“Liouville Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Liouville Function

释义

Liouville Function

The function

(1)

where

is the number of not necessarily distinct Prime Factors of

, with

. The first few values of

are 1,

, 1,

, 1, 1,

, .... The Liouville function is connected with theRiemann Zeta Function by the equation

(2)

(Lehman 1960).

The Conjecture that the Summatory Function

(3)

satisfies

for

is called the Pólya Conjecture and has been proved to befalse. The first

for which

are for

, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Sloane's A028488),and

is, in fact, the first counterexample to the Pólya Conjecture (Tanaka1980). However, it is unknown if

changes sign infinitely often (Tanaka 1980). The first few values of

are 1,0,

, 0,

, ... (Sloane's A002819).

also satisfies

(4)

where

is the Floor Function (Lehman 1960). Lehman (1960) also gives the formulas


	(5)

and

(6)

where

, and

are variables ranging over the Positive integers,

is the MöbiusFunction,

is Mertens Function, and

, and

are Positive real numbers with

See also Pólya Conjecture, Prime Factors, Riemann Zeta Function

References

Fawaz, A. Y. ``The Explicit Formula for .'' Proc. London Math. Soc. 1, 86-103, 1951.

Lehman, R. S. ``On Liouville's Function.'' Math. Comput. 14, 311-320, 1960.

Sloane, N. J. A. SequencesA028488 andA002819/M0042in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Tanaka, M. ``A Numerical Investigation on Cumulative Sum of the Liouville Function.'' Tokyo J. Math. 3, 187-189, 1980.

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