Analytic Number Theory/Printable version


Analytic Number Theory

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The Chebychev ψ and ϑ functions

Proposition (the Chebychev ψ function may be written as the sum of Chebyshev ϑ functions):

We have the identity

.

Proposition (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.)

Proof: 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.