Gram's law 3. Turing 3. A simple algorithm for the zeta function 3. Further reading 4. Empirical evidence 4. Verification in an interval 4. A brief history of computational evidence 4. The Riemann hypothesis and random matrices 4. The Skewes number. Equivalent statements 5. Number-theoretic equivalences 5. Analytic equivalences 5.
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Other equivalences 6. Extensions of the Riemann hypothesis 6.
Special order items
The Riemann hypothesis 6. The generalized Riemann hypothesis 6. The extended Riemann hypothesis 6. An equivalent extended Riemann hypothesis 6. Another extended Riemann hypothesis 6. The Grand Riemann hypothesis 7.
Assuming the Riemann hypothesis and its extensions 7. Another proof of the prime number theorem 7. Goldbach's conjecture 7. More Goldbach 7. Primes in a given interval 7. The least prime in arithmetic progressions 7. Primality testing 7. Artin's primitive root conjecture 7. Bounds on Dirichlet L-series 7. Titchmarsh's [delta] T function 7. Mean values of [cedilla] s.
Failed attempts at proof 8. Stieltjes and Mertens' conjecture 8. Hans Rademacher and false hopes 8. Louis de Branges's approach 8.
The Riemann Hypothesis
No really good idea 9. Formulas Timeline pt. Original papers Expert witnesses Bombieri Sarnak Conrey The experts speak for themselves Chebyshev Riemann Hadamard Hardy Hardy and J.
Littlewood Weil Selberg Skewes Haselgrove Montgomery Newman Korevaar Daboussi Hildebrand Goldston and H. Agrawal, N. That is what has happened here. The original derivation of Equation 8. The result would have been a tautology: rhs of Equation 8. Alternatively, if the original derivation of Equation 8.
The result would be, and is, again a tautology: rhs of Equation 8. Sans contradiction, neither of these can prove or disprove RH. However, in this case, it is a useful exercise to demonstrate the validity of Equation 6. First, the foregoing analysis provides a simple demonstration that the sum appearing in Equation 8. The transition from Equation 8. A complicated proof of the convergence of Equation 8. Edwards, Edwards, H.
The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike
Secondly, it suggests that a careful numerical evaluation of Equation 8. On equalities involving integrals of the logarithm of the Riemann-function with exponential weight which are equivalent to the Riemann hypothesis. International Journal of Analysis. Thirdly, it appears to validate counting theorems that place bounds on the location of the zeros see Section 12 by the presence of discontinuities of the argument.
It will now be shown that all of the above are compromised. Figure 7. As noted above, verification of Equation 8. In general, a discontinuity of the arg operator for any analytic function h z can be associated with either of two events. The first is associated with the presence of a zero of order n , viz. The LogGamma function demonstrates how such a possibility arises—see Figure 1. This means that, since one expects see Equation 6. Such is the perceived reality incorporated into many well-accepted results, and it is difficult to visualize how this could fail to be the case.
I shall refer to this configuration by the term "anomalous zero".
Banks , Kang : On Repeated Values of the Riemann Zeta Function on the Critical Line
Figure 8. Figure 9. Figure Before considering the implications of these observations, it is fair to ask if there are other such anomalies. Here, we see that the situation is the exact opposite of the previous one. Tables of zeros of the Riemann zeta function. Odlyzko, private communication].
At this point, it is worthwhile to recall three formulae that locate zeros. The first is the well-known Backlund Edwards, Edwards, H. A third relevant result taken from Milgram Milgram, M. A simple asymptotic form is easily obtained from Equation This also shows that the counting formula Equation 9. The first missing term of Equation 9. The advantage of using Equation 9.
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In this way, the calculation will not involve possible cancellation of large numbers as might occur if Equation In addition, the Lehmer zeros Edwards, Edwards, H. As well as affecting counting formula such as Equation 9. The analysis given in Section 8 depended on Equation 8.
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Each consecutive segment was labelled by unit increments of k. In fact, the equivalence of Equation 8. However, Figures 9 and 11 demonstrate that such is not the case due to anomalous zeros. Thus, over this one segment, Equation 8. Is this the long-sought contradiction in a RH equivalent discussed in Section 8?
Unfortunately not. To understand why, it is necessary that the derivation of Equation 8. Referring to Sekatskii et al. The integral is then related to the residue of enclosed singularities by the residue theorem, and it is simplified by taking one section of the contour of integration to coincide with a cut in the function F z. And therein lies a problem. If f z is analytic over a region, it does not follow that F z will be analytic over that same region. And as we have seen in the previous sub-section, it is not true. Thus, if the derivation of Equation 8. Thus, any RH equivalence of the Volchkov genre stated in the form Equation 8.
In a later work Sekatskii et al. As has been discovered, the existence of anomalous zeros demonstrates that discontinuities can occur at points other than a zero, and this eventuality must be incorporated into the resulting proofs.