15
votes
Computing hypergeometric function at 1
Carlo's answer is correct but doesn't show why the identity holds, so let me explain how to do this easily by hand.
Look more generally at
$$S={}_3F_2\left(\begin{matrix}-m,a,b\\a+1,b+1\end{matrix};1\...
- 1,514
12
votes
Accepted
Is there any use of logarithmic derivatives of modular forms?
Application 1: in the chapter on modular forms in Serre's Course in Arithmetic, he integrates the logarithmic derivative $f'/f$ of a modular form $f$ on ${\rm SL}_2(\mathbf Z)$ around a contour (...
- 49.1k
10
votes
On the number of distinct prime factors of $p^2+p+1$
There is the following theorem of Halberstam, "On the distribution of additive number-theoretic functions. III." Let $\omega(n)$ be the number of prime factors of $n$. Given any irreducible ...
- 10.3k
10
votes
Accepted
On the number of distinct prime factors of $p^2+p+1$
Yes. At first, there exist $c$ distinct primes $q_1,...,q_c$ which divide some $m_i^2+m_i+1$ for $i=1,\ldots,c$ respectively (induction on $c$: if you found $c-1$ such primes, take $m_c$ being equal ...
- 99k
7
votes
Raising positive integer to $c\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!
To expand on a comment of Lucia, when $c$ is irrational, we can show that there are at most $O((\log N)^2)$ values of $n\leq N$ such that $N^c$ is rational, let alone an integer.
Let $\mathcal{A}$ be ...
- 1,778
6
votes
Accepted
Computing hypergeometric function at 1
For questions like this, Mathematica is your friend:
$$\, _3F_2\left(-m-\tfrac{1}{2},-m,k-m+\tfrac{1}{2};\tfrac{1}{2}-m,k-m+\tfrac{3}{2};1\right)$$
$$=\tfrac{1}{2}(k+1)^{-1}\Gamma (m+1) \left(\frac{(2 ...
- 173k
5
votes
Accepted
'$\times$' or '$\otimes$' when writing $L$-functions?
The symbol $\times$ on the left-hand side is the Rankin-Selberg product. If $\pi$ and $\rho$ are automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(n)$, respectively, then one can define ...
- 95.7k
5
votes
Which algebraic groups are generated by (lifts of) reflections?
Let $M$ be a division algebra of degree 3 (i.e., dimension 9) over $\mathbf{Q}$ that splits over $\mathbf{R}$, and $M_1$ its norm 1 subgroup. So $M_1$ is a $\mathbf{Q}$-anisotropic simple algebraic ...
- 59.4k
4
votes
Accepted
Large sets of nearly orthogonal integer vectors
Let me prove the bounds
$$2^k{n\choose k+1}+\sum_{j=0}^{k}2^{j}{n\choose j}\leqslant a(n,k)\leqslant 2^{k}{n\choose k+1}+\sum_{j=0}^{k}2^{j}{k\choose j}{n\choose j}$$
which differ by $O_k(n^{k-1})$ ...
- 99k
4
votes
Four new series for $\pi$ and related identities involving harmonic numbers
a bit long for a comment.
The "four new series for $\pi$" are examples of relationships between hypergeometric functions $_pF_{p-1}$ with rational arguments, for example, the first series is ...
- 173k
3
votes
Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?
As in the question $K$ is a number field and $E/K$ an elliptic curve.
Let me start by saying that I think the best analogues are the following two short exact sequences (the first two "$/n$" ...
- 7,961
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