On the distribution of the power generator

We present a new method to study the power generator of pseudorandom numbers modulo a Blum integer . This includes as special cases the RSA generator and the Blum-Blum-Shub generator. We prove the uniform distribution of these, provided that the period with fixed 0$"> and, under the same condition, the uniform distribution of a positive proportion of the leftmost and rightmost bits. This sharpens and generalizes previous results which dealt with the RSA generator, provided the period . We apply our results to deduce that the period of the binary sequence of the rightmost bit has exponential length.

     

On the power generator and its multivariate analogue

We obtain a new estimate on the discrepancy of the power generator over a part of the period that improves several previous results. We also introduce a multidimensional analogue and show that the corresponding vector sequence is uniformly distributed, provided it is of a sufficiently large period. This result is based on a recent estimate of T. Cochrane and C. Pinner on binomial exponential sums. Our construction extends the class of nonlinear pseudorandom number generators for which a power saving against the trivial bound is possible in estimates of their discrepancy. It has several additional properties such as high nonlinearity and inhomogeneity which may be useful for its cryptographic applications.     

关于位数函数高次均值的计算

针对1993年美国数论专家Smarandache提出了初等数论及集合论中的105个未解决的问题中的5个关于自然数列的性质问题,就自然数列的位数函数问题进行了研究,给出了在一个正整数的n进制表示中的位数函数定义,采用了归纳、猜想的方法得出了位数函数a（m,n）的高次均值的精确计算公式.     

Square-free values of the Carmichael function

We obtain an asymptotic formula for the number of square-free values among p−1, for primes p?x, and we apply it to derive the following asymptotic formula for L(x), the number of square-free values of the Carmichael function λ(n) for 1?n?x,

     

Rational values of the Riemann zeta function

We prove the existence of a constant C such that for any D?3 there are at most rational numbers s with 2<s<3 and denominator at most D such that ζ(s) is also rational with denominator at most D. This is done by combining elements of the works of Bombieri-Pila, Pila, and Surroca with a new zero estimate.     

Probability model of Gunther generator

This paper constructs the probability model of Gunther generator at first,and the finite dimension union distribution of the output sequence is presented.The result shows that the output sequence is an independent and uniformly distributed 0,1 random variable sequence. It gives the theoretical foundation about why Gunther generator can avoid the statistic weakness of the output sequence of stop-and-go generator,and analyzes the coincidence between output sequence and input sequences of Gunther generator.The conclusions of this paper would offer theoretical references for designers and analyzers of clock-controlled generators.     

Uniqueness of entire function sharing a small function with its derivative

This paper is devoted to studying the relationship between an entire function and its derivative when they share one small function. We generalize some previous results of Gundersen and Yang [G. Gundersen, L.Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998) 85–95], Chang and Zhu [J. Chang, Y. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl. 351 (2009) 491–496].     

整函数的公共值与奇异方向

设f(z)是复平面上的超越整函数，本文在f(z)的级满足一定限制下证明了复平面上存在一条从原点出发的射线OR，使得以OR为分角线的任意小角域内f(z)与其导函数f(z)至多只有一个IM公共值。     

全纯函数族涉及公共值的正规定则

研究涉及公共值的全纯函数正规族问题．设F为单位圆△内的全纯函数族,a,b为两个判别的有穷复数且b≠0．文中证明了若对每一个,f∈F,Ef(a)=Ef(a),E↑-f′（b)包含于E^-f(b),则F在单位圆△内正规．     

Subdifferential calculus for a quasiconvex function with generator

Recently, we discussed optimality conditions for quasiconvex programming by introducing ‘Q-subdifferential’, which is a notion of differential of quasiconvex functions. In this paper, we investigate basic and fundamental properties of the Q-subdifferential. Especially, we show results of a chain rule for composition with non-decreasing functions, monotonicity of the Q-subdifferential, mean-value theorem, a sufficient condition for a global minimizer for quasiconvex programming, and the calculus of the Q-subdifferential of the supremum of quasiconvex functions.     

On the power values of power sums

Using several currently available techniques, including Baker's method, Frey curves and modular forms, we prove that for odd values of k with 1?k<170, the equation

k¹+k²+?+xk=y2n     

The Herglotz-Zagier function, double zeta functions, and values of L-series

In this paper, we define L-series generalizing the Herglotz-Zagier function (Ber. Verhandl. Sächsischen Akad. Wiss. Leipzig 75 (3-14) (1923) 31, Math. Ann. 213 (1975) 153), and the double zeta function (First European Congress of Mathematics, Vol. II, Paris, 1992, pp. 497-512; Progress in Mathematics, Vol. 120, Birkhäuser, Basel, 1994) and evaluate them after meromorphic continuation at integer points in their extended domains. This is accomplished in three steps. First, when is a periodic function and are the harmonic numbers, we establish identities relating these series to the L-series

     

On conditional irrationality measures for values of the digamma function

The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the principles of Chudnovsky and then apply them to obtain conditional irrationality measures for values of the digamma function.     

Asymptotically good towers of function fields with small p-rank

Over any quadratic finite field we construct function fields of large genus that have simultaneously many rational places, small p-rank, and many automorphisms.     

On the Nevanlinna direction of an algebroid function dealing with multiple values

In this paper, by using Ahlfors' theory of covering surfaces, we prove that for an algebroid function w(z) satisfying , there exists at least one Nevanlinna direction dealing with multiple values.     

Sharp power mean bounds for the Gaussian hypergeometric function

Sharp inequalities are established between the Gaussian hypergeometric function and the power mean. These results extend known inequalities involving the complete elliptic integral and the hypergeometric mean.     

On the Linear Complexity of the Power Generator

We obtain a lower bound on the linear complexity of the powergenerator of pseudo-random numbers, which in some special cases is alsoknown as the RSA generator and as the Blum–Blum–Shubgenerator. In some very important cases this bound is essentially thebest possible. In particular, this implies that lattice reductionattacks on such generators are not feasible.     

整函数与其微分多项式IM分担两个值

设 f(Z)为非常数整函数 ,L (f )为 f (Z)的具有小函数系数的线性微分多项式 ,本文研究当 f(Z)与 L (f ) IM分担两个有穷复数时的唯一性问题 ,给出了 f (Z)与L (f )完全确定的表达式 ,改进了李平、杨重骏最近获得的一个结果 ,从而也改进了先前的一些结果 .     

An integral representation of the Mordell-Tornheim double zeta function and its values at non-positive integers

A surface integral representation of the Mordell-Tornheim double zeta function is given, which is a direct analogue of a well-known integral representation of the Riemann zeta function of Hankel’s type. As an application, we investigate its values and residues at integers, where generalizations of a generating function of Bernoulli numbers naturally appear.