Engineering DNA logic systems is considered as one of the most promising strategies for next-generation molecular computers. Owing to the inherent features of DNA, such as low cost, easy synthesis, ...and controllable hybridization, various DNA logic devices with different functions have been developed in the recent decade. Besides, a variety of logic-programmed biological applications are also explored, which initiates a new chapter for DNA logic computing. Although this field has gained rapid developments, a systematical review that could not only elaborate the logic principles of diverse DNA logic devices but also outline recent representative works is urgently needed. In this review, we first elaborate the general classification and logical principle of diverse DNA logic devices, in which the operating strategy of these devices and representative examples are selectively presented. Then, we review state-of-the-art advancements in DNA computing based on different non-canonical DNA-nanostructures during the past decade, in which some classical works are summarized. After that, the innovative applications of DNA computing to logic-controlled bioanalysis, cell imaging, and drug load/delivery are selectively presented. Finally, we analyze current obstacles and suggest appropriate prospects for this area.
For a wide class of sequences of integer domains
D
n
⊂
N
d
,
n
∈
N
, we prove distributional limit theorems for
F
(
X
1
(
n
)
,
…
,
X
d
(
n
)
)
, where
F
is a multivariate multiplicative function and
...(
X
1
(
n
)
,
…
,
X
d
(
n
)
)
is a random vector with uniform distribution on
D
n
. As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set
{
X
1
(
n
)
,
…
,
X
d
(
n
)
}
. This generalizes previously known limit results for
D
n
being either a discrete cube or a discrete hyperbolic region.
We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may ...have their own interest.
Let
N
be the set of positive integers, and denote by
λ
(
A
)
=
inf
{
t
>
0
:
∑
a
∈
A
a
-
t
<
∞
}
the convergence exponent of
A
⊂
N
. For
0
<
q
≤
1
,
0
≤
q
≤
1
, respectively, the admissible ideals
I
...<
q
,
I
≤
q
of all subsets
A
⊂
N
with
λ
(
A
)
<
q
,
λ
(
A
)
≤
q
, respectively, satisfy
I
<
q
⊊
I
c
(
q
)
⊊
I
≤
q
, where
I
c
(
q
)
=
{
A
⊂
N
:
∑
a
∈
A
a
-
q
<
∞
}
.
In this note we sharpen the results of Baláž et al. from (J Number Theory 183:74–83, 2018) and other papers, concerning characterizations of
I
c
(
q
)
-convergence of various arithmetic functions in terms of
q
. This is achieved by utilizing
I
<
q
- and
I
≤
q
-convergence, for which new methods and criteria are developed.
In this paper, we shall establish a rather general asymptotic formula in short intervals for a class of arithmetic functions and announce two applications about the distribution of divisors of ...square‐full numbers and integers representable as sums of two squares.
In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in Fqt. The method is the function field analogue of a result first proved by Stefan A. ...Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions dr(f) with an error term that improves the one given by Burr.
The main result of this paper describes the meromorphic continuation of a class of Bateman-like totient zeta function associated to a special class of arithmetical multiplicative functions, ...discovering a comb-like form of the region of meromorphic continuation, and verifying the expected natural boundary by clustering logarithmic type singularities. Then, a standard application of the Selberg-Delange classical method allows us to derive the distribution of values of a class of multivariate multiplicative functions. We also use Balazard-Diamond convolution method to improve in some cases the error term given by Selberg-Delange classical method.
In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended ...Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.
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