Pólya Conjecture
المؤلف:
Haselgrove, C. B.
المصدر:
"A Disproof of a Conjecture of Pólya." Mathematika 5
الجزء والصفحة:
...
15-9-2020
847
Pólya Conjecture
Let 
be a positive integer and 
the number of (not necessarily distinct) prime factors of 
(with 
). Let 
be the number of positive integers 
with an odd number of prime factors, and 
the number of positive integers 
with an even number of prime factors. Pólya (1919) conjectured that
is 
, where 
is the Liouville function.
The conjecture was made in 1919, and disproven by Haselgrove (1958) using a method due to Ingham (1942). Lehman (1960) found the first explicit counterexample, 
, and the smallest counterexample 
was found by Tanaka (1980). The first 
for which 
are 
, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Tanaka 1980, OEIS A028488). It is unknown if 
changes sign infinitely often (Tanaka 1980).
REFERENCES:
Haselgrove, C. B. "A Disproof of a Conjecture of Pólya." Mathematika 5, 141-145, 1958.
Ingham, A. E. "On Two Conjectures in the Theory of Numbers." Amer. J. Math. 64, 313-319, 1942.
Lehman, R. S. "On Liouville's Function." Math. Comput. 14, 311-320, 1960.
Pólya, G. "Verschiedene Bemerkungen zur Zahlentheorie." Jahresber. deutschen Math.-Verein. 28, 31-40, 1919.
Sloane, N. J. A. Sequence A028488 in "The On-Line Encyclopedia of Integer Sequences."
Tanaka, M. "A Numerical Investigation on Cumulative Sum of the Liouville Function" [sic]. Tokyo J. Math. 3, 187-189, 1980.
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