Home » Posts tagged 'Real Analysis'
Tag Archives: Real Analysis
Let denote Euler’s totient function. Prove that for it holds that:
where stands for the Riemann zeta function.
Well by Euler’s product we have,
Combining we get the result.
Note: It also holds that
Let such that . Prove that
We are invoking the Hermite – Hadamard Inequality for the convex function . Thus,
Evaluate the integral
Successively we have:
We used the simple observation that
Let be a function such that and is differentiable at . Let us set
Evaluate the limit .
Since is differentiable at , there is some such that
and is of course continuous.
Let . There exists such that which in return means that . Hence , for larger than it holds that
On the other hand , the sum is a Riemann sum and converges to .
Let denote one of the Jacobi Theta functions. Prove that
We have successively,
The sum is evaluated as follows. Consider the function
and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,
and the exercise is complete.