On the addition of residue classes modp

Given distinct residue classesa ₁ ,...,a _k modulo a primep, we consider the setS _j of all sums , whereli ₁<...<i _j k. We give a sufficient condition for the inequality |S _j |j(k–j)+1 to hold.     

Counting primes in residue classes

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing can be used for computing efficiently , the number of primes congruent to modulo up to . As an application, we computed the number of prime numbers of the form less than for several values of up to and found a new region where is less than near .

     

On the distribution of the summands of partitions in residue classes

Summary It is proved that the summands of almost all partitions of nare well-distributed modulo dfor dup to d= n^1/2-ε.     

Pairing conjugate partitions by residue classes

We study integer partitions in which the parts fulfill the same congruence relations with the parts of their conjugates, called conjugate-congruent partitions. The results obtained include uniqueness criteria, weight lower-bounds and enumerating generating functions.     

Divisors in residue classes, constructively

Let be integers satisfying , , , and let . Lenstra showed that the number of integer divisors of equivalent to is upper bounded by . We re-examine this problem, showing how to explicitly construct all such divisors, and incidentally improve this bound to .

     

On the distribution of the summands of unequal partitions in residue classes

Summary It is proved that the summands of almost all unequal partitions of nare well-distributed modulo dfor d=o(n^1/2).     

Explicit bounds for primes in residue classes

Let be an abelian extension of number fields, with . Let and denote the absolute discriminant and degree of . Let denote an element of the Galois group of . We prove the following theorems, assuming the Extended Riemann Hypothesis:

(1)

There is a degree- prime of such that , satisfying .

(2)

There is a degree- prime of such that generates
the same group as , satisfying .

(3)

For , there is a prime such that , satisfying
.

In (1) and (2) we can in fact take to be unramified in . A special case of this result is the following.

(4)

If , the least prime satisfies
.

It follows from our proof that (1)--(3) also hold for arbitrary Galois extensions, provided we replace by its conjugacy class . Our theorems lead to explicit versions of (1)--(4), including the following: the least prime is less than .

     

The divisor function on residue classes III

The Ramanujan Journal - We combine an extended version of Bailey’s transform with an identity of Bressoud and with some identities of Berkovich and Warnaar to prove a variety of positivity...     

The distribution of sequences in residue classes

We prove that any set of integers with lies in at least many residue classes modulo most primes . (Here is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below which are multiplicatively generated by the coprime integers (i.e. whose counting function is also ) lie in at least residue classes, modulo most small primes , where as .

     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

Distribution of values of arithmetic functions in residue classes

We give a criterion for weakly uniform distribution of integral multiplicative functionsf(n) of the class CS modulo N, generalizing a result of Narkiewicz (W. Narkiewicz, Acta Arithm.,12, 269–279 (1967)). We obtain an asymptotic formula for N(n<x|f(n)=a(mod N)). We consider particular cases for the functionf(n):r ₂(n) the number of integral points in the circlex ²+y ²n, and others.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 176–186, 1983.     

Catalan and Apéry numbers in residue classes

We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p^13/2(logp)⁶ elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring p^O(p) elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apéry numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p.     

On precover classes

Sunto LetG andH be abstract classes of modules. The classH is said to have theG-property if to each infinite cardinal λ there exists a cardinal κ>λ such that for everyF∈H with |F|≥κ and every its submoduleK with |F/K|≤λ there exists a submoduleL ofK such thatF/L/teG and |F/L|<κ. This condition is stronger than the condition (P) requiringL≠0 instead of |F/L|<κ, which was introduced and investigated in [8]. In this note we are going to study the relations of this more general condition to the existence of precovers with respect to some classes of modules. As an application we obtain some sufficient conditions for the existence of σ-torsionfree precovers related to a given hereditary torsion theory σ for the categoryR-mod. This result is closely related to and in some sense extends that of [5]. The research has been partially supported by the Grant Agency of the Czech Republic, grant #GAČR 201/03/0937 and also by the institutional grant MSM 113 200 007.     

On torsionfree classes which are not precover classes

In the class of all exact torsion theories the torsionfree classes are cover (pre-cover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite type is presented. This research has been partially supported by the Grant Agency of the Czech Republic, grant #GAČR 201/06/0510 and also by the institutional grant MSM 0021620839.     

Distribution of the full rank in residue classes for odd moduli

The distribution of values of the full ranks of marked Durfee symbols is examined in prime and nonprime arithmetic progressions. The relative populations of different residues for the same modulus are determined: the primary result is that k-marked Durfee symbols of n equally populate the residue classes a and bmod2k+1 if gcd(a,2k+1)=gcd(b,2k+1). These are used to construct a few congruences. The general procedure is illustrated with a particular theorem on 4-marked symbols for multiples of 3.