What is the value of Chebyshev $\psi(x)$ function at non-integer values ?
For example, what is the value of $\psi(3.56)$? I have seen, in same place, it seems that $$\psi(3.56)=\psi(3)$$
And in some other place, $$\psi(3.56)=0$$ and $\psi(x)$ always equals to $0$ at non-integers.
The reasons why I need the values of Chebyshev at non-integer, because I need do some Fourier transform with this function.
By definition $$\psi\left(x\right)=\sum_{n\leq x}\Lambda\left(n\right) $$ where $$\Lambda\left(n\right)=\begin{cases} \log\left(p\right), & n=p^{k},\,p\textrm{ is a prime and }k\geq1\\ 0, & \textrm{otherwise} \end{cases} $$ then $$\psi\left(3.56\right)=\sum_{n\leq3.56}\Lambda\left(n\right)=\sum_{n\leq3}\Lambda\left(n\right)=\psi\left(3\right)=\log\left(6\right).$$ Maybe there are some text that define $\psi(x)$ only for integers, but I always see the definition I used.