Divisors of Bernoulli sums
نویسنده
چکیده
Let B n = b 1 +. .. + b n , n ≥ 1 where b 1 , b 2 ,. .. are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic order of the sums N n=1 d θ,D (B n), where d θ,D (B n) = #{d ∈ D, d ≤ n θ : d|B n } and D is a sequence of positive integers.
منابع مشابه
Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers
We investigate sums of products of Bernoulli numbers including poly-Bernoulli numbers. A relation among these sums and explicit expressions of sums of two and three products are given. As a corollary, we obtain fractional parts of sums of two and three products for negative indices.
متن کاملElliptic analogue of the Hardy sums related to elliptic Bernoulli functions
In this paper, we define generalized Hardy-Berndt sums and elliptic analogue of the generalized Hardy-Berndt sums related to elliptic Bernoulli polynomials. We give relations between the Weierstrass ℘(z)-function, Hardy-Bernd sums, theta functions and generalized Dedekind eta function. 2000 Mathematical Subject Classification: Primary 11F20, 11B68; Secondary 14K25, 14H42.
متن کاملCongruences and exponential sums with the sum of aliquot divisors function
We give bounds on the number of integers 1 6 n 6 N such that p | s(n), where p is a prime and s(n) is the sum of aliquot divisors function given by s(n) = σ(n) − n, where σ(n) is the sum of divisors function. Using this result we obtain nontrivial bounds in certain ranges for rational exponential sums of the form Sp(a,N) = ∑ n6N exp(2πias(n)/p), gcd(a, p) = 1.
متن کاملBernoulli numbers and generalized factorial sums
We prove a pair of identities expressing Bernoulli numbers and Bernoulli numbers of the second kind as sums of generalized falling factorials. These are derived from an expression for the Mahler coefficients of degenerate Bernoulli numbers. As corollaries several unusual identities and congruences are derived.
متن کاملExplicit formulae for sums of products of Cauchy numbers including poly-Cauchy numbers
Recently, K. Kamano studied sums of products of Bernoulli numbers including poly-Bernoulli numbers. A relation among these sums was given, and an explicit expression of sums of two products was also given, reduced to the famous Euler’s formula. The concept of poly-Cauchy numbers is given by the author as a generalization of the classical Cauchy number and an analogue of poly-Bernoulli number. I...
متن کامل