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 whose reference I need. Does there exist any better bound unconditionally? Any help will be highly...

I have been doing a computational/visual exploration of the Riemann zeta function and its completed xi function. I am a software developer and mathematics enthusiast rather than a professional mathematician, and my main aim here is to identify the established theory behind the following...

If $$S=\sum_{b\geq 2, \ \mu^2(b)=1} \frac{\mu(b)}{b^2}\sum_{c|b, c<\sqrt{b}}\frac{1}{\left(\frac{1}{c^2}+\frac{c^2}{b^2}\right)^{3/2}}$$ Prove that $S$ is conditionally convergent. Since $\mu(b)\neq0$ only for squarefree $b$, write $b=ck$, $c<k$, $(c,k)=1$. Then $ \mu(b)=\mu(ck)=\mu(c)\mu(k), $...

I have a question regarding this paper by Tenenbaum and Bréteché. They define $$ S_\alpha(\mathcal{M}) = \sum_{m,n\in\mathcal{M}} \frac{(m,n)^\alpha}{[m,n]^\alpha} = \sum_{m,n\in\mathcal{M}} \biggl( \frac{(m,n)^2}{mn} \biggr)^\alpha \quad\text{and}\quad \Gamma_{\alpha}(N) =...

I know that Schur and Murty proved that an Euclidean proof (hence a "non-analytic" proof) for the existence of infinite primes $p \equiv \ell \mod q$ with $q$ and $\ell$ coprime can be given if and only if $\ell^2 \equiv 1 \mod q$. Are there any cases where $\ell^2 \not\equiv 1 \mod q$, but we...

I had posted this question in MO that has remained unanswered in MO for more than two years now. While working on it, I accidently found an unexpected result. Let $0<x\leq y\leq z$ be the ordered side lengths of the triangle determined by three independent uniformly distributed points on a...

We consider the $L$-function associated with a nonzero Maass cusp form $f$ of weight $0$, level $N$, and Laplace eigenvalue $1/4+r^2$. Let $t(n)$ be the normalized Fourier coefficient corresponding to the Maass cusp form $f$. I need the estimate of $$\sum_{n\le T}t(n)e^{2\pi i n x},$$ where $x...

Fix $\frac12<\sigma<1$, and define the signed logarithmic integral $$ A_\sigma(T) \int_2^T \log |\zeta(\sigma+it)|,dt. $$ I am interested in transferring recent zero-density estimates into bounds for $A_\sigma(T)$. Applying Littlewood's lemma to $\zeta(s)$ in the rectangle $$\sigma\le...

The following functions were originally proposed in a Reddit discussion on r/googology Define $$ \operatorname{LCD}(a,b)= \begin{cases} \min\{\,d>1:\ d\mid a,\ d\mid b\,\}, & \text{if such integer divisor exists},\\ 0, & \text{otherwise}. \end{cases} $$ For each positive integer $n$, let $$...

Let $p \geq 3$ be an odd prime. I would like to know whether the following conjecture is true. Conjecture For every odd prime $p \geq 3$, there exist integers $a$ and $b$ such that: $a+b+3$ is prime; $4a+2b+3=p$ $\gcd(a,b,3)=1$ $b^2\leq 4a$ $a\geq 1$. Here, $\mathbb{P}$ denotes the set of prime...

Let $s \ge 4$ be a composite integer with $s \ne 0 \pmod3$. Computationally, for every such $s \le 2000$, I can find a prime $p$ and integer $k \ge 1$ such that $p^k - s$ is prime (usually with $p \in \{2,3\}$ and small $k$). Heuristically this seems unsurprising: for fixed small $p$, the values...

This is different from asking whether the Prime Number Theorem implies the asymptotic. I am specifically asking whether this asymptotic has an explicit published reference (journal article, book, or monograph), rather than whether it follows from standard methods. I am looking for a literature...

Let $b\ge 2$. Partition $[0,1)$ into the $b$ equal half-open intervals $$ I_j=\left[\frac{j}{b},\frac{j+1}{b}\right), \qquad 0\le j\le b-1. $$ Define the same-bin indicator $$ H_b(x,y)= \begin{cases} 1, & x,y\text{ lie in the same }I_j,\\ 0, & \text{otherwise}, \end{cases} $$ and the centered...

Consider the sequence defined by $a_1 = 1$ and the recurrence relation for $n \ge 2$: $$a_n = \sum_{k=1}^{n-1} a_{\gcd(n,k)}$$ Grouping the terms by their divisors $d = \gcd(n,k)$, the number of integers $k < n$ such that $\gcd(n,k) = d$ is given by $\varphi(n/d)$, where $\varphi$ is Euler's...

Let $$B:=\{n\geq 13: n \text{ odd and there exists a prime } q \text{ such that } 9n^2+1-q \text{ is prime, where either } q=3, \text{or } q\geq 11,q \equiv 2\pmod 3, q-1 \text{ is not a square}\}$$ I am trying to understand whether the following “almost-all Goldbach” statement is known /...

I am working on moments of the Riemann zeta function, and want to get a numerical lower bound for the $k$th moment of $\zeta(s)$ of the form $$\int_1^T|\zeta(1/2+it)|^{2k}dt>C(k)T(log T)^{k^2}.$$ Soundararajan (https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300011438)...

I am reading the paper Oscillatory Integrals with Uniformity in Parameters by Kiral, Petrow, and Young, and I am having trouble understanding the notion of an inert function introduced in Definition $2.1$. These are my confusions Since $X=X_T \in [1,\infty]$.Then if we consider a family of...

$\newcommand{\e}[1]{\exp\left(#1\right)}$ $\newcommand{\le}{\leqslant}$ In Terry Tao's Math 254A Notes 5 (Bounding exponential sums and the zeta function), Exercise 5 outlines the derivation of the Vinogradov-Korobov bound for Dirichlet $L$-functions. Let $\chi$ be a non-principal character of...

Given a real polynomial $f(x)=a_0x^d+\cdots+a_1 x$, I want to give a sharp estimate for $\sum_{n\le X} e(f(n))$. If $f(x)=ax$ is a linear function, we have $$\sum_{n\le X} e(\alpha n)\ll \min (X,\|\alpha\|^{-1});$$ If $f(x)\in\mathbb{Z}[x]$ where $p$ is a prime, using Weil's bound for...

Let $A\in GL(r,\mathbb{Z})$ be a positive definite matrix, $Q(x)=\dfrac{1}{2} x^t A x$ be the quadratic form associate to $A$, $Q^*(x)=\dfrac{1}{2} x^t A^{-1} x$ be the adjoint form of $Q(x)$ . Given $(c,d)=1,m\in\mathbb{Z}^r,$ we define $$G_m\left(\dfrac{d}{c}\right)=\sum_{h\,\text{mod}\,c}...