质数的平方的倒数和是多少?
2 3 5 7 11............先平方,
再倒数
再求和
那么和是多少呢?
看结果是收敛的
Clear["Global`*"];(*Clear all variables*)li =
Table, {k, 1, PrimePi}]; Print@
N, 10];
li = Table, {k, 1, PrimePi}]; Print@
N, 10];
li = Table, {k, 1, PrimePi}]; Print@
N, 10];
0.4483700608
0.4519741916
0.4522263342
那么是多少呢? 二次方 三次方 四次方呢??????? 这个问题一百多年前就有人提出来了。
参考文献：
C. W. Merrifield, The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers, Proc. Roy. Soc. London 33 (1881) 4–10. doi:10.1098/rspl.1881.0063. JSTOR 113877
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25 (1891) 347–362.
$1$楼要求的是$P(2)=0.4522474200...$，
$2$楼要求的是$P(3)$、$P(4)$、……
其中$P$是“prime zeta function”。
参考链接：
http://mathoverflow.net/questions/53443/sum-of-the-reciprocal-of-the-primes-squared Column[{#, N, 50]} & /@ Range]
{2,0.45224742004106549850654336483224793417323134323989}
{3,0.17476263929944353642311331466570670097541212192615}
{4,0.076993139764246844942619295933157870162041059714843}
{5,0.035755017483924257132818242538855711131697276726651}
{6,0.017070086850636512954133673266059399209585941874544}
{7,0.0082838328561335925351241387294487230891833288853078}
{8,0.0040614053665178305605234391426830805229771445120717}
{9,0.0020044675749624506630735851407831175368229203497386}
{10,0.00099360357443698021785585070014773941630187254528520}
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