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Sums of Higher Divisor Functions with Almost Equal Variables
Let k, ℓ ≥ 3 be integers, and let τ k ( n ) denote the k -th divisor function. In this paper, we apply the circle method to obtain an asymptotic formula...
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Low moments of Dirichlet series
We determine the maximum possible size of the q th moment of a Dirichlet series, for 1 ≦ q ≦ 2.
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On Ramanujan sums over the Gaussian integers
This note deals with Ramanujan sums c m ( n ) over the ring ℤ[ i ], in particular with asymptotics for sums of c m ( n ) taken over both variables m , n .
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Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms
Let l and m be two integers with l > m ≥ 0, and let a and b be integers with a ≥ 1 and a + b ≥ 1. In this paper, we prove that log lcm ...
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A central limit theorem for Ramanujan’s tau function
A central limit theorem is established for the absolute value of the modular Fourier-coefficient function defined by Ramanujan, and for that of the...
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On a generalization of the Euler totient function
For a general polynomial Euler product F ( s ) we define the associated Euler totient function φ ( n , F ) and study its asymptotic properties. We prove...
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On the divisor function in short intervals
Let d ( n ) denote the number of positive divisors of the natural number n . The aim of this paper is to investigate the validity of the asymptotic...
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The least common multiple of a sequence of products of linear polynomials
Let f ( x ) be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate log lcm ( f (1),…, f ( n ))∼ An as n →∞...
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Shifted fourth moment of the Riemann zeta-function
For the Riemann zeta-function we present an asymptotic formula of a shifted fourth moment in an unbounded shift range along the critical line.
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Two remarks on iterates of Euler’s totient function
Let φ k denote the k th iterate of Euler’s φ -function. We study two questions connected with these iterates. First, we determine the average order of φ ...
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Limit-periodic arithmetical functions and the ring of finite integral adeles
In this paper, we show that the ring of finite integral adeles, together with its Borel field and its normalized Haar measure, is an appropriate...
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Inequalities for divisor functions
We give certain optimal inequalities for the divisor function. Such inequalities are useful in estimating the sums of divisor functions which are...
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On common values of φ(n) and σ(m). I
We show, conditional on a uniform version of the prime k -tuples conjecture, that there are x /(log x ) 1+ o (1) numbers not exceeding x common to the...