Sign changes of error terms related to certain arithmetic functions

AbstractLetH(x) =Pnx`(n) n6 2x. Motivated by a conjecture of P. Erdos, Y.-K. Lau developed a new method and proved #f1nT:H(n)H(n+ 1)<0g T: We consider arithmetic functionsf(n) =Pdjnbd dwhose summation can be expressed asPnxf(n) =fix+P(log(x)) +E(x), wherefiis a real number,P(x) is a polynomial and the error termE(x) is of the fromE(x) =Xny(x)bn n x n+O 1 k(x); for(x) =x[x]1 2, and wherey(x),k(x) andbnsatisfy some general conditions. We generalize Laus method and prove results about the number of sign changes for these error terms. We illustrate our results with a list of well known arithmetic functions. In particular, we prove the following generalization of Laus result: Letf(n) =Pdjnbd dbe a rational valued arithmetic function and suppose the sequencebnsatisflesPnxbn=Bx+Ox logAxandPnxb4nxlogDx;for someBreal,D >0 andA >6 +D 2, respectively. Letfi=1Xn=1bn n2; b= limx!1Xnxbn nBlogx!; E(x) =Xnxf(n)fix+Blog 2x 2 + b 2: Then, except whenfi= 0, orB= 0 andfiis rational, we have#f1nT:E(n)E(n+ 1)<0g T,#f1nT:fiE(n)<0g T: We also study the error term (x) =Pnx(n)xlogx(2 1)xand prove #f1nT: (n)(n+ 1)<0g>p T+O(1):Index words: sign changes, error term, Euler function, divisor function.