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Riemann Hypothesis: Riemann Zeta Function (1)
The reciprocals of these zeta functions are known to be polynomials, and the functions themselves are known to satisfy Riemann hypotheses precisely when the graph in question is Ramanujan.
zeta functions associated with curves over finite fieldsArtin conjectured that the Riemann hypothesis holds for the zeta function associated with any curve over a finite field.
“The failure of the Riemann Hypothesis would create havoc in the ..
includes very helpful discussion of the Riemann Hypothesis, Extended Riemann Hypothesis, Grand Riemann Hypothesis, Weil Conjectures, Langlands Programme, the functional equations of zeta and functions, modularity of theta functions, follows this up, framing certain issues concerning the bewildering array of zetaand in terms of .
3, 703731 "It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the NumberTheoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains...The general idea is to explain the pseudorandom features of certain number theoretical functions by considering them as observables of a spin chain of statistical mechanics."[17] C.
Bernhard Riemann; Brian Conrey;
Moronianu,
zeta functions of energy of PTsymmetric quantum systems
Epstein zeta functionHeilbronn proved that the Riemann Hypothesis fails for the Epsteinzeta function.
In particular, there is a 'zetafunction' associated in a natural numbertheoretic way to any function field overa finite field, and Weil has shown that the analog of theRiemann hypothesis is true for such 'zeta functions'"H.M.
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Under the Riemann Hypothesis, ..
A solution to the greatest unresolved problem in pure mathematics – Riemann’s hypothesis – could be coming closer, thanks to a remarkable international collaboration involving mathematicians from the University of Bristol and University of Warwick, partfunded by a £2.24million grant from the Engineering and Physical Sciences Research Council (EPSRC).
Brian Conrey, "The Riemann Hypothesis ..
Professor John Cremona of the University of Warwick, Principal Investigator on the EPRSC grant said: “While the database was built to provide tools that could help tackle the Riemann hypothesis, one of the Millennium Prize Problems nominated by the Clay Mathematics Institute, I also hope it will be a useful tool for researchers of another of the Clay Millennium problems: the Birch and SwinnertonDyer conjecture on the set of rational solution to equations defining an elliptic curve. I was privileged to be one of Professor Bryan Birch’s doctoral students and would be delights if this work was used to help prove the Birch and SwinnertonDyer conjecture.”
LMFDB  Generalized Riemann hypothesis
David Farmer (American Institute of Mathematics) and Eric Saias (Université Paris 6) have a wager that reflects theiropposing views of the subject:The RH Wager: Eric Saias will pay 10 euros to David Farmerevery year until the Riemann Hypothesis is proven.
Hilbert quote on the Riemann hypothesis  Google …
However, it seems that theyprovide some of the best reasons for believing that the Riemann hypothesisis true  for believing, in other words, that there is a profound andas yet uncomprehended numbertheoretic phenomenon, one facet of which is that the roots all lie on Re[s] = 1/2.
Society for his article entitled The Riemann Hypothesis
There is nothing like thenontrivial zeros a la Riemann Hypothesis...note that it is just a historical accident that it wascalled a zeta function and has nothing to do with the rest...As a curiousity, the Jacobi theta function is involvedwith the functional equation of the Riemann zeta functionvia the Mellin transform.
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