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  1. The Riemann hypothesis (RH) states that all non-trivial zeros of the zeta function have real part $\frac{1}{2}$.

  2. The zeta function is intimately connected with the Gamma function via the functional equation.

The second fact suggests that there is an equivalent form of RH which is expressed solely in terms of the Gamma function.

Question: What is the most natural form to translate RH as directly as possible (without mentioning the zeta function) into a hypothesis on the behaviour of the Gamma function?

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    Or Can there be an integral representation of the $\zeta$ function? Because for example there are certain integrals, whose values equal $\frac{\pi^{2}}{6} = \zeta(2)$, may be we should and can think of this question in that manner.2010-08-31

3 Answers 3

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Since

$\zeta(z)=\frac{\Gamma (1-z) \left(2^{-z} \left(\psi \left(z-1,1\right)+\psi \left(z-1,\frac{1}{2}\right)\right)-\psi(z-1,1)\right)}{\ln(2)}$

where $\psi(x,z)$ is the generalized polygamma following Espinosa's generalization, whatever we say about Zeta function we can also say about the right hand part of this identity. It consists only of Gamma function, its (fractional) derivatives and integrals.

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    The formula by Espinosa is obtained from fractional integration of polygamma by Adamchik (Adamchik, V. (1998). Polygamma functions of negative order. Jour. Comp. Appl. Math., 100, 191–199.) whose modified (by introducing proper integration constants for balancing) definition is used by Espinosa.2010-12-05
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The Wikipedia article gives a Mellin transform

$\Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1} dx.$

The Dirichlet series over the Möbius function gives the reciprocal

$ \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} .$

Thus we may write

$\Gamma(s) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} \int_0^\infty\frac{x^{s-1}}{e^x-1} dx .$

This holds true for every complex number s with real part greater than $1$. Now let's try to enlarge the domain of validity of this representation. Riemann showed (see the book of H. M. Edwards, Riemann Zeta Function, for the details) that modifying the contour gives a formula valid for all complex s.

$ 2\sin(\pi s)\Gamma(s)\zeta(s) = i \oint_C \frac{(-x)^{s-1}}{e^x-1}dx .$

This leads to

$ \sin(\pi s) \Gamma(s) = \frac{i}{2} \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} \oint_C \frac{(-x)^{s-1}}{e^x-1} dx . \qquad (*) $

However, this formula is again only valid for s with real part greater than $1$ because of the use of the Dirichlet series. Wikipedia remarks:

"The Riemann hypothesis is equivalent to the claim that [this representation of the reciprocal of the zeta function] is valid when the real part of $s$ is greater than $\frac{1}{2}$."

Thus a possible answer to my question is:

The representation $\,*\,$ is valid for all $s$ with real part greater than $\frac{1}{2}$ if and only if the RH holds.

Perhaps someone can elaborate further to give this relation a more geometric meaning? Where are the non-trivial zeros of the zeta function to be spotted in this setup?

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    The remarks are delightful, but do they aid to answer the question which was: "What is the most natural form to translate RH as directly as possible (without mentioning the zeta function) into a hypothesis on the behaviour of the Gamma function?" Qiaochu Yuan's remark "I would expect that this is not possible" is the remark most agreed upon; however it does not help me to see the reason, as "the philosophy that the Gamma function is the term in the Euler product "at infinity"" is above my head and perhaps better suited for MO than for SE.2010-10-31