I am self studying analytic number theory from An introduction to Sieve Methods and its applications by Alina Carmen and M Ram Murthy .
I have a doubt on page 7 in Theorem of chebyscheff whose proof was given by S. Ramanujan .
Authors defines $\psi(x) = \sum_{n\leq x} \Lambda(n) $ and then $T(x)=\sum_{n\leq x} \psi(x/n) $.
Then Authors after some steps just writes without giving any argument $T(x) – 2T(x/2) = \sum_{n\leq x} (-1)^{n-1} \psi(x/n) $.
I don't understand how authors derive it and I could not think about it despite trying many times. Can someone please tell how to derive it?
Best Answer
We are given that
$$ T(x) := \psi(x) + \psi\Big(\frac{x}2\Big) + \psi\Big(\frac{x}3\Big) + \psi\Big(\frac{x}4\Big) + \dots $$
and thus
$$ T\Big(\frac{x}2\Big) = \psi\Big(\frac{x}2\Big) + \psi\Big(\frac{x}4\Big) + \psi\Big(\frac{x}6\Big) +\dots. $$
Subtract twice the second equation from the first to get
$$ T(x) \!-\! 2\,T\Big(\frac{x}2\Big) = \psi(x)\!-\! \psi\Big(\frac{x}2\Big) \!+\! \psi\Big(\frac{x}3\Big) - \psi\Big(\frac{x}4\Big) \!+\! \dots $$
where we have just changed the signs of every term with $\,n\,$ even but all of the terms with $\,n\,$ odd are not changed because the $\,T(\frac{x}2)\,$ summation only includes terms with $\,n\,$ even.