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6 votes

A basic conjecture/observation on the Riemann $\xi$-function

Consider any analytic function $f$ and a zero $s_0 \ne 0$ say simple, though things do not really change if there is a multiplicity as below. Then for some $c \ne 0$ we have $f(s)=f(s_0)+c(s-s_0)+O(|s-...
Conrad's user avatar
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4 votes
Accepted

Classification of L functions and Dirichlet series by poles

The answer to your question is "no". Let $\chi$ be a nontrivial Dirichlet character. Then for the Dirichlet series $\sum_{n=1}^\infty \chi(n)/n^{2s}$ we have $\sigma_c=0$ and $\sigma_a=1/2$, ...
GH from MO's user avatar
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4 votes

Equidistribution on $\mathrm{SU}_2$

It was asked in the comments that I provide some details. I prove slightly more: if $\mu_n$ denotes uniform probability on the sphere of radius $n$ and if $\rho:F_2 \to \mathrm{SU}_2$ is a ...
Mikael de la Salle's user avatar
8 votes
Accepted

Equidistribution on $\mathrm{SU}_2$

In the article "On the spectral gap for finitely-generated subgroups of SU(2)" by Jean Bourgain and Alex Gamburd (Invent. Math. 171, No. 1, 83-121 (2008)), they show that free subgroups of $...
Lucas Kaufmann's user avatar
4 votes
Accepted

Modified Gauss Sum when the characters have different period

One approach is to use the Polya-Vinogradov method: i.e. using a Fourier transform mod $2q$, find $c_m$ such that $$\sum_{m=1}^{2q} c_m e \left( \frac{mn}{2q} \right)= \begin{cases} 1 & \textrm{if ...
Will Sawin's user avatar
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5 votes
Accepted

Generators of the ideal class group

Let $G$ be the ideal class group, and let $H$ be the subgroup generated by the prime ideals of norm at most $3\log^2(d^2)=12\log^2(d)$. Assume that $H$ is a proper subgroup of $G$. Then there is a ...
GH from MO's user avatar
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4 votes

Density of a set of numbers whose prime factors are defined by congruences

Sean Eberhard has already answered your question in the comments, but perhaps it's worth mentioning that one can find quite precise information about the general class of problems you are interested ...
Anurag Sahay's user avatar
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8 votes

Reference request - Pillai-Selberg Theorem

The original references are: S. Selberg [Math. Z. 44 (1939), 306–318; zbMATH:0019.39308] S. S. Pillai [Proc. Indian Acad. Sci. Sect. A. 11 (1940), 13–20; zbMATH:66.0168.01, MR0001761]. Interestingly ...
Anurag Sahay's user avatar
  • 1,191
4 votes

Reference request - Pillai-Selberg Theorem

Sigmund Selberg (1939) proved this for square-free numbers. You can find the original proof here. For the general statement with a better error term than $o(x)$, see Addison (1957).
GH from MO's user avatar
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3 votes

Limit of an infinite series with quadratic arguments

We will use the following well known fact (two proofs of this fact can be found at the end of the post): Given $f(x)$ with period $1$, its Fourier series $$ f(x)=\sum_{j=0}^\infty a_j\cos(2\pi jx), $$ ...
Cave Johnson's user avatar
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0 votes

Is there a non-constructive proof that a specific integer satisfies the Goldbach conjecture?

We assume that for the largest known prime $p_\text{max}$ it holds that $p_\text{max} < n$ and $2n = p_1 + p_2$. Then it follows that $2p_\text{max} < 2n$. The Bertrand's theorem states that ...
Martin Pribyl's user avatar
2 votes

A question about the setup of zero density estimates for Dirichlet $L$-functions

0. You probably mean that $\chi$ runs through primitive Dirichlet characters modulo $q$. 1. Changing $1/2\leq\sigma$ to $1/2<\sigma$ would not make any difference. Indeed, for $\sigma=1/2$ better ...
GH from MO's user avatar
  • 96.9k
2 votes

Conjectured upper bound on the maximum value of the absolute value of the Möbius function in the poset of multiplicative partitions under refinement

This is far from an answer, I am just trying to summarize some of my previous comments, in order to be able to delete them. The upshot is: I guess that your Conjecture is true, possibly even with $\...
Sam Hopkins's user avatar
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