How to show that there exist no complex valued Riemann integrable function $f$ on $[-\pi , \pi]$ such that $\hat f (n)=1/n , \forall n \ge 1$ , $\hat f (n)=0 , \forall n \le 0$ , where $\hat f (n):=\dfrac 1{2\pi} \int_{-\pi}^\pi f(t)e^{-int} , \forall n \in \mathbb Z$ ?
If such a Riemann integrable function exist then since $f \in L^1[-\pi , \pi]$ , I can conclude from Uniqueness of Fourier series that $f(x)=\sum_{n=1}^\infty e^{inx}/n $ a.e. on $[-\pi, \pi]$ . If this equality would hold everywhere then calculating its Abel sum at $x=0$ , I could get a contradiction , but unfortunately the equality holds only a.e. How should I proceed ? Please help . Thanks in advance