Analytic Number Theory/The Chebychev ψ and ϑ functions

the Chebychev ψ function may be written as the sum of Chebyshev ϑ functions We have the identity . estimate of the distance between the Chebychev ψ and ϑ functions Whenever , we have . Note: The current proof gives an inferior error term. A subsequent version will redeem this issue. (Given the Riemann hypothesis, the error term can be made even smaller.) We know that the formula holds. Hence, . By a result obtained by Pierre Dusart (based upon the computational verification of the Riemann hypothesis for small moduli), we have whenever . If is in that range, we hence conclude . By Euler's summation formula, we have . Certainly and . Moreover, . Now derivation shows that is an anti-derivative of the function of . By the fundamental theorem of calculus, it follows that for real numbers such that . This integral is not precisely the one we want to estimate. Hence, some analytical trickery will be necessary in order to obtain the estimate we want. We start by noting that if only the bracketed term in the integral were absent, we would have the estimate we desire. In order to proceed, we replace by the more general expression (where ), and obtain . The integrand is non-negative so long as . Moreover, if is strictly within that range, we obtain . We now introduce a constant and obtain the integrals and . The first integral majorises the integral , whereas the second integral majorises the integral . We obtain that . Now we would like to set . To do so, we must ensure that is sufficiently large so that resp. is strictly within the admissible interval. The two summands on the left are now estimated using our computation above, where is replaced by for the first computation: Indeed, and . Putting the estimates together and setting , we obtain whenever and . We now choose the ansatz and for constants and . These equations are readily seen to imply and . Note though that and is needed. The first condition yields . The equations for and may be inserted into the above constraints on and ; this yields and , that is, and . If all these conditions are true, the ansatz immediately yields . We now amend our ansatz by further postulating . This yields and . From this we deduce that in order to obtain an asymptotically sharp error term, we need to set . But doing so yields the desired result.