The Universality of Some Compositions on Short Intervals

Siberian Mathematical Journal - We obtain approximation theorems for analytic functions by the shifts $ F(\zeta(s+i\tau)) $ with $ \tau\in{??} $ , where $ \zeta(s) $ is the Riemann $...     

Analytic continuation ofζ 3 (S,K) to the critical strip. Arithmetic part

In this paper we study the zeta-function

Hermite Expansion of the Riemann Zeta Function

Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction, it is crucial to regard the zeta function as Fourier transfomations of Schwartz' distributions.     

Zeta function of a convolution

The zeta function of the convolution ,convergent in the domain s > 1, is extended meromorphically into the left half-plane. The explicit form of this extension is found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 22–48, 1990.     

ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.     

On the mean square value of Hurwitz zeta function

R Balasubramanian has shown that $$\mathop \smallint \limits_1^{\rm T} |\zeta (\tfrac{1}{2} + it)|^2 dt = T\log \tfrac{T}{{2\pi }} + (2\gamma - 1)T + O(T^{\theta + \in } )$$ with θ = 1/3. In this paper we develop a hybrid analogue for the mean square value of the Hurwitz zeta function ζ (s, a) and show that (i) new asymptotic terms arise in the expression for ζ (s, a) which are not present in the above expression for the ordinary zeta function and (ii) the corresponding error term is given by $$O(T^{5/12} log^2 T) + O\left( {\frac{{logT}}{{\left\| {2a} \right\|}}} \right)$$ for 0 <a < 1.     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

Mean value of weyl sums

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$      

Representations of $$\zeta (2n + 1)$$ and Related Numbers in the Form of Definite Integrals and Rapidly Convergent Series

Doklady Mathematics - Let $$\zeta (s)$$ and $$\beta (s)$$ be the Riemann zeta function and the Dirichlet beta function. The formulas for calculating the values of $$\zeta (2m)$$ and $$\beta (2m -...     

A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS

Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.     

Sums of the higher divisor function of diagonal homogeneous forms

The Ramanujan Journal - We give an asymptotic formula for $$\sum _{1\le n_1.n_2, \ldots ,n_l\le x^{1/r}}\tau _k(n^r_1+n^r_2+\ldots +n^r_l)$$ , where $$\tau _k(n)$$ represents the k-th divisor...     

Hyperbolic stochastic differential equations: Absolute continuity of the LKW of the solution at a fixed point

LetW be the Wiener process onT=[0, 1]². Consider the stochastic integral equation $$\begin{gathered} X_\zeta = x_0 + \int_{R_\zeta } {a_1 (\zeta \prime )X(s\prime ,dt\prime )ds\prime + } \int_{R_\zeta } {a_2 (\zeta \prime )X(ds\prime ,t\prime )dt\prime } \hfill \\ + \int_{R_\zeta } {a_3 (X_{\zeta \prime , } \zeta \prime )W(ds\prime ,dt\prime ) + } \int_{R_\zeta } {a_4 (X_{\zeta \prime , } \zeta \prime )ds\prime ,dt\prime ,} \hfill \\ \end{gathered} $$ whereR _ζ =(s, t) ∈ T, andx ₀ ∈ ?. Under some assumptions on the coefficients a_i, the existence and uniqueness of a solution for this stochastic integral equation is already known (see [6]). In this paper we present some sufficient conditions for the law ofX _ζ to have a density.     

The approximate solution of singular integral equations

A computational scheme of collocation type is proposed for a singular linear integral equation with a power singularity in the regular integral and the justification is given. The results obtained are used to justify the approximate solution of the singular integral equation $$Kx \equiv a(t)x(t) + \frac{{b(t)}}{{\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{x(\tau )d\tau }}{{\tau - t}} + \frac{1}{{2\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{h|t,\tau )x(\tau )}}{{\left| {\tau - t} \right|^\delta }}d\tau = f(t)$$ by a modification of the method of minimal residuals.     

On the Oscillation of Partial Difference Equations Generated by Deviating Arguments

We offer sufficient conditions for the oscillation of all solutions of the partial difference equations y(m - 1,n) + β(m,n)y(m, n - 1) -δ(m,n)+ P(m,n,y(m + k,n + l)) = Q(m,n,y(m + k,n + l)) and (y(m - 1,n)+ β(m,n)y(m,n - 1) - δ(m,n)y(m,n) + $$\mathop \Sigma \limits_{i = 1}^\tau $$ Pi(m,ny(m + ki,n + li)) = $$\mathop \Sigma \limits_{i = 1}^\tau $$ Qi(m,n,y9m + ki,n + li)). Several examples which dwell upon the importance of our results are also included.     

