Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together w...

Freiman’s Theorem describes the structure of a set A under the condition that A+ A has size close to that of A. If P is a generalized arithmetic progression, then |P +P | is close to |P |. Freiman’s Theorem states the partial converse: if |P + P | is close to P then P must be contained in a small generalized arithmetic progression. The theorem may be stated as follows, and we will give the rema...

Let U be an initial segment of ∗N closed under addition (such U is called a cut) with uncountable cofinality and A be a subset of U , which is the intersection of U and an internal subset of ∗N. Suppose A has lower U -density α strictly between 0 and 5 . We show that either there exists a standard real > 0 and there are sufficiently large x in A such that |(A+ A)∩[0, 2x]| > ( 10 3 + ) |A∩[0, x]...

These are lecture notes for the summer semester 2008 of the course Analytic and combinatorial number theory (NDMI045, Analytická a kombinatorická teoriě císel) which I have been teaching on the Faculty of Mathematics and Physics of the Charles University in Prague. The first booklet (the second one, [25], is for summer semester 2010) covers four major theorems, three on the distribution of prim...

Let k and n be positive integers, and let d(n;k) be the maximum density in f0;1;2 : : : ;kn 1g of a set containing no arithmetic progression of k terms with first term a = ∑aik and common difference d = ∑εik, where 0 ai k 1, εi = 0 or 1, and εi = 1 ) ai = 0. Setting βk = limn!∞ d(n;k), we show that limk!∞ βk is either 0 or 1. Throughout, we shall use the notation [a;b) = fa;a+ 1;a+ 2; : : : ;b ...

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an n-point subset of the plane is shown to be no more than ⌊(4n− 1)(n− 1)/18⌋. The number of k-term arithmetic progressions that lie within an n-point subset of the line is shown to be at most (n...

The following is a quick set of notes of some properties of Dirichlet characters, in particular, how they are used to prove the infinitude of primes in arithmetic progressions. These notes are from from An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash. As this is a modified snippet from the book, references to other parts of the book are displayed as ??. 1. Dirichlet Ch...

has only the solution (n, k, b, y, l) = (48, 3, 6, 140, 2) in positive integers n, k, b, y and l, where k, l ≥ 2, P (b) ≤ k and P (y) > k. Here, P (m) denotes the greatest prime factor of the integer m (where, for completeness, we write P (±1) = 1 and P (0) = ∞). Rather surprisingly, no similar conclusion is available for the frequently studied generalization of this equation to products of con...

Integers have many interesting properties. In this paper it will be shown that, for an arbitrary nonconstant arithmetic progression {an}TM=l of positive integers (denoted by N), either {an}TM=l contains infinitely many palindromic numbers or else 10|aw for every n GN. (This result is a generalization of the theorem concerning the existence of palindromic multiples, cf. [2].) More generally, for...