# Tagged Questions

115 views

### 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 $n$, ...
235 views

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 $x\in ... 1answer 892 views ### How good is “almost all” when it comes to the Riemann Hypothesis? Let$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 319 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 827 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 683 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 819 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$$F(\alpha) = \frac{1}{N(T)} \sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} ... 3answers 934 views ### On Robin's criterion for RH [closed] $$\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 ...