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May
1 |
comment |
An introduction to sieve method and their application, Cojocaru & Murty
Actually, wait. Unless I'm missing something, their claimed error term is wrong, and it should be $O(1/\log x)$ rather than $O(1/x^{1/4})$ (this is fine for their purposes, as the error on the next line is $O(1/\log x)$ anyway). To see it has to be at least this big, just look at the integral of the error from 3 to 4. To see that it's no bigger, split the integral into two parts, integrating from 3 to $x^{1/2}$, say, and from $x^{1/2}$ to $x$. The latter decays like $x^{-1/8}$, while the former can be bounded by $O(1/\log x)$. |
May
1 |
comment |
An introduction to sieve method and their application, Cojocaru & Murty
Plugging in the lemma, you'd get an $A_1 \log t$ term. This is the same as $-A_1(1+\log(x/t))+A_1\log x + A_1$. Is that your question? |
Mar
18 |
awarded | Yearling |
Mar
17 |
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Question about a certain class of primes
Greg: Good point. I was pretty sure there was an easy check, but I didn't bother to think about it. |
Mar
17 |
revised |
Question about a certain class of primes
deleted 159 characters in body |
Mar
17 |
revised |
Question about a certain class of primes
added 851 characters in body |
Mar
17 |
answered | Question about a certain class of primes |
Mar
7 |
awarded | Enlightened |
Mar
7 |
awarded | Nice Answer |
Mar
7 |
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The behavior of a certain greedy algorithm for Erdős Discrepancy Problem
Fixed! Sorry about that. Apparently all the files I'd hosted at Emory before moving have gone away. |
Mar
7 |
revised |
The behavior of a certain greedy algorithm for Erdős Discrepancy Problem
added 11 characters in body |
Feb
27 |
awarded | Revival |
Nov
18 |
comment |
Long gaps between primes
Following up on @GerhardPaseman's comment, the expected order of magnitude is $c \log^2 n$ (though proving this is hard!). See, for example, this paper of Granville: www.dms.umontreal.ca/~andrew/PDF/cramer.pdf. It discusses Cramer's conjecture, why it's probably wrong (at least in a precise form), and what the correct modification should be. |
Nov
18 |
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Long gaps between primes
The Wikipedia article en.wikipedia.org/wiki/Prime_gap on prime gaps covers this topic. In particular, C can be taken to be arbitrarily large. |
Nov
3 |
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What is known about the sum x^{n^2}/n?
I'm not an expert, but the Eichler integral can be defined, purely formally, as the operator that sends, with your notation, $x^n \mapsto x^n/n$ (recall that for modular forms $x=exp(2\pi i z)$, so this is integration $dz$). Your series is then a formal ``half-integral'' of the standard theta function. I've seen such things arise in talks, though I don't know anything about them myself (and I trust your ability to google as much as my own). I'd recommend searching for "half-derivative" rather than "half-integral", since the latter appears quite frequently with another meaning. |
Oct
29 |
awarded | Informed |
Oct
23 |
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On extended Riemann Hypothesis and coefficients of Selberg Class L-functions
Even with an infinite sum, I think it's impossible if $d>1$. Let $G(s)=L(s,f)H(s)$. We need to balance three things: 1) The sum of $f*h$ being bounded, 2) $H(s)$ analytic in $\Re(s)>0$, maybe by asking for $S_h(X) \ll X^\epsilon$, and 3) The non-existence of zeros of $H(s)$ in $\Re(s)>0$. It's not even obvious to me that 1 and 2 are compatible once $d>1$: each is equivalent to a system of inequalities on $h(n)$. 1 has coefficients $S_f(X/n)$, whereas 2 has coefficients all one. Once $S_f(X)$ has size (as in $d>1$), these systems are on different scales, so a common solution is not obvious. |
Oct
21 |
answered | On extended Riemann Hypothesis and coefficients of Selberg Class L-functions |
Apr
17 |
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Off critical line zeros for half integer weight $L$-functions
Have you done any computations yourself? While I'm dubious that this should be true for almost any form, it's worth noting that $L(s,\theta_\chi)=L(2s-1/2,\chi)$ for a non-trivial Dirichlet character $\chi$, so RH presumably holds in this case. In general, though, the multiplicative structure of half-integral weight eigenforms is more complex, and I'd be very surprised if it were to hold if the form is orthogonal to the space of unary theta functions. |
Nov
30 |
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Is this extension of the Selberg class trivial?
You're absolutely right that there are issues with $L(s,f)^{1/2}$, which is why it's not actually something I want to consider. I brought it up mostly to clarify points 1 and 2. Maybe a prototype question would be this: Can $L(s,f)^{1/2}L(s,g)^{1/2}$ ever be sensibly continued to an entire function, where $L(s,f)$ and $L(s,g)$ are primitive elements of the Selberg class? It's known that each has zeros disjoint from the other, but maybe all the simple zeros coincide, or are there are no simple zeros, or... Conjecturally, this can't happen, but that's the sort of thing I'm imagining. |