Hi it is about mathematical analysis (Riemann Stieljes Integral)

note: the first image is just informative

-If X21 let us denote by :r(x) to the number of prime numbers less than or equal to x, that is 9113:) =21, pgx where the sum extends over all prime numbers p g x. The Prime Number Theorem states that logx:1 lim afar) IF} x This is usually proved by studying a function V given by V(x)=zlog p, pix where again, the sum is extended overall prime numbers p g x. Both functions, V y TI." are step functions with jump discontinuities at the prime a) If x 2 2, prove that x(x) and V(x) can be expressed as the following Riemann-Stieltjes integral: I ( xx ) = dV (t), V(x)-3 log tdn (t) log t Note: The lower bound can be replaced by any number in the open range (1, 2). b) If x 2 2 use integration by parts to prove that x. V(x) = m(x)log x- 7( ) at 2 t IT (x ) C X V (t) 7 (x) = + dt log x J2 tlog' t