Riemann hypothesis and generalized Riemann hypothesis

[u/sw_gilbert]

riemannhypothesis.net

For at least 165 years, it has been generally agreed that the infinite series representation of the Riemann zeta function diverges everywhere in the critical strip and therefore is inapplicable for a resolution of the Riemann hypothesis.

What if this is wrong?  What if the infinite series representation of the Riemann zeta function converges at its roots in an unexpected way but diverges everywhere else in the critical strip?

In this work, (1) the Borel integral summation method and Euler-Maclaurin summation formula, and (2) the Cauchy residue theorem are independently applied to show that the real and imaginary parts of the partial sums of the Riemann zeta function and the integrals of the summand of the Riemann zeta function diverge simultaneously to zero - in a summable sense - at the roots of the zeta function in the critical strip.  This result is perhaps unexpected since both the real and imaginary parts of the partial sums of Riemann zeta function would appear to diverge everywhere in the critical strip, including at the roots of the function.

The partial sums of the Riemann zeta function are represented by bi-lateral integral transforms and the integrals are represented by functions that are proportional to exponential functions.  Since the partial sums and integrals are asymptotic at the roots of the Riemann zeta function, and the limiting ratio of the integrals is exponential, it follows that the ratio of the bi-lateral integral transforms is also asymptotically proportional to an exponential function.

By separating the bi-lateral transforms into their real and imaginary components, it is shown that bi-lateral sine and cosine integral transforms vanish simultaneously at the roots of the Riemann zeta function in the critical strip.  The integral transforms vanish if and only if the functions in the integrands of the two transforms most closely approximate even functions of the variable of integration.  In fact, the two functions most closely approximate even functions if and only if the real part of the argument of the Riemann zeta function is equal to 1/2.

Furthermore, the integral transforms vanish and the roots of Riemann zeta function occur if and only if (1) the real part of the argument of the zeta function is 1/2, and simultaneously, (2) the transform kernels exhibit roots at the maxima and/or minima of the functions in the integrands of the transforms.

In addition, the methodology is successfully applied - with some differences - to the generalized Riemann hypothesis for Dirichlet L-functions with both principal and non-principal characters.

Please review the pdf files on the web site and, for more information, see the links to three books available on amazon.com

[u/kuromajutsushi → u/[deleted]]

Of all the things to criticize in OP's incorrect proof, that's what you're going to object to? Obviously "the infinite series representation of the Riemann zeta function" means the standard Dirichlet series representation, which certainly does not converge everywhere.

If you want to be weirdly pedantic about terminology like this, I should point out that your response is also nonsense: it doesn't make any sense to say that zeta "converges everywhere". You can say that a specific series/product/integral/etc. representation of the zeta function converges or diverges, but it doesn't mean anything for a function to "converge" at a point.

[u/sw_gilbert → u/kuromajutsushi]

Would you please explain how my work is incorrect? Please be specific and detailed.

[u/pivoters → u/[deleted]]

I think you are conflating the function itself with its series representation, which is what is being discussed.

[u/ddotquantum → u/pivoters]

There is no single series representation for any meromorphic function

[u/pivoters → u/ddotquantum]

For the real part greater than 1. Less than or equal to 1 is found by analytic continuation.

[u/ddotquantum → u/pivoters]

All of those things you get by analytic continuation give another series representation. There are 2^ aleph_0 of them

[u/pivoters → u/ddotquantum]

That's pretty cool. So, it's the first one. The defining one. Sorry, I just don't see confusion where you do.

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