ve#7-oct
Let be a Arithmetic FunctionArithmetic Function then I define average as
For example for
Let be real numbers and ( . exist and is continuous) then
where is the fractional part of t
i)
where means something is absolute value .i
for some and Large Here is known as the Euler Maclaurin constant
proof: Now by Euler Maclaurin summation FormulaEuler Maclaurin summation Formula
complete the madness
proof of E-M
we have
the rest is exercise decide if you should complete it
We have
Proof.
□
Let be arithmetic function and ,, be positive real numbers such that then
Proof. we have
Hence choosing we have