Rigidity theorems for multiplicative functions
We establish several results concerning the expected general phenomenon that, given a multiplicative function f : N → C , the values of f ( n ) and f ( n + a ) are “generally” independent unless f is of a “special” form. First, we classify all bounded completely multiplicative functions having unifo...
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Veröffentlicht in: | Mathematische annalen 2018-10, Vol.372 (1-2), p.651-697 |
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Sprache: | eng |
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Zusammenfassung: | We establish several results concerning the expected general phenomenon that, given a multiplicative function
f
:
N
→
C
, the values of
f
(
n
) and
f
(
n
+
a
)
are “generally” independent unless
f
is of a “special” form. First, we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values. This implies the solution of the following folklore conjecture: for any completely multiplicative function
f
:
N
→
T
we have
lim inf
n
→
∞
|
f
(
n
+
1
)
-
f
(
n
)
|
=
0
.
Second, we settle an old conjecture due to Chudakov (On the generalized characters. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, p. 487. Gauthier-Villars, Paris) that states that any completely multiplicative function
f
:
N
→
C
that: (a) takes only finitely many values, (b) vanishes at only finitely many primes, and (c) has bounded discrepancy, is a Dirichlet character. This generalizes previous work of Tao on the Erdős Discrepancy Problem. Finally, we show that if many of the binary correlations of a 1-bounded multiplicative function are asymptotically equal to those of a Dirichlet character
χ
mod
q
then
f
(
n
)
=
χ
′
(
n
)
n
it
for all
n
, where
χ
′
is a Dirichlet character modulo
q
and
t
∈
R
. This establishes a variant of a conjecture of H. Cohn for multiplicative arithmetic functions. The main ingredients include the work of Tao on logarithmic Elliott conjecture, correlation formulas for
pretentious
multiplicative functions developed earlier by the first author and Szemeredi’s theorem for long arithmetic progressions. |
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ISSN: | 0025-5831 1432-1807 1432-1807 |
DOI: | 10.1007/s00208-018-1724-6 |