774.5337 17700.1158 1.500000e+13 511905471132 125904.1735 -12522.4150 1.600000e+13 544830816681 193903.6546 51277.1576 1.700000e+13 577688618018 144607.2552 -2081.2814 1.800000e+13 610482945548 146117.6119 -4507.3708 1.900000e+13 643217611048 138409.5740 -16036.8298 2.000000e+13 675895909271 170356.4223 12194.4469 2.100000e+13 708520996055 84243.3484 -77536.3752 2.200000e+13 741095263696 190877.2552 25570.5392 2.300000e+13 773621469799 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