The Niven constant , named after the Canadian-American mathematician Ivan M. Niven , is a mathematical constant from number theory . It is defined as the limit of the arithmetic mean of the maximum exponents of the prime factorization of the first natural numbers for .
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definition
Let be an integer with the prime factorization with and for , besides and the maximum of the exponents in the prime factorization of (sequence A051903 in OEIS ), for example the numbers with exactly the square-free numbers. The Niven constant is thus defined as
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{\ displaystyle m = p_ {1} ^ {a_ {1}} p_ {2} ^ {a_ {2}} p_ {3} ^ {a_ {3}} \ cdots p_ {k} ^ {a_ {k} }}
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{\ displaystyle H (m) = 1}
lim
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{\ displaystyle \ lim _ {n \ to \ infty} {\ frac {1} {n}} \ sum _ {j = 1} ^ {n} H (j).}
properties
The Niven constant can be expressed using the Riemann zeta function and can be approximately calculated in this way (Niven 1969):
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lim
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{\ displaystyle \ lim _ {n \ to \ infty} {\ frac {1} {n}} \ sum _ {j = 1} ^ {n} H (j) = 1 + \ sum _ {k = 2} ^ {\ infty} {\ biggl (} 1 - {\ frac {1} {\ zeta (k)}} {\ biggr)}}
=
1.705
21st
11401
05367
76428
85514
53434
50816
07620
27651
65346
.
.
.
{\ displaystyle = 1 {,} 70521 {\ text {}} 11401 {\ text {}} 05367 {\ text {}} 76428 {\ text {}} 85514 {\ text {}} 53434 {\ text {}} 50816 {\ text {}} 07620 {\ text {}} 27651 {\ text {}} 65346 {\ text {}} ...}
(Follow A033150 in OEIS )
At the suggestion of Erdős , Niven proved for the asymptotic behavior of the minima of the exponents
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{\ displaystyle \ sum _ {j = 1} ^ {n} h (j) = n + {\ frac {\ zeta ({\ tfrac {3} {2}})} {\ zeta (3)}} {\ sqrt {n}} + o ({\ sqrt {n}}),}
where and the minimum of the exponents in the prime factorization of (sequence A051904 in OEIS ) and is a Landau symbol . Thus, in particular
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{\ displaystyle h (m) = \ min \ {a_ {1}, ..., a_ {k} \}}
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{\ displaystyle \ lim _ {n \ to \ infty} {\ frac {1} {n}} \ sum _ {j = 1} ^ {n} h (j) = 1.}
literature
Steven R. Finch: Niven's constant . Chapter 2.6 in Mathematical constants . Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , pp. 112-115 (English)
Web links
Individual evidence
↑ Ivan Niven : Averages of exponents in factoring integers . (June 18, 1968), Proceedings of the AMS 22, 1969, pp. 356-360 (English)
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