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This paper concerns the visibility properties of lattice points in multiple random walks on N k , where k ≥ 2 is an integer. We study two aspects of the visibility: simultaneous visibility in multiple random walkers; and that only some of these walkers are visible. Combining tools from number theory and probability theory, we prove the corresponding densities of the above two parts. [ABSTRACT FROM AUTHOR]

T. M. Apostol introduced a certain Möbius function μ k (·) of order k, where k ≥ 2 is a fixed integer. Let k=1, then μ 1 (·) coincides with the Möbius function μ (·) , in the usual sense. For any fixed k ≥ 2 , he proved the asymptotic formula ∑ n ≤ x μ k (n) = A k x + O k (x 1 / k log x) as x → ∞ , where A k is a positive constant. Later, under the Riemann Hypothesis, D. Suryanarayana showed the O-term is O k (x 4 k 4 k 2 + 1 exp (D log x log log x ) ) with some positive constant D. In this paper, without using any unproved hypothesis we shall prove that the O-term obtained by Apostol can be improved to O k (x 1 / k exp (- D k (log x) 3 / 5 (log log x) 1 / 5 ) ) with some positive constant D k . [ABSTRACT FROM AUTHOR]

We establish an asymptotic formula of the Montgomery–Hooley type describing the variance of the distribution of the number of Goldbach representations in arithmetic progressions. [ABSTRACT FROM AUTHOR]

We consider the simultaneous sign changes of normalized Fourier coefficients associated to two distinct Hecke eigenforms supported at positive integers represented by a primitive integral binary quadratic form with negative discriminant whose class number is 1. We provide a quantitative result for the number of sign changes of such sequence in the interval (x , 2 x ] for sufficiently large x . [ABSTRACT FROM AUTHOR]

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga [2] on the number of k -compositions of n satisfying some coprimality conditions. We substantially refine the error term concerning the number of k -compositions of n with pairwise relatively prime summands. We use a different approach, based on properties of multiplicative arithmetic functions of k variables and on an asymptotic formula for the restricted partition function. [ABSTRACT FROM AUTHOR]

Let x → ∞ be a parameter. Feng [5] proved that the Deshouillers–Dress–Tenenbaum's arcsine law on divisors of the integers less than x also holds in arithmetic progressions for ''non-exceptional moduli" q ⩽ exp { (1 4 - ε) (log 2 x) 2 } , where ε is an arbitrarily small positive number. We show that in the case of a prime-power modulus ( q : = p ϖ with p a fixed odd prime and ϖ ∈ N ) the arcsine law on divisors holds in arithmetic progressions for q ≤ x 15 / 52 - ε . [ABSTRACT FROM AUTHOR]

Assume that K is Gaussian field, and a K (n) is the number of non-zero integral ideals in Z [ i ] with norm n . We establish an Erdős–Kac type theorem weighted by a K (n 2) l (l ∈ Z +) in short intervals. We also establish an asymptotic formula for the average behavior of a K (n 2) l in short intervals. [ABSTRACT FROM AUTHOR]

We study a square mean value problem and two general divisor problems related to Hecke eigenvalues of classical holomorphic cusp forms and classical Maass cusp forms, respectively. We improve previous results. [ABSTRACT FROM AUTHOR]

We study the mean value estimate of coefficients of product L-function related to a primitive holomorphic cusp form f(z) of weight k for the full modular group SL2(Z). Upper bounds and asymptotic formulas are established. [ABSTRACT FROM AUTHOR]

We study the distribution of the sequence of vectors $${(p^{{\alpha}_1},\ldots,p^{{\alpha}_k})}$$ modulo one. Here p runs over prime numbers, k ≥ 2 is a fixed integer, α, . . . , α are fixed real numbers lying in the interval ( s, s + 1) with s being a sufficiently large integer. [ABSTRACT FROM AUTHOR]

We study two general divisor problems related to Hecke eigenvalues of classical Maass cusp forms. We give the relevant estimation and improve previous results. [ABSTRACT FROM AUTHOR]

Let f, g be completely multiplicative functions, $${|f(n)| = |g(n)| = 1, (n \in \mathbb{N})}$$ such that $${|f ([\sqrt{2n}]) - Cg(n)| \leqq \varepsilon(n),\,\varepsilon(n) \downarrow 0}$$ and We prove that $${f(n) = g(n) = n^{i\tau} , C = (\sqrt{2})^{i\tau}}$$ . [ABSTRACT FROM AUTHOR]

We prove that the multiplicative group generated by $${\{ \frac{[\sqrt{2}n]}{n} | n \in \mathbb{N} \}}$$ is the group of positive rational numbers. It is proved that if a completely additive function f satisfying f $${([\sqrt{2}n]) - f(n) \rightarrow C (n \rightarrow \infty)}$$ for some real number C, then $${f(n) = A{\text{log}} n}$$ , where $${A = \frac{2C}{log 2}}$$ . [ABSTRACT FROM AUTHOR]