Mertens Conjecture - Connection To The Riemann Hypothesis

Connection To The Riemann Hypothesis

The connection to the Riemann hypothesis is based on the Dirichlet series for the reciprocal of the Riemann zeta function,

valid in the region . We can rewrite this as a Stieltjes integral

and after integrating by parts, obtain the reciprocal of the zeta function as a Mellin transform

\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s) = \int_0^\infty x^{-s} M(x)\, \frac{dx}{x}.

Using the Mellin inversion theorem we now can express M in terms of 1/ζ as

which is valid for 1 < σ < 2, and valid for 1/2 < σ < 2 on the Riemann hypothesis. From this, the Mellin transform integral must be convergent, and hence M(x) must be O(xe) for every exponent e greater than 1/2. From this it follows that

for all positive ε is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that

.

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