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The whole ordeal of "at least X% satisfy BSD" is to my opinion a rather kitschy way of stating the Bhargava-school results in the first place (though for publicity reasons I can understand it). Though for that matter, I can't think that upping 40% to 41% for zeta-zeros is numerically that interesting either, yet the novelty in methods of course has some regard.
... ie the semidirect product of the half line by multiplication by positive integers. This site comes endowed with the structure sheaf of piecewise affine convex functions with integral slopes: a geometric structure of tropical type. We use mathematics in characteristic one to obtain Riemann-Roch formulas in this context, and generalizations of the Jensen formula to almost periodic functions in order to relate the existence part in Riemann-Roch to known results in complex geometry.
Here is Connes' abstract for next week (2 comments): I will explain in my talk the slow progression in the understanding of a geometry allowing one to get the explicit formulas as a trace formula, the zeta function as a Hasse-Weil counting and to start transposing the proof of Weil using a Riemann-Roch formula following Mattuck, Tate and Grothendieck. The work extends over twenty years and is joint work with Consani in the last ten years. During this last period we found the topos theoretic interpretation of the adele class space as the space of points of the scaling site ...
I have my own opinions on the matter, but Connes himself (perhaps in preparation for the Bristol workshop next week) put a preprint on arXiv a few days ago. arxiv.org/abs/1805.10501
If you are asking about in the half-plane of (conditional) convergence, then see 9.33 of Titchmarsh. archive.org/stream/TheTheoryOfFunctions/… If not, can't you just exponentiate something like the $L$-series of the Dirichlet character of conductor 4? This is still a Dirichlet series, and it has an entire continuation, and routinely it grows faster than you like vertically. You can of course iteratively exponentiate....
The OP used the word "ideas", while you talked about "intuition". In the interest of precision, I don't think these are exactly the same, as the latter can be much more hand-wavey, and really only a philosophical reason why something should be true, while the former IMO is more grounded in how the proof proceeds, though details glossed over. As for intuition in analysis, I don't know the Strichartz book, but the experience of 19th century analysis would lead me to think that some "intution" is ex post facto in nature (which is necessarily not bad, just could mislead the student at times).
