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1.
For each integer n 2, let
be the index of composition of n, where
. For convenience, we write (1)=(1)=1. We obtain sharp estimates for
and
, as well as for
and
. Finally we study the sum of running over shifted primes.Research supported in part by a grant from NSERC.Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. 相似文献
2.
D. P. Dryanov 《Constructive Approximation》1994,10(3):377-409
Let
n–1 be the linear space of algebraic polynomials of degreen–1. We prove that the extremal problem
相似文献
3.
Dr. Wolfgang Woess 《Monatshefte für Mathematik》1980,90(4):339-345
Several conditions are shown to be equivalent to the aperiodicity of a regular probability measure on a locally compact, separated topological GroupG. In particular, is aperiodic if and only if the sequence (
(
(n)
denoting then-th convolution power of ) is convergent for any nonvoid open subsetU ofG with compact closure. It is always assumed that the support of generatesG as a closed semigroup. 相似文献
4.
Yisheng Song 《Positivity》2009,13(4):643-655
In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by
5.
M. I. Gil’ 《Acta Appl Math》2007,99(2):117-159
Let f be an entire function. Denote by z
1(f),z
2(f),… the zeros of f with their multiplicities. In the paper, estimates for the sums
6.
Christer Borell 《Probability Theory and Related Fields》2008,140(1-2):195-205
Let be an integer, let γ be the standard Gaussian measure on , and let . Given this paper gives a necessary and sufficient condition such that the inequality is true for all Borel sets A
1,...,A
m
in of strictly positive γ-measure or all convex Borel sets A
1,...,A
m
in of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn–Minkowski type
for γ which are true for all convex sets but not for all measurable sets.
相似文献
7.
Irrationality measures are given for the values of the series
, where
and Wn is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality
of all the numbers
8.
Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies
the constructions of bent functions having the form
for odd n and
for even n, over the finite field
of odd characteristic p, where
. Based on the irreducibility of some polynomials on
, we focus on characterizing the bent functions for n=p
v
q
r
and n=2p
v
q
r
, where
is an odd prime and p a primitive root modulo q
2. Moreover, the enumerations of those functions are also considered.
Partially supported by the NSF of China under Grants No. 60603012 and No. 60573053. 相似文献
9.
We establish a new 3G-Theorem for the Green’s function for the half space
We exploit this result to introduce a new class of potentials
that we characterize by means of the Gauss semigroup on
. Next, we define a subclass
of
and we study it. In particular, we prove that
properly contains the classical Kato class
. Finally, we study the existence of positive continuous solutions in
of the following nonlinear elliptic problem
10.
J.-P. Allouche 《The Ramanujan Journal》2007,14(1):39-42
We answer a question of Berndt and Bowman, asking whether it is possible to deduce the value of the Ramanujan integral I from the value of the Ramanujan integral J, where
11.
In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier
transforms in the following way: If
and
, and
is locally integrable, then
distributionally if and only if there exists k such that
, for each a > 0, and similarly in the case when
is a general distribution. Here
means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional
point value given in [5] by
. We also show that under some extra conditions, as if the sequence
belongs to the space
for some
and the tails satisfy the estimate
,\ as
, the asymmetric partial sums\ converge to
. We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We
apply these results to lacunary Fourier series of distributions. 相似文献
12.
J.-C. Schlage-Puchta 《The Ramanujan Journal》2006,12(3):455-460
Denote by σ
k
(n) the sum of the k-th powers of the divisors of n, and let
. We prove that Schinzel’s conjecture H implies that S
k
is irrational, and give an unconditional proof for the case k = 3.
2000 Mathematics Subject Classification Primary—11A25, 11N36, 11J72 相似文献
13.
Let q be a complex number satisfying |q| < 1. The theta function (q) is defined by (q) =
. Ramanujan has given a number of Lambert series expansions such as
14.
Let X,X
n
;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b
n
=B(n),n1. If
15.
Wen-Bin Zhang 《Mathematische Zeitschrift》2009,261(1):233-234
Let be the generalized integers n
j
associated with a set of generalized primes p
i
in Beurling’s sense. On the basis of the general mean-value theorems, established in our previous work, for multiplicative
function f(n
j
) defined on , we prove extensions, in functional form and in mean-value form, of the Elliott–Daboussi theorem to high order mean-values.
For the main result, let α,ρ, and τ be positive real constants such that α > 1,ρ≥1 and . Then a multiplicative function f satisfies the following conditions, with some constant , (1) All four series
16.
Ankur Basu 《The Ramanujan Journal》2008,16(1):7-24
A new method in the study of Euler sums is developed. A host of Euler sums, typically of the form
, are expressed in closed form. Also obtained as a by-product, are some striking recursive identities involving several Dirichlet
series including the well-known Riemann Zeta-function.
相似文献
17.
Dr. Erhard Braune 《Monatshefte für Mathematik》1976,82(1):17-26
Let (itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |(itk)(s)|, |(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt. 相似文献
18.
R. Nair 《Monatshefte für Mathematik》1998,125(3):241-253
Supposek
n denotes either (n) or (p
n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp
k denotes thek
th rationals prime. Also let
denote the sequence of convergents to a real numberx and letc
n(x))
n=1
be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants
n(x))
n=1
by
19.
Rudolf Grübel 《Probability Theory and Related Fields》1986,71(3):393-404
Summary Let be a probability and
the corresponding harmonic renewal measure. Complementing earlier results where is concentrated on a halfline we investigate the behaviour ofv
h
([x, x + 1]) and the harmonic renewal functionG(x) =v
h((–,x])asx ifm
1=x(dx)>0. We also consider the casem
1=0. 相似文献
20.
G. A. Kalyabin 《Functional Analysis and Its Applications》2004,38(3):184-191
We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities
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