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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