Search Results

Fleshing out Ofir Gorodetsky's comment: if we define $G(s) = \sum_{n=1}^\infty \mu(n)\phi(n)n^{-s}$, then we have $G(s) = F(s)/\zeta(s-1)$ where $$ F(s) = \prod_p \bigg( 1 - \frac1{p^s-p} \bigg) $$ is …

Forgetting the squarefree condition for a moment, the number of integers up to $N$ that are not divisible by any primes less than $N^{1/k}$ is asymptotic to $$ \omega(k) \frac N{\log N} \sim e^\gamma …

I don't believe your new conjecture is true. Taking $b=a+\sqrt a$ for concreteness: standard sieve results show that the number of integers in $[a,b]$ that can be represented as the sum of two squares …

For (2), the answer is yes - see for example this paper of Baker and Harman.

Let $\chi(n)$ denote the quadratic character modulo $p$ (so $\chi(n) = 1$ if $n$ is a quadratic residue modulo $p$, and $\chi(n)=-1$ if $n$ is a quadratic nonresidue modulo $p$). The difference betwee …

Under a very slight strengthening of the hypothesis, this can be done by elementary methods, namely the "Wirsing–Odoni method". The following version is Proposition 4 in this paper of mine with Finch …

I feel like this is an error. It is true that $1-1/r^\epsilon$ is a known upper bound for the real zeros of Dirichlet $L$-functions corresponding to characters $\chi\pmod r$ (for $r$ sufficiently larg …

The standard approach would use the Möbius function: \begin{align*} \sum_{\substack{N<n<M\\ (n,q)=1}}e^{2\pi i n/q} &= \sum_{N<n<M}e^{2\pi i n/q} \sum_{d\mid(n,q)} \mu(d) \\ &= \sum_{d\mid q} \mu(d) \ …

Good question, and I agree that the authors should have been more explicit here. However, I think I can reconstruct their argument: note that \begin{align*} \sum_{q \leq x^{2\delta}}\tau(q)^2 \bigg | …

The difference between $x\ge3$ and $x\ge2$ is, in my opinion, completely unimportant. When $2\le x<3$, the sum being considered is just $a_1+a_2$, and the right-hand side of the asymptotic formula is …

One should always consider Rankin's method: for $\varepsilon>0$, $$ \sum_{n\le x} \frac{\mu^2(n)\tau_k(n)}{\phi(n)} \le \sum_{n=1} ^\infty \frac{\mu^2(n)\tau_k(n)}{\phi(n)} \frac{x^\varepsilon}{n^\var …

Kaczorowski has written a few papers on this topic. One of his more recent papers gives almost this result, assuming (something somewhat weaker than) the Selberg orthogonality conjecture. The result i …

For fixed $a$, the function $\big( \frac ap \big)$ defines a Dirichlet character (mod $4a$) (and often modulo a smaller modulus). More precisely, the Jacobi symbol $\big( \frac an \big)$ defines such …

We can show at least that your expression tends to a constant strictly greater than $0$. (Well, what we'll actually show is that its lim inf is positive.) First we claim that $$ \sum_{d\le w} \frac{\ …

Nature wants to count the primes up to some cutoff point $x$; when we humans insist on labeling the $n$th prime as the $n$th prime, we are destined to have very large error terms. Here, I don't know t …