# Tag Info

### If for the first $\|n\|$ primes $p_i, \left(\frac{p_i}n\right)=+1$, then $n$ is a square

This is not known. However, it may be provably false under the Generalized Riemann Hypothesis (GRH), depending on a constant calculated in a paper of Montgomery. Least Quadratic Non-Residue You are ...
• 69.9k
Accepted

### Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?

We have \begin{align*} \prod\limits_{2<p \le n} {\left( {1 - \frac{2}{p}} \right)^{ - 1} } & = \exp \left( { - \sum\limits_{2<p \le n} {\log \left( {1 - \frac{2}{p}} \right)} } \right) \\ &...
• 20.4k
Accepted

### Do we have the following property about positive integer sequences?

The answer is no, and $P=\{$all primes$\}$ provides a tidy counterexample: we know that $\sum_{p\in P} \frac1p$ diverges. However, given any $U\ge1$, the number of integers in $[U,2U)$ that are the ...
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### Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?

Asymptotically, $$g(n) = \prod_{2 < p < n}^{p \text{ prime}} 1-\frac2p \sim \frac{4C_2e^{-2\gamma}}{(\log n)^2} \approx \frac{0.83244}{(\log n)^2}$$ And presumably your expression is simply the ...
• 1,385
Accepted

### Show that $\sum_{n\le x}\max(n)=O(x)$

Miracles often happen when we turn this expression into a multiple sum. Let $p,q$ denote primes. Then we have \begin{aligned} \sum_{n\le x}\max(n) &=\sum_{t\le\log_2x}\sum_{\substack{n\le x\\\max(...
• 3,635
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### Mertens' theorem with the numerator other than $1$

That first asymptotic relation is incorrect. For $p \geq 3$, $$\frac{(1-1/p)^2}{1-2/p} = \frac{1-2/p+ 1/p^2}{1-2/p} = 1 + \frac{1}{p(p-2)}$$ and $\prod_{p\geq 3} (1 + \frac{1}{p(p-2)})$ converges by ...
• 30.7k

### Inverse Laplace transform of the Riemann zeta function

$\mathcal{L}$ means the bilateral Laplace transform. For each $\sigma\ne 1$, $t\mapsto \zeta(\sigma+it)$ is a tempered distribution, so its inverse Fourier transform makes sense in the sense of ...
• 68.4k
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### The composition of an algebraic function and a transcendental function

Is it true that the result is transcendental if I precompose a transcendental function with a non-constant algebraic function? Yes, it is true. Statement: Let $X,Y\subseteq\mathbb{R}$ (or $\mathbb{C}$)...
• 4,208
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• 30.9k
1 vote

### An integral formula valid for $\zeta(s)$ for all $s\neq 1$

Wikipedia (https://en.wikipedia.org/wiki/Riemann_zeta_function#Integral) says, $$\zeta(s)={1\over s-1}+{1\over2}+2\int_0^{\infty}{\sin(s\arctan t)\over(1+t^2)^{s/2}(e^{2\pi t}-1)}\,dt$$ for all ...
• 169k
1 vote
Accepted

• 3,635
1 vote
Accepted

### Help needed in deducing an inequality in a lemma in the proof of linnik's theorem.

So if $h\in \Omega_p$ that means $h$ is a part of one of the excluded residue classes mod $p$. By definition of $S$, any $n\in S$ will not be a part of $\Omega_p$ and thus we can never have \$n\equiv h\...
• 1,616

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