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By J. P. Buhler

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A Second Glimpse of the Prime Number Theorem We have concluded Chapter 1 by providing “an idea for an idea” on how to obtain the Gauß conjecture of the prime number theorem. Unfortunately, at this stage we could not really identify the leading contribution since we did not have enough knowledge about the function ϕ. In the present section we initiate another attempt to verify the Gauß conjecture Eq. 1), which is based on the expression Eq. 8) for the sieve of Eratosthenes. Unfortunately, we again fall short of a proof, since the remainder in our estimate grows exponentially.

N=−∞ It is for this reason that we refer to ϑ as theta function, although strictly speaking theta functions are a much larger class of functions as illustrated for example in Whittacker and Watson A Course of Modern Analysis. According to Eq. : Representations of the Riemann ζ-function in the various domains of complex space s ≡ σ + it. For σ > 1 (downwards stripes) the Dirichlet series Eq. 1) is uniformly convergent. In the critical strip, that is for 0 < σ < 1, as well as for σ > 1 (darkened zone) we can use the integral representation Eq.

N=1 In order to prove this claim we substitute this expression for wD into the right hand side of Eq. 19), interchange summation and integration. With the help of the substitution u ≡ πn2 x we find the formula π s/2 Γ 2s ∞ ∞ n=1 ∞ 2 dx e−πn x xs/2−1 = d(n) d(n)n−s = D(s), n=1 0 which is indeed Eq. 18). 2. Finite Sums and Inverse Mellin Transform Now we go a step further and represent D not just as the product of two functions with one being the Mellin transform but as the inverse Mellin transform of a single function.

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