니븐 상수

정수론에서 니븐 상수(Niven constant)는 이반 니븐(Ivan Niven)의 이름을 따서 이름붙여진 수학 상수이다.

양의 정수 n에 대하여, $H(n)$ 을 n의 소인수분해에 나타나는 가장 큰 지수를 나타내는 수론적 함수로 정의하자. 또한 $H(1)=1$ 로 정의하자. 이 $H(n)$ 함수에 대해, 니븐 상수는 다음과 같이 정의된다:

\lim _{n\to \infty }{{1} \over {n}}\sum _{k=1}^{n}H(k)=1+\sum _{k=2}^{\infty }\left(1-{{1} \over {\zeta (k)}}\right)=1.705211\dots

여기서 $\zeta (s)$ 는 리만 제타 함수이다(Niven, 1969).

같은 논문에서, 니븐은

\sum _{j=1}^{n}h(j)=n+c{\sqrt {n}}+o({\sqrt {n}})

임도 보였다. 여기서 $h(n)$ 은 양의 정수 n의 소인수분해에 나타나는 가장 작은 지수를 나타내는 수론적 함수로 정의되며, $h(1)=1$ 이다. 또한, $o$ 는 작은 o 표기법을 나타내며 상수 $c$ 의 값은

c={{\zeta \left({{3} \over {2}}\right)} \over {\zeta (3)}}

이다. 니븐의 식으로부터,

\lim _{n\to \infty }{{1} \over {n}}\sum _{j=1}^{n}h(j)=1

을 얻을 수 있다.

리만 제타 함수의 특수 값

\zeta ({-1})=-{1 \over 12}

\zeta ({0})=-{1 \over 2}

\zeta \left({{1} \over {2}}\right)\approx -1.4603545....(OEISA059750)

\zeta (1)=1+{1 \over 2}+{1 \over 3}+....=\infty

\zeta \left({{3} \over {2}}\right)\approx 2.612....(OEISA078434)

\zeta ({2})={\pi ^{2} \over {6}}=1.645....(OEISA013661)

\zeta ({3})=1.202....(OEISA002117)

같이 보기

참고

Niven, Ivan M. (August 1969). “Averages of Exponents in Factoring Integers”. 《Proceedings of the American Mathematical Society》 22 (2): 356–360. doi:10.2307/2037055. JSTOR 2037055.
Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2003

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