# Tag Info

1 vote

### Can every positive integer eventually be expressed in this form?

Based on John Omielan's outline, I will prove below that every integer $a\geq 122$ has a representation with exponents $s_k\in\{1,2\}$. Let us use the fact that every integer exceeding $33$ is a sum ...
• 100k

• 152k
Accepted

### Bounding the sum $f(x)=-\frac{x}{2}+\sum_{p\le x}\log(p)-\frac{1}{x}\sum_{p\le x}p\cdot \log(p)$

Just an $\epsilon$-expansion of my comment on the question. Recall the usual notation for the Chebyshev $\vartheta$-function: $$\vartheta(x) = \sum_{p\leqslant x} \log p.$$ Then, using the notation ...
• 1,276
Accepted

### Bounds for Dirichlet L-functions

As Peter Humphries said in a comment, the best known bound for $\sigma=1/2$ (applying to all $\chi$) is due to Petrow and Young: $$L(1/2 + it,\chi) \ll_{\varepsilon} (q(|t| + 1))^{1/6 + \varepsilon}.$$...
• 100k

### 1-1 map on the $\{0,1\}^k$

I assume words “more” and “less” to be non-strict, otherwise shift $k_0$ by 1. Assume that the weight (=number of 1s) of $x$ is $k/2-k_0$. There are plenty of $a$s which agree with $x$ only at $2k_0$ ...
• 22.4k
Accepted

### Bounding $H^4_{\text{ėt}}$ of a surface

Are you absolutely sure you want to compute $p$-adic etale cohomology for a smooth proper $\mathbb{Z}[1/S]$-scheme with $p \notin S$, so $p$ is not invertible on $X$? This will be painful, and I ...
• 36.4k

### Bloch–Beilinson conjecture for varieties over function fields of positive characteristic

This may not be precisely what you want, but a function field analogue of Beilinson's conjectures is formulated in R. Sreekantan, Non-Archimedean regulator maps and special values of $L$-functions, ...
• 1,369

### Is it possible to have square-free order(s) in $\mathbb{Z}^\times_N$?

It is impossible. The proof goes here. Denote $d$ as the order of $p_0$, so $d$ divides the order of $\mathbb{Z}^\times_N$ that is $4p'q'$. Let, $e=p_1\times p_2\times\cdots\times p_m$ be the exponent....
1 vote

### Automorphism groups in class sets of ternary lattices

I put lots of references at http://zakuski.math.utsa.edu/~kap/ I've got an early version working. At first I thought it would be just class number one or two. The six coefficients $a,b,c,d,e,f$ ...
• 25.4k
1 vote
Accepted

• 100k
Accepted

### Test for pair of odd primes $(p, 2p^2-1)$

Below I will prove that the proposed test is necessary, that is, if $k\in\text{A106483}$ then $b(2k+1)=6k$. Following the simplification proposed by Will Sawin in the comments, the test for a given ...
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• 6,210

### Integer Points on an Elliptic Curve

The sage code is an one-liner: EllipticCurve(QQ, [-4, 9]).integral_points(both_signs=True, verbose=True) This gives some few details on the computations and lists ...
• 1,946

### Finding rational points on intersection of quadrics in affine 3-space

If you want to solve a particular pair of these equations locally, and you know the intersection isn't singular, then you can always create a genus one model for the intersection on Magma and use the ...
Accepted

### Quantum probabilistic method?

The Hilbert-Polya approach to the Riemann hypothesis follows this path, by attempting to relate the zeroes of the Riemann zeta function to a quantum mechanical scattering problem. The probability ...
• 180k

### Sequence derived from transform of a given vector (with Fibonacci as partial sums)

Not a complete answer, but too long for a comment and addressing the conjecture which I take to be the most important part of the question. The double-loop transformation process seems familiar to me ...
• 6,776
### Convergence of a product in $\mathbb Q_2[[X]]$
The coefficients you use are all in $\mathbf Z_2$, so I advise working in $\mathbf Z_2[[x]]$ rather than in $\mathbf Q_2[[x]]$. You are using the wrong topology on the power series, as mentioned ...
Are you asking for a proof of existence, or an explicit construction? These are very different things! It is immediate from the definition that there exists a finite family $(W_i, W_i')_{i \in I}$ ...