On the Periodic Zeta‐Function. II

We obtain a more precise asymptotic formula than in [Lith. Math. J., 41(2), 168–177 (2001)] for the mean square of the periodic zetafunction.     

On Small Values of the Riemann Zeta‐Function on the Critical Line and Gaps Between Zeros

Small values of are investigated, using the value distribution results of A. Selberg. This gives an asymptotic formula for Some related problems involving values of and gaps between ordinates of consecutive zeros of (s) are also discussed.     

Properties of the Appell–Lerch function (I)

The Ramanujan Journal - A number of equations involving the Appell–Lerch function, $$ mu $$ , are derived. Emphasis is placed on equations which are analogous to certain linear relations...     

On the Mean Square Value Formula of Lerch zeta-function

For complex number s =σ + it and real number 0<α,x<1, let φ(x,α,s) be Lerchzeta-function defined by φ(x,α,s)=sum from n=0 to ∞ (e~(2πinx))/(n+α)~s for Re(S)>1and its analytic continuation, and let of φ_1(x,α,s) =α(x,α,s) -α~(-s). The main purposeof this paper is to study the asymptotic properties of the mean square value     

A Discrete Limit Theorem for the Matsumoto Zeta‐Function in the Space of Meromorphic Functions

In the paper, a discrete limit theorem for the Matsumoto zetafunction in the space of meromorphic functions is proved.     

A Discrete Limit Theorem for the Matsumoto Zeta‐Function in the Space of Analytic Functions

In the paper, a discrete distribution of the Matsumoto zetafunction is considered. It is proved that the probability measure , converges weakly as .     

On the Error Function of the Square-full Integers

A positive integer n is called square-full, if p|n implies p~2|n (p is a primenumber).Let L(x) denote the number of the square-full integers not exceedingx.It is well-known that     

On the Relations Between the Numbers of Deficient Function and the Growth of an Algebroidal Function

Let f(z) be an algebroidal function defined by the following irreducible eq-uation A_ny~n A_(n-1)y~(n-1) … A_1y A_0=0 (1)where A_j(j=0,1,…,n) are entire functions in the finite complex plane.     

On the Favard Class of Cosine Operator Function

61.IntroductionLetXbeaBanachspace,B(X)thealgebraofallboundedlinearoPeratoronXandC(t),t6R,thestronglycontinuouscosineoperatorfunctionsonXwithinfinitesimalgeneratorA,TheFavard(orSaturation)classorgeneralizedFav(A)ofcosineoperatorfuctionC(t)withgeneratorAisdefinedbyFav(A)={x:supl,1<,Il2t~'(C(t)-I)xllX oo}.Theestimater'P,llC(t)x-xIl=IlAJoJoC(p)xdedsIl相似文献   

On the Uniformity of Distribution of the Naor–Reingold Pseudo-Random Function

We show that the new pseudo-random number function, introduced recently by M. Naor and O. Reingold, possesses one more attractive and useful property. Namely, it is proved that for almost all values of parameters it produces a uniformly distributed sequence. The proof is based on some recent bounds of character sums with exponential functions.     

On a Positivity Property of the Riemann ξ‐Function

We consider a quantitative version of the positivity condition of the Riemann function. J. C. Lagarias obtained that for >10. The main aim of this paper is to prove the equality for 1.     

On the Approximation of Continuous Function by Harmonic Means

Let f(x) be a continuous and periodic function with period 2π and [f]=a_o/2+sum from n=1 to ∞(a_ncos nx+b_nsin nx) be its Fourier series. Denote by s_n(f,x) the n-th partial sums of [f] and by ω(f,δ) the moduls of continuity of f(x). When ω(δ) is a modul of continuity, we denote by H[ω]。the class of all functions for which ω(f,t)≤ω(t). It is well known that the n-th harmonic means and the n-th Cesro means of f∈C_2, are defined as##属性不符     

On the dual nature of partial theta functions and Appell–Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell–Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell–Lerch sums. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers–Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions.     

Some discrete Fourier transform pairs associated with the Lipschitz–Lerch Zeta function

It is shown that there exists a companion formula to Srivastava’s formula for the Lipschitz–Lerch Zeta function [see H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77–84] and that together these two results form a discrete Fourier transform pair. This Fourier transform pair makes it possible for other (known or new) results involving the values of various Zeta functions at rational arguments to be easily recovered or deduced in a more general context and in a remarkably unified manner.     

A Product Formula for the Twisted Lefschetz Zeta Function

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On the Properties of the Solution of a Strongly Degenerate Parabolic Equation

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On the Extensions Preserving the Shape of a Semi-Hölder Function

We present some results concerning the extension of a semi-Hölder real-valued function defined on a subset of a quasi-metric space, preserving some shape properties: the smallest semi-Hölder constant, the radiantness and the global minimum (maximum) of the extended function.     

Schrödinger Operator with Morse Potential and Zeros of the Riemann Zeta Function

Mathematical Notes -     

On the Lucas Base and Computation of Counting Function Mean Value

In this paper, we introduce a new counting function a(m) related to the Lucas number, then use conjecture and induction methods to give an exact formula Ar(N)=α(n), (r=1,2,3) and prove them.     

On the Geometric Properties of the Poisson Kernel for the Lamé Equation

Computational Mathematics and Mathematical Physics - It is shown that the Poisson kernel for the Lamé equation in a disk can be interpreted as a bi-univalent mapping of the projection of an...