Abstract not found

A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x

The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135-137). For any non-principal Dirichlet character χ (mod q) we let M(χ): = max

We consider several generalizations and variations of the character sum inequalities of Weil and Burgess. A number of incomplete character sum inequalities are proved while further conjectures are formulated. These inequalities are motivated by extremal graph theory with applications to problems in computer science. 1 1.

constants. Z, N and N0 denote the set of integers, positive integers and non-negative integers respectively. The cardinality of a set S is denoted by |S|. ⌊x ⌋ and {x} denote the integer part and the fractional part of x and ‖x ‖ denotes the distance from x to the nearest integer: ‖x ‖ = min({x}, 1 − {x}). We write e 2πiα = e(α). If f(n) = O(g(n)), then we write f(n) ≪ g(n); if the implied constant depends on a certain parameter c, then we write f(n) ≪c g(n). A, B,... denote subsets of N0 and A + B denotes the set of the non-negative integers n that can be represented in the form n = a + b with a ∈ A, b ∈ B. ω(n) denotes the number of distinct prime factors of n and Ω(n) denotes the number of prime factors of n counted with multiplicity. λ(n) is the Liouville function: λ(n) = (−1) Ω(n). The divisor function is denoted by τ(n).

In this talk we consider in an elementary way some simple problems which relate to incomplete sums and can be studied by appealing to a classical method of Vinogradov and its modifications. Vinogradov's idea was to use finite Fourier transforms in order to estimate an incomplete sum by means of complete but in some respects more complicated sums. A natural application was to estimate an incomplete sum of multiplicative characters by means of Gaussian sums. This application can be generalized in many natural ways; e.g., we may find bounds for the least nonnegative residue or nonresidue of a polynomial modulo a prime. Mordell and some others modified Vinogradov's method in order to find small solutions of congruences or small boxes containing solutions of a system of equations over finite fields. We survey these results briefly and in a very elementary way. Thereafter we consider a new interesting application. Code division multiple access systems require large families of sequences with...

father Matvei Avraam'evich was priest at the graveyard church (pogost) of the village of Milolyub in the Velikie Luki district of Pskov province in western Russia. His mother was a teacher. He early showed an aptitude for drawing and, instead of an ecclesiastical school (as would have been normal for a son of the clergy), his parents sent him in 1903 to the modern school {reaVnoe uchilishche: that is, with a scientific as opposed to a classical orientation) in Velikie Luki, whither his father had moved with his family on his translation to the Church of the Holy Shroud (Pokrovskaya Tserkov 1) there. In 1910, on completing school, Vinogradov entered the mathematical section of the Physico-mathematical Faculty of the University at the Imperial capital, St. Petersburg. Amongst the staff were A. A. Markov, whose lectures on probability he is said to have known by heart, and Ya. V. Uspenskil ( = J. V. Uspensky, later of Stanford University, U.S.A.), both with interests in number theory and probability theory. There had been a long tradition in these subjects (Chebyshev in both, Korkin, Zolotarev, Voronoi in number theory). Vinogradov was attracted to number theory,

λ3(n) = (−1) ω3(n) where ω3(n) is the number of distinct prime factors congruent to −1 mod 3 in n (with multiple factors counted multiply). We give explicit closed form evaluations of the following variety. Theorem 0.1. λ3(1) + λ3(2) + · · · + λ3(n) = Dn where Dn be the number of 1’s in the base three expansion of n. Note that above sum grows logarithmically, equals k for the first time when n = 3 0 + 3 1 + 3 2 + · · · + 3 k and is never negative. More generally let χp denote the Legendre character and let λp(n): = (−1) ωp(n) where ωp(n) is the number of distinct prime factors q with χp(q) = −1 (with multiple factors counted multiply). We give analogous formulae for λp(1) + λp(2) +... + λp(n). Theorem 0.2. For p = 5 λ5(1) + λ5(2) +... + λ5(n) = Dn where Dn be the number of 1’s in the base five expansion of n minus the number of 3’s in the base five expansion of n. While the analysis, as usual, conceals the approach all these results where found experimentally.

this paper we do not focus on the extreme values of jL(1; )j, but rather on the distribution of the set of values

Davenport and Erdős showed that the distribution of values of sums of the form ∑x+h