Murata's constant is defined as
 |  | ![product_(p)[1+1/((p-1)^2)]](/images/equations/MuratasConstant/Inline3.svg) |
(1)
|
 |  |  |
(2)
|
(OEIS A065485), where the product is over the primes 
.
It can also be written as the sum
 |  | ![exp[-sum_(n=2)^(infty)((1+i)^n+(1-i)^n-2)/nP(n)]](/images/equations/MuratasConstant/Inline10.svg) |
(3)
|
 |  | ![exp[sum_(n=2)^(infty)(2-2^(n/2+1)cos(1/4npi))/nP(n)],](/images/equations/MuratasConstant/Inline13.svg) |
(4)
|
where 
is the prime zeta function.
Murata's Constant
See also
Artin's Constant, Barban's Constant, Feller-Tornier Constant, Heath-Brown-Moroz Constant, Prime Products, Quadratic Class Number Constant, Sarnak's Constant, Taniguchi's Constant, Twin Primes ConstantExplore with Wolfram|Alpha
References
Finch, S. R. "Artin's Constant." §2.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 104-106, 2003.Murata, L. "On the Magnitude of the Least Prime Primitive Root." J. Number Th. 37, 47-66, 1991.Niklasch, G. "Some Number-Theoretical Constants." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml.Sloane, N. J. A. Sequence A065485 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Murata's ConstantCite this as:
Weisstein, Eric W. "Murata's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MuratasConstant.html