# Tagged Questions

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### Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions

For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, ...
1answer
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Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space $$l^2_{k^{-2}}:=\{z=\{z(k)\}_{k=1}^\infty:\sum\limits_{k=1}^\infty z(k)^2k^{-2}<\infty\}.$$ It is known that for some $x\in ... 1answer 886 views ### How good is “almost all” when it comes to the Riemann Hypothesis? Let$N(T)$be the number of zeroes of the Riemann zeta function$\zeta$having imaginary part strictly between$0$and$T$, and let$N_0(T)$be the number of those zeroes that also have real part ... 2answers 315 views ### Reference and best bounds of$\sum_{n\leq x}\frac{\mu(n)}{n}$Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally,$A(x)=O(e^{-c\sqrt{\log x}})$is known to me. Does ... 0answers 818 views ### Questions on de Branges' work on the Riemann hypothesis According to Wikipedia, Louis de Branges de Bourcia has obtained some notable results, such as a proof of the Bieberbach conjecture in 1985, which is now known as de Branges' theorem. Initially, his ... 1answer 311 views ### On link between Riemann hypothesis and partial GRH Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character$\chi(1)$which could be qualified of ... 1answer 923 views ### A reformulation of the Riemann Hypothesis I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates$\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where$N$is product of distinct primes. Let's define ... 1answer 364 views ### Heuristic for Montgomery's conjecture This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true. ... 1answer 231 views ### Estimate on the prime-counting function$\psi(x)$. There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ... 2answers 681 views ### Effective Chebotarev without Artin's conjecture Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Artin's$L$-function and ... 4answers 1k views ### Good uses of Siegel zeros? The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "expectional zeros" of course first ... 1answer 894 views ### Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ? Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ... 1answer 818 views ### A question about Speiser's 1934 result on the Riemann hypothesis A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies$\zeta'(s)\neq 0$for all$0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) ... 1answer 1k views ### What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers? Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis? I've heard Freeman Dyson say that ... 1answer 572 views ### Montgomery's pair correlation function without RH? In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as$$F(\alpha) = \frac{1}{N(T)} \sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} ... 3answers 933 views ### On Robin's criterion for RH [closed] $$\sigma(n) < e^\gamma n \log \log n$$ In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ... 1answer 1k views ### Exceptional zeros and Liouville's$\lambda\$ function

This originated from an textbook exercise (recently posted to math.stackexchange http://math.stackexchange.com/questions/62883/quadratic-characters-and-liouvilles-function with no success) but I think ...
3answers
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### The Hardy Z-function and failure of the Riemann hypothesis

David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...
4answers
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### Modular forms and the Riemann Hypothesis

Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions? What I'm thinking of is this: under the Mellin transform, the Riemann zeta function ...