Additive representations of integers

In this paper we shall mainly study additive representations of integers prime to the firstm primes as a sum of some integers having a peculiar property. The conjectures of Goldbach and twin primes are also observed in connection with these representations of integers.     

Additive representations of integers

In this paper we shall mainly study additive representations of integers prime to the first m primes as a sum of some integers having a peculiar property. The conjectures of Goldbach and twin primes are also observed in connection with these representations of integers. Received: 20 October 1997     

Multiplicative properties of sets of positive integers

It is proved that every set of positive integers with upper Dirichlet density greater than 1/2 contains three distinct elements whose product is a square. Several similar problems are considered.     

On binary signed digit representations of integers

Applications of signed digit representations of an integer include computer arithmetic, cryptography, and digital signal processing. An integer of length n bits can have several binary signed digit (BSD) representations and their number depends on its value and varies with its length. In this paper, we present an algorithm that calculates the exact number of BSDR of an integer of a certain length. We formulate the integer that has the maximum number of BSDR among all integers of the same length. We also present an algorithm to generate a random BSD representation for an integer starting from the most significant end and its modified version which generates all possible BSDR. We show how the number of BSD representations of k increases as we prepend 0s to its binary representation.     

On optimal binary signed digit representations of integers

Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is and how to generate them entirely. We also show which kinds of integers have the maximum number of optimal BSD-Rs.     

Kolmogorov complexity and set theoretical representations of integers

We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations) in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple representation of integers that we suitably effectivize in order to develop an associated Kolmogorov theory. Such effectivizations are particular instances of a general notion of “self‐enumerated system” that we introduce in this paper. Our main result asserts that, with such effectivizations, Kolmogorov theory allows to quantitatively distinguish the underlying semantics. We characterize the families obtained by such effectivizations and prove that the associated Kolmogorov complexities constitute a hierarchy which coincides with that of Kolmogorov complexities defined via jump oracles and/or infinite computations (cf. [6]). This contrasts with the well‐known fact that usual Kolmogorov complexity does not depend (up to a constant) on the chosen arithmetic representation of integers, let it be in any base n ≥ 2 or in unary. Also, in a conceptual point of view, our result can be seen as a mean to measure the degree of abstraction of these diverse semantics. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplicative character sums and products of sparse integers in residue classes

We estimate multiplicative character sums over the integers with a fixed sum of binary digits and apply these results to study the distribution of products of such integers in residues modulo a prime p. Such products have recently appeared in some cryptographic algorithms, thus our results give some quantitative assurances of their pseudorandomness which is crucial for the security of these algorithms.     

Multiplicative factorizations of integral representations by quadratic forms

Multiplicative dependence of integral representations of integers by positive definite quadratic forms in an odd number of variables on square factors of the represented numbers is studied. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 15–55. Translated by A. S. Goloubeva.     

Multiplicative functions supported on the k-free integers with small partial sums

The Ramanujan Journal - We provide examples of multiplicative functions f supported on the k-free integers such that at primes $$f(p)=\pm 1$$ and such that the partial sums of f up to x are...     

On the distribution of digits in Cantor representations of integers

Asymptotic formulas on the average values of the “sum of digits” function and the average numbers of occurrences of fixed subblocks in Cantor representations of integers are established. The theorems generalize a result by H. Delange [Enseign. Math.21 (1975), 31–47].     

Hardy-Littlewood system and representations of integers by an invariant polynomial

Let f be an integral homogeneous polynomial of degree d, and let be the level set for each . For a compact subset in ), set

Number of the representations of integers by certain ternary quadratic forms

Exact formulas are derived for the number of the representations of positive integers by certain positive ternary quadratic forms, belonging to multiclass genera. The Fourier coefficients of the corresponding cusp forms (on the basis of the known results of Waldspurger and Tunnell) are expressed in terms of the values of the Hasse-Weil L-functions of certain elliptic curves at the center of the critical strip.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 154–162, 1986.