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Your heuristic is wrong: $G(x)=x\exp{(-a\sqrt{\log{x}}})$ follows from $f=\frac{c}{\log{(|t|+3)}}$ for some fixed real $c>0$. I really don't want to tell you the answer, because this is a great exerc …

This case, at least, you can do by hand; if $c(n)$ is the $n$th coefficient of $\zeta(s)\zeta(2s)$, then $\sum_{n\leq X}c(n)=\sum_{nm^2\leq X}1=\sum_{m}\sum_{n\leq X/m^2}1 = \sum_{m < \sqrt{X}}(m^{-2 …

For any fixed real $\alpha$, the fractional parts of the numbers $\alpha \gamma$, where $\beta+i\gamma$ runs over all zeros of $\zeta(s)$ in the critical strip with $0<\gamma < T$, become uniformly d …

This may be a stretch, but what about the Burgess bound for character sums? The Riemann hypothesis over finite fields is a ubiquitous tool in number theory now, and this seems as a good an introducti …

Well, suppose pi is a cuspidal automorphic representation of GL(n)/Q. This has the structure of a tensor product, indexed by primes p, of representations pi_p of the groups GLn(Qp). The Satake isomo …

If $r_k(n)$ is the number of representations of $n$ as a sum of $k$ squares, and $k\geq 5$, then the estimate $\sum_{n \leq X}r_k(n)=C_k X^{\frac{k}{2}}+O(X^{\frac{k}{2}-1})$ is known, and is best p …

(2017-11-26 edit by j.c.: earlier versions of this answer consisted of David Hansen's screenshot of the following, with the text "Here is a screenshot of a semi-answer which froze my computer when I h …

Hi Franz, Unfortunately I doubt this Euler product has very good behavior. If you believe the Hardy-Littlewood conjectures, then $\sum_{n\leq X}\Lambda(n^2+1) \sim cX$ where $c=\prod_{p>2}(1-\chi_{ …

I would be willing to wager vast sums of money that this function has the unit circle as a boundary of essential singularities. As $x\to 1$ it diverges like $u(x) \sim c (1-x)^{-2}\log{(1-x)}$. Edit …

Something small, but maybe useful, which no one seems to have pointed out: as $p\to\infty$, $(\mathbb{Z}/p\mathbb{Z})^{\times}$ contains arbitrarily long strings of consecutive quadratic residues. In …

I believe you can extract this from a paper of Andrew Granville, "Prime divisors are Poisson distributed". There is an electronic copy of this on his website.

I believe the following statement is a folklore conjecture: Let pi1 and pi2 be a pair of cuspidal automorphic representations of GL2 whose local constituents agree at a set of places of positive Diric …

Sorry, I meant to say that furthermore one of the pi's should be genuine in Shahidi's sense, or in otherwords not twist-equivalent to itself. Automorphic inductions of grossencharakters, as in Kevin' …

Hi Anweshi, Since Emerton answered your third grey-boxed question very nicely, let me try at the first two. Suppose $L(s,f)$ is one of the L-functions that you listed (including the first two, which …

To elaborate on my comment, I really do think this won't be resolved until RH itself is resolved, or at least until new tools are brought onto the scene, because it's so far from what we know about th …