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1.
Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
We also show that
where ω(n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9]. Received: 23 November 2006  相似文献   

2.
Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let
R. de la Bretèche studied the maximum values of τ P (n) in intervals. Here the following is proved: if P(X)∈ℤ[X] is not of the form a(X+b) k with a,b∈ℚ, and k∈ℕ then
This improves partially on La Bretèche’s results. Research partially supported by Hungarian National Foundation for Scientific Research, Grants T043631, T043623 and T049693.  相似文献   

3.
We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in with “degree” (a)=m satisfies
with constants q>1, ρ 1<ρ 2<⋅⋅⋅<ρ r =ρ, ρ≥1, γ>1+ρ. For the main result, let α,τ,η be positive constants such that α>1,τ ρ≥1, and τ α ρ≥1. Then for a multiplicative function f(a) on the following two conditions (A) and (B) are equivalent. These are (A) All four series
converge and
and (B) The order τ ρ mean-value
exists with m f ≠0 and the limit
exists with M v (α)>0.   相似文献   

4.
In this paper, the authors present some new results for the oscillation of the second order nonlinear neutral differential equations of the form
. Easily verifiable criteria are obtained that are also new for differential equations without neutral term i.e. for p(t)≡0.  相似文献   

5.
Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB n(n = 1,2,3,...) assert that (1-p m-1)B m/m isp-integral and
((1))
ifm ≡ n (mod (p-1)p n). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤jp-l.Then for any non-negative integerN,
((2))
ifp-1 is not a divisor ofm andm ≡ n (mod (p-1)p n). HereB n(x) (n = 0,1,2,...) are Bernoulli polynomials. This of course contains the Kummer’s congruences. Furthermore, it contains new congruences for Bernoulli polynomials of odd indices.  相似文献   

6.
Let r k(n) denote the number of representations of an integer n as a sum of k squares. We prove that
where
Here n = 2 p p p is the prime factorisation of n, n is the square-free part of n, the products are taken over the odd primes p, and ( ) is the Legendre symbol.Some similar formulas for r 7(n) and r 9(n) are also proved.  相似文献   

7.
Let p be a prime number, n be a positive integer, and ƒ(x) = axk + bx. We put
where e(t) = exp(2πit). This special exponential sum has been widely studied in connection with Waring’s problem. We write n in the form n = Qk + r, where 0 ≤ r ≤ k − 1 and Q ≥ 0. Let α = ord p(k), β = ord p(k − 1), and θ = ord p(b). We define
and J = [ζ]. Moreover, we denote V = min(Q, J). Improving the preceding result, we establish the theorem. Theorem. Let k ≥ 2 and n ≥ 2. If p > 2, then
. An example showing that this result is best possible is given. Bibliography: 15 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 63–75.  相似文献   

8.
§ 1 IntroductionConsiderthenonautonomousdelaylogisticdifferenceequationΔyn =pnyn( 1 - yτ(n) )  ,n =0 ,1 ,2 ,...,( 1 1 )wherepn ∞n =0 isasequenceofpositiverealnumbers ,τ(n) ∞n =0 isanondecreasingsequenceofintegers,τ(n) <nandlimn→∞τ(n) =∞ ,Δyn=yn +1- yn.Motivatedbyplausibleapplications…  相似文献   

9.
Oscillatory and asymptotic behaviour of solutions of a class of fourth order nonlinear neutral difference equations of the form
and
are studied under the assumption < ∞, for different ranges of p(n). Sufficient conditions are obtained for the existence of positive bounded solutions of (E).   相似文献   

10.
A detailed analysis is made of the structure of positive solutions of fourth-order differential equations of the form
under the assumption that α, β are positive constants, p(t), q(t) are positive continuous functions on [a,∞), and p(t) satisfies
Mathematics Subject Classification (2000) 34C10, 34D05  相似文献   

