25
votes
Accepted
A generalisation of theorem of Landau on sum of two squares?
The general result you seek is due to Paul Bernays, who studied under E. Landau in Göttingen and proved in his dissertation (1912) that $$\tag{$\star$}\sum_{n \le x} b_Q(n) \sim C_Q \frac{x}{\sqrt{\...
- 11.4k
23
votes
$\psi(x)-x$ on average
In Theorem 1 of
Brent, Richard P.; Platt, David J.; Trudgian, Timothy S., The mean square of the error term in the prime number theorem, ZBL07569752.
it is shown that for sufficiently large $x$ one ...
- 96.2k
19
votes
Accepted
Does the sum of reciprocal of integers with average power at least two converge?
$\DeclareMathOperator{\ap}{ap}$ $\DeclareMathOperator{\rad}{rad}$
I think so. Fix $\rad(m)=p_1\ldots p_k=:P$. Denote by $\Omega$ the set of positive integers with all prime divisors in $\{p_1,\ldots,...
- 93.7k
15
votes
Accepted
Why are Deligne-type exponential sum estimates so hard to use?
There are a lot of subtle reasons such exponential sums can fail to exhibit square-root cancellation. First let me comment on two reasons suggested in your answer:
(1) trying to have an explicit ...
- 124k
8
votes
Accepted
Large values of $\zeta(1/2+it)$ from sums of short moments
I wrestled with this question for an annoyingly long time. Now understanding things much better, I'm a bit embarrassed that this took me so long to figure out (such is the process of learning though). ...
- 1,381
7
votes
Accepted
$\psi(x)-x$ on average
Impossible as this would imply that $\frac{\zeta'(s)}{\zeta(s)}+\frac1{s-1}$ is analytic on $\Re(s)\ge 1/2$.
Your bound
$$\int_1^x |\psi(y)-y|^2 dy = O(x^a)$$ for some $a < 2$ implies (with Cauchy-...
- 3,275
5
votes
Accepted
Proof of an asymptotic formula by Tricomi
$\newcommand{\si}{\sigma}
\newcommand{\Z}{\mathbb Z}$As noted in Pietro Majer's comment, the meaning of $\sim$ in the claim that, if $n$ and $N$ are sufficiently large, then
\begin{equation*}
P_{n,...
- 93.7k
4
votes
Accepted
Estimating a sum involving the von Mangoldt function
Following Joshua Stucky's remark, the sum can be rewritten as the following sum over prime numbers:
$$\sum_{p\leq x}(\log p)\left(1-\frac{1}{p}\right)\sum_{k=1}^\infty\left\lfloor\frac{x}{p^k}\right\...
- 89.9k
4
votes
Artin's conjecture for polynomials and rational functions over finite fields
Your formulation of Artin’s primitive root conjecture over $\mathbf F_q[x]$ is incorrect: you need to avoid not just that $a(x)$ is a square (for odd $q$), but also that $a(x)$ is a $d$th power where $...
- 46.5k
4
votes
The function $f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}$ for $g\in \text{SL}(2,\mathbb R)$
I will show the infimum is $0$ and give a bound for the supremum. Since $f$ is clearly positive, its supremum is positive, so $f$ is not constant. I don't know if there is any way to estimate the ...
- 124k
3
votes
What are the best known upper bounds for $\frac{1}{L(s, \chi)}$?
For unconditional results, see Theorem 11.4 of Montgomery-Vaughan, which states the following. Let $\chi$ be a primitive Dirichlet character modulo $q > 1$. Then there exists a constant $c > 0$ ...
- 7,639
3
votes
Accepted
Solutions of a linear diophantine equation
As Max Alekseyev observed, $N(h)$ is the number of partitions of $6h-6$ into at most $h-1$ parts. A complicated asymptotic formula exists for this quantity, as a special case of a result of Szekeres (...
- 89.9k
2
votes
Equidistribution of $(2^nx)$, $x$ irrational
Not always. The sum
$x=2^{-1}+2^{-2}+2^{-4}+2^{-8}+...$
with the exponent doubling on each successive term is irrational, but $(2^nx)$ is not uniform as $n\to\infty$. It clusters around each negative-...
- 1,014
2
votes
Bound for some trigonometric polynomials
Consider $\displaystyle g_1(t) = \frac{t^{10} - t}{t -1}$ and $\displaystyle g_2(t) = \frac{t^{110} - t^{11}}{t^{11} - 1}$, so that $g_j(e^{2\pi i x}) = f_j(x)$ for $j = 1,2$. We have
$$
g_1(t) = t\...
- 485
1
vote
Proof of an asymptotic formula by Tricomi
This is not a new answer but more properly a complement to the question itself and the Iosif Pinelis's answer above.
Thanks to my librarian, I found a digital copy of the paper by Tricomi in the ...
- 4,961
1
vote
Solutions of a linear diophantine equation
$N(h)$ can be expressed via partition function $q$ as $$N(h)=q(6h-6,h-1).$$
- 28k
1
vote
Inversion shift of a Galois radius
Only a partial answer for the special case of an integer $n$ with Galois radius $r$ of type $(a,b)$ where $\min(a,b)=1$.
Denote by $\mathbb{P}_{k}$ the set of $k$-th powers of primes and suppose $b>...
- 6,912
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