### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Matthew Russell, Rutgers University, russell2 {at} math [dot] rutgers [dot] edu)
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Zeta Function-Like Sums Over Lucas Numbers

Speaker: William Kang, Bergen County Academies

Date: Thursday, November 7, 2013 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

The study of sequences extends in a wide range from the golden ratio to the current financial trading algorithm. Although not realized by many, both the Fibonacci and Lucas sequences are incorporated into the algorithm today. In addition, among the far-reaching applications of these numbers are the Fibonacci search method and the Fibonacci heap data structure in computer science. Hence, mathematicians seek to find more results and further studies into these sequences for the purpose of greater potential benefits. In the area of mathematics, Fibonacci and Lucas numbers are used in connection to efficient primality testing of Mersenne numbers; the method revolves around the fact that if Fn is prime, then n is prime with the exception for F(4)=3. Some of the largest known primes were discovered via this process. Yet, despite the extensive literature on Fibonacci and Lucas numbers, there are still many questions, which are still unanswered. Recently, several authors considered the problem of estimating expressions of the classical Fibonacci and Lucas sequences given by F0 = 0, F1 =1,Fn =Fn.ANb.FN"1+Fn.ANb.FN"2 and L0 =2,L1 =1,Ln =Ln.ANb.FN"1+Ln.ANb.FN"2, respectively. In this paper, we find the exact asymptotic behavior for a class of infinite partial sums whose general term is a negative integer power of L(k). Most of the previous works focused on the relatively simple cases s = 1 and s = 2 where s represents the power. For this paper, we find the general idea in which expressions could be found for cases of s from 1 to 6.