11.
Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the type we prove the following prime analogue of a conjecture of Erd?s & Pomerance (1985) concerning the exponent function fa(p) (defined to be the minimal exponent e for which ae ≡ 1 modulo p):
((‡))
where
The main result is obtained by computing all the higher moments corresponding to ω(fa(p)), and by comparing them, via the Fréchet-Shohat theorem, with estimates due to Halberstam concerning the moments of ω(p − 1). Received: 25 October 2004; revised: 12 February 2005  相似文献   

12.
Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral difference equations of the form
and
are studied under the assumption , for various ranges of p(n). Sufficient conditions are obtained for the existence of bounded positive solutions of (NH).   相似文献   

13.
The paper studies the function L(τ;·) defined by the Dirichlet series
, where τ(υ) is the υth Fourier coefficient of the Kubota-Patterson cubic theta function. For this function, an exact and an approximate functional equations are derived. It is established that the function does not vanish in the halfplane Re s ≥ 1.3533 and has no singularities except for a simple pole at the point 5/6. Issues related to computing the coefficients τ(υ) and values of the special functions arising in the approximate functional equation are considered. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 212–232.  相似文献   

14.
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
converge and (2)
if and only if the order τρ mean-value
exists with and the limit
exists with . The proof is deduced from an intrinsic connection between m f and . An erratum to this article can be found at  相似文献   

15.
The paper considers the function defined by the Dirichlet series
, where τ(ν) is the νth Fourier coefficient of the cubic Kubota-Patterson theta function. The zeros of the function L(τ ·) in the domain |Im s| < 444 are computed. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 173–186.  相似文献   

16.
For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p r)≤A r,A>0, and for anyε>0 there exist constants ,α>0 such that and Σ p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:
. Hereτ(n) is the number of divisors ofn andC(ƒ) is a constant. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 443–456, September, 1998. The work of the first author was supported by the Russian Foundation for Basic Research.  相似文献   

17.
We remark that an easy combination of two known results yields a positive answer, up to log(n) terms, to a duality conjecture that goes back to Pietsch. In particular, we show that for any two symmetric convex bodies K, T in , denoting by N(K, T) the minimal number of translates of T needed to cover K, one has:
, where are the polar bodies to K, T, respectively, and C  ≥ 1 is a universal constant. As a corollary, we observe a new duality result (up to log(n) terms) for Talagrand’s functionals.  相似文献   

18.
Denote by B(τ) the class of all complex functions of the form
$ f(z) \equiv \tau + \sum\limits_{n = 1}^\infty {(a_n (f)z^n + \overline {b_n (f)} \bar z^n )} $ f(z) \equiv \tau + \sum\limits_{n = 1}^\infty {(a_n (f)z^n + \overline {b_n (f)} \bar z^n )}   相似文献   

19.
20.
Let h(d) be the class number of properly equivalent primitive binary quadratic forms ax2+bxy+cy2 with discriminant d=b2-4ac. The behavior of h(5p2), where p runs over primes, is studied. It is easy to show that there are few discriminants of the form 5p2 with large class numbers. In fact, one has the estimate
x^{1 - \delta } \} \ll x^{2\delta } ,$$ " align="middle" vspace="20%" border="0">
where is an arbitrary constant number in (0;1/2). Assume that (x) is a positive function monotonically increasing for x and (x). If
, then (assuming the validity of the extended Riemann hypothesis for certain Dedekind zeta-functions) it is proved that
\alpha (x)} \right\} \asymp \frac{{\pi (x)}}{{\alpha (x)}}.$$ " align="middle" vspace="20%" border="0">
It is also proved that for an infinite set of p with one has the inequality
where log k p is the k-fold iterated logarithm (k is an arbitrary integer, k3). Results on mean values of h(5p 2 ) are also obtained. Similar facts are true for the residual indices of an integer a2 modulo p:
where o(a,p) is the order of a modulo p. Bibliography: 13 titles.  相似文献   

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