Question about Abel summation

Question

Now in this sum.

$A(y) =\sum_{n=1}^y a_n$

or

$A(y) =\sum_{n=0}^y a_n$

Which one is true where does n starts?

Answer 1

You can find Abel's identity in the book "Introduction to analytic number theory" written by Tom Apostol. (It is theorem 4.2):

For any arithmetical function a(n) let $A(x)=\sum_{n \leq x} a(n)$ where A(x)=0 if $x < 1$. Assume that f has a continuous derivative on the interval [y,x], where 0 < y < x. Then we have $\sum_{y<n\leq x} a(n)f(n) = A(x)f(x)-A(y)f(y) - \int_y^xA(t)f'(t)dt$