Inequalities concerning the function π(x): Applications

Laurenţiu Panaitopol

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A note on the equation $a x^{n} - b y^{n} = c$

Maurice Mignotte (1996)

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On sums of two cubes: an Ω₊-estimate for the error term

M. Kühleitner, W. G. Nowak, J. Schoissengeier, T. D. Wooley (1998)

Acta Arithmetica

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The arithmetic function $r_{k} (n)$ counts the number of ways to write a natural number n as a sum of two kth powers (k ≥ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of $r_{k} (n)$ leads in a natural way to a certain error term $P_{_{k}} (t)$ which is known to be $O (t^{1 / 4})$ in mean-square. In this article it is proved that $P_{₃} (t) = Ω ₊ (t^{1 / 4} {(l o g l o g t)}^{1 / 4})$ as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a...

Refinement of an estimate for the Hurwitz zeta function in a neighbourhood of the line σ = 1

Mieczysław Kulas (1999)

Acta Arithmetica

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The well-known estimate of the order of the Hurwitz zeta function $ζ (s, α) - α^{- s} ≪ t^{c {(1 - σ)}^{3 / 2}} l o g^{2 / 3} t$ 0. The improvement of the constant c is a consequence of some technical modifications in the method of estimating exponential sums sketched by Heath-Brown ([11], p. 136).

On the frequencies of large values of divisor functions

Karl K. Norton (1994)

Acta Arithmetica

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Effective solution of families of Thue equations containing several parameters

Clemens Heuberger, Robert F. Tichy (1999)

Acta Arithmetica

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A generalization of Zeeman’s family

Michał Sierakowski (1999)

Fundamenta Mathematicae

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E. C. Zeeman [2] described the behaviour of the iterates of the difference equation $x_{n + 1} = R (x_{n}, x_{n - 1}, . . ., x_{n - k}) / Q (x_{n}, x_{n - 1}, . . ., x_{n - k})$ , n ≥ k, R,Q polynomials in the case $k = 1, Q = x_{n - 1}$ and $R = x_{n} + α$ , $x_{1}, x_{2}$ positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.

On the representation of integers as sums of distinct terms from a fixed set

Norbert Hegyvári (2000)

Acta Arithmetica

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A lower bound for the rank of $J_{0} (q)$

E. Kowalski, P. Michel (2000)

Acta Arithmetica

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On the diophantine equation $x^{2} - p^{m} = \pm y^{n}$

Yann Bugeaud (1997)

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Bounds for the solutions of unit equations

Yann Bugeaud, Kálmán Győry (1996)

Acta Arithmetica

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On the greatest prime factor of (ab + 1) (ac + 1) (bc + 1)

C. L. Stewart, R. Tijdeman (1997)

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Sumsets of Sidon sets

Imre Z. Ruzsa (1996)

Acta Arithmetica

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1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that $N^{1 / 3}$ is possible. In [1] they show that...