On the error term $$\Delta _{(k,l)}(x)$$ Δ ( k,l ) ( x )

The Ramanujan Journal - Let k, l be non-negative integers and $$\zeta ^{(k)}(s)$$ denote the kth derivative of the Riemann zeta function $$\zeta (s).$$ Further let $$d_{(k,l)}(n)$$ be the nth...     

EXISTENCE CONDITIONS OF A SPECIAL TYPE OF LIAPUNOV FUNCTIONAL FOR TWO-DIMENSIONAL DELAY SYSTEM

In this paper, the author considers the two-dimensional delay systems $$\[\mathop x\limits^ \cdot (t) = Ax(t) + Bx(t - r),A,B \in {R^{2 \times 2}},x \in {R^2},r = const \ge 0\]$$ and gives the necessary and sifficient conditions under which where exists a simple type of positive definite Liapunov functional $$\[V(\varphi ) \buildrel \Delta \over = {\varphi ^''}(0){T_\varphi }(0) + \int_{ - \tau }^0 {{\varphi ^''}(\theta )E\varphi (\theta )d\theta } \]$$ and $\[\alpha (s)\]$(where T , E are positive definite 2x2 matrices, $\[\varphi \in C([ - \tau ,0],{R^n})\]$, "." stands for transpose, $\[\alpha (s)\]$ is continuous and $\[\alpha (0) = 0,\alpha (s) > 0,s > 0\]$. such that $\[{V_{(*)}}(\varphi ) \le - \alpha (\left| {\varphi (0)} \right|).\]$.     

On extensions of the generalized quadratic functions from “large” subsets of semigroups

Let $$(G,+)$$ be a commutative semigroup, $$\tau $$ be an endomorphism of G and involution, D be a nonempty subset of G, and $$(H,+)$$ be an abelian group, uniquely divisible by 2. Motivated by the extension problem of J. Aczél and the stability problem of S.M. Ulam, we show that if the set D is “sufficiently large”, then each function $$g{:} D\rightarrow H$$ such that $$g(x+y)+g(x+\tau (y))=2g(x)+2g(y)$$ for $$x,y\in D$$ with $$x+y,x+\tau (y)\in D$$ can be extended to a unique solution $$f{:} G\rightarrow H$$ of the functional equation $$f(x+y)+f(x+\tau (y))=2f(x)+2f(y)$$.     

On a characteristic of cosine functions

This paper is concerned with the property of cosine function. It is proved that a family {T(t)} _t≥0 of strongly continuous linear operators is a cosine function on Banach space X if and only if T(0)=I and there holds $$\begin{aligned} \int_0^{t+s}T(\tau)d\tau=T(t)\int _0^s T(\tau)d\tau+\int_0^t T(\tau)d\tau T(s),\quad t,s \geq0, \end{aligned}$$ where all the integrals concerning operator valued functions are understood to be in the strong operator topology.     

Asymptotics of solutions of a system of linear inhomogeneous singularly perturbed differential equations

Solutions with asymptotics in integral and fractional powers of the parameter ? are constructed for the vector differential equation $$\varepsilon ^h \dot X = A(t,\varepsilon ) X + \varepsilon ^{\alpha _1 } p(t,\varepsilon ) \exp \left( {\varepsilon ^{ - h} \int\limits_0^t {\lambda (\tau )d\tau } } \right)$$ in the case of resonance and multiple spectrum of the limit matrix. $$\varepsilon ^h \dot X = A(t,\varepsilon ) X + \varepsilon ^{\alpha _1 } p(t,\varepsilon ) \exp \left( {\varepsilon ^{ - h} \int\limits_0^t {\lambda (\tau )d\tau } } \right)$$      

Eine Erweiterung des Riemann-Lebesgue-Lemmas

It is proved that iff∈L ₁(?),f'∈L ₁(?) and ∫∣x∣ ⁱ ∣f(x)∣dx<∞ fori=1, ...,k?1 and ifA=(a _ij ) is a (k×k)-matrix with non-vanishing determinant, for $$\tilde f_A (\zeta ): = \smallint \exp (i\zeta _1 \sum\limits_{j = 1}^k {a_{1j} x^j } + ... + i\zeta _k \sum\limits_{j = 1}^k {a_{kj} x^j } )f(x)dx$$ the following relation holds: $$\tilde f_A (\zeta ) = O(\left\| \zeta \right\|)^{ - b_k } with b_k : = (\sum\limits_{j = 1}^k {j!)^{ - 1} } for k \in \mathbb{N}$$ .