I have proved the twin number but no journal would like to review, ti is too scary !!

I will be happy if any one can read my Sketch of the proof’s section Page 5 and 6 only

https://arxiv.org/abs/1512.00970

Best regards

Samir

Page 33: eq (5.7): integration should be over instead of , and it should be the -fold differential with respect to instead of all variables.

Paragraph after (5.9): I think the reference (5.5) should be to the set defined in the equation above.

Page 34: after (5.10): maybe mention is fixed.

After (5.13): maybe recall definition (3.6) of

Page 36: equation after (5.14): again, integration over , and derivative (wrt variables other than ).

Page 38: first line: not .

Line 3: instead of .

Eq 5.19: there is an accidental in the second sum.

Line -7: for instead of

Line -3: should this be instead of (obviously the second is also true). Also for page 39 line 2.

Page 39: line 5: `.. we prove Theorem 3.14’ instead of Theorem 3.12.

Page 40: equations after 5.22: I think we use EH to prove the lower bound for the prime sums, and GEH to prove the asymptotic for the non-prime sums, so these should be the other way round. Also (as above) should it be rather than .

Page 42: line -4: instead of .

Page 46: line 10: maybe mention that we are extending to by .

inequality before (6.20): instead of .

Page 47: In various places I think we need to swap strict and weak inequalities, since the equation doesn’t hold if (and is not defined at 0). This occurs on line 1 and twice on line 5.

Line 7: instead of .

Line 11: `third equality’ instead of `second equality’.

Line -3: Maybe note we are using (6.12) here.

(6.24): There should be a factor of on the right hand side.

Page 48: line 6: not

Line 8: Maybe note we are using (6.10) here.

Line -6: The use of for general hasn’t been defined; maybe better to stick with .

Page 49: line 14: any not .

Page 50: line 8: ds missing from integral.

Page 52: line 5: I get as the final constraint.

Page 53: before section 7: Maybe mention that the constraints (6.10)-(6.12) hold for each of these choices of parameters.

*[Added, thanks – T.]*

*[Corrected, thanks – T.]*

Very nice! I’ve added a summary of your results to Section 6 of the draft paper. I’m going through the paper now to try to clean it up and move it closer to a publishable state; hopefully I can be done in a few days (modulo some remaining gaps, such as Appendix A) and then I will roll over the thread.

]]>This very interesting solution has the properties:

1. For each , is analytic on

– implying that the eigenfunction is real analytic on its support, except on the jump singularities on the intersection of the lines and with support.

2. The threshold is probably related to the same threshold above which has a reduced support (i.e. vanishes on a certain inner triangle ).

]]>I have now done some analytical calculations for the eigenfunction associated with (I do hope I did not make any obvious mistakes). It turns out that the calculations do not become more difficult, but just more involved. You can find it at the following link: http://users.ugent.be/~ibogaert/KrylovMk/M2eps.pdf

]]>*[Corrected, thanks – T.]*

Ah, you’re right, we do get every even number within 2 of a sum of two primes from this argument. (What we almost, but don’t quite get, though, is that every odd number is within 1 of a sum of two primes.) I’ve modified the paper slightly to indicate this.

]]>*[Corrected, thanks – T.]*