Gowers Norm, Pseudorandom Binary Sequences And D.H.Lehmer Problem

Recently there has been much progress in the study of arithmetic progressions, for example B. Green and T. Tao showed that primes contain arbitrarily long arithmetic progressions. An important tool in these developments is the Gowers uniformity norm. Therefore it would be interesting to investigate the Gowers norm further. On the other hand, stream ciphers play a crucial role in cryptography, and they are based on the use of pseudorandom binary sequences. This means that the search for new approaches and new constructions should be continued. In this dissertation we study the Gowers norm, pseudorandom binary sequences, and D. H. Lehmer problem, and establish some connections between these subjects. Furthermore, we study the character sum, Dedekind sum, Dirichlet L-function, exponential sum, and give some new results. The main achievements contained in this dissertation are as follows:1. Some new estimates forare given, and an estimate for the upper bound of Dirichlet L-function L(1,χ) is obtained. By using M. Toyoizumi's identitywe give some applications to the mean values of Cochrane sum, the difference between an integer and its inverse modulo a prime p, and a problem of D. H. Lehmer.2. Dedekind sum and related sums, and Dirichlet L-function are studied. First we give some formulae on the mean value of the Dedekind sum and Hardy sum, and generalize the results of J. B. Conrey, E. Fransen, R. Klein, C. Scott, C. Jia and W. Zhang. The hybrid mean value of Dedekind sum and primitive character is studied, and a new asymptotic formula is given. We obtain a new upper bound for the high-dimensional Cochrane sum, which improves the result of Z. Xu and W. Zhang. By establishing some connection between some generalized Dedekind sum and Dirichlet L-function, the mean value is studied. Finally we give a few generalized Subrahmanyam's identities and Knopp's theorems for the generalized Dedekind sum, Hardy sum and Cochrane sum.3. Mean values of generalized Gauss sum, Kloosterman sum and exponential sum are studied, and a few asymptotic formulae and identities are given. For details, two identities for the fourth power mean of the generalized Gauss sumare obtained, and the hybrid mean value of Gauss sum and generalized Bernoulli number are studied. Moreover, the hybrid mean value of generalized Bernoulli number, Kloosterman sum and Gauss sum are investigated. Some identities for the fourth power mean of the mixed exponential sumare obtained, and some new results on the generalized Kloosterman sum are given. Finally we study the fourth power mean of the exponential sumand give some applications to the fourth power mean of the hyper-Kloosterman sum.4. Some generalizations on the problem of D. H. Lehmer are given. For details, we study the distributions of D. H. Lehmer numbers which are the rth residue modulo p or the primitive roots modulo p~α, and give two asymptotic formulae. Furthermore, the relationship between the D. H. Lehmer problem and hyper-Kloosterman sum is studied.5. By using the D. H. Lehmer problem, multiplicative inverse, exponential sum, character sum and Dirichlet L-function, the following pseudorandom binary sequences are given: Moreover, it is proved that these sequences are "good" pseudorandom binary sequences.6. The Gowers norm for pseudorandom binary sequences are studied, and some connection between these two subjects are established. Some examples are given to show that the "good" pseudorandom sequences have small Gowers norm. Moreover, a generalization on the D. H. Lehmer problem is given.