be defined by 
(a) Show that 
has exactly one point of discontinuity.
![$f \in C_{\text{per}}([-\pi,\pi]; \mathbb{C})$](http://latex.artofproblemsolving.com/7/a/a/7aa8f551ba6422af33ab978aaf842329b6c2aced.png)
such that its Fourier series diverges at 
.![$t \in [-\pi,\pi]$](http://latex.artofproblemsolving.com/d/1/3/d1331fab5626e64ec75c6c40edc6379bc591c803.png)
,![$f_t \in C_{\text{per}}([-\pi,\pi]; \mathbb{C})$](http://latex.artofproblemsolving.com/e/1/e/e1ef8ac797a9743ebd9e71bf1682b0246b221086.png)
such that the Fourier series of 
diverges at 
,
be a Banach space and for every 
we have a normed space 
.
and 
such that![\[
\sup_{k \geq 1} \| T_{n,k}x_n \|_{Y_n} = \infty.
\]](http://latex.artofproblemsolving.com/a/1/b/a1b7a2453200f95f61c2451fc91a8fc3a81fd8fd.png)
Show that![\[
B = \left\{ x \in X : \sup_{k \geq 1} \| T_{n,k}x \|_{Y_n} = \infty \ \forall n \geq 1 \right\}
\]](http://latex.artofproblemsolving.com/4/1/a/41a671b6c40492f454bf79d1a3205cd5592a94e6.png)
is of second category. (I am given the hint to write 
as![\[
A = \bigcup_{n \geq 1} A_n = \bigcup_{n \geq 1} \left\{ x \in X : \sup_{k \geq 1} \| T_{n,k}x \|_{Y_n} < \infty \right\}
\]](http://latex.artofproblemsolving.com/b/9/1/b9174ec68dd2a503c6d68deaf281c5e110637cbd.png)
and show that 
is of first category.)
.![$f_D \in C_{\text{per}}([-\pi,\pi]; \mathbb{C})$](http://latex.artofproblemsolving.com/8/a/b/8ab83356343e4c5e30778770377027d0b2d57735.png)
such that the Fourier series of 
diverges at 
for all ![\[
T_{n,k} : (C_{\text{per}}([-\pi,\pi]; \mathbb{C}), \| \cdot \|_\infty) \to \mathbb{C}, \quad f \mapsto S_k(f)(t_n),
\]](http://latex.artofproblemsolving.com/2/2/5/225a5eec7a9582981841918fc28de96379b31404.png)
where![\[
S_k(f)(x) = \sum_{|j| \leq k} c_j(f) e^{ijx}.
\]](http://latex.artofproblemsolving.com/a/c/3/ac37ad818424867f4504e1472b2abe0883b6284b.png)
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.
and trying to solv
Y by Mathematicsislovely, Mango247, Mango247
Let 
be a fixed positive integer. Prove that for every 
, there exist infinitely many 
and 
such that there are atleast 
primes among the following numbers given by
![\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]](//latex.artofproblemsolving.com/1/4/5/145bf6662b94ac521b905b028dff4ede7fd5fd18.png)
Proposed by Bimit Mandal
be a fixed positive integer. Prove that for every 
, there exist infinitely many 
and 
such that there are atleast 
primes among the following numbers given by![\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]](http://latex.artofproblemsolving.com/1/4/5/145bf6662b94ac521b905b028dff4ede7fd5fd18.png)
Proposed by Bimit Mandal
This post has been edited 1 time. Last edited by kapilpavase, Jan 25, 2021, 9:12 AM
be the first 
primes that are of the form 
.
.
and 
,
,
,
such that, there does not exist any 
with 
.
such that 
is not a 
th power modulo any 
.
where 
.
,
.
)
is asymptotically equivalent to,
Hence, for sufficiently large 
Thus, the total number of primes(counted with multiplicity) among the numbers of the form,
will be at least,
Therefore by pigeonhole principle there exists 
such that, the numbers of the form,
has at least,
primes.
be a fixed positive integer. Prove that for every 
,
and 
primes among the following ![\[m + 1, m + 2^t, m + 3^t, \cdots, m + N^t.\]](http://latex.artofproblemsolving.com/5/7/f/57f5384a94aa0df9cfdce5504e342dbabe78ffd6.png)
![\[\frac{n}{\varphi{n}} \ge 100Ct\]](http://latex.artofproblemsolving.com/1/1/b/11bd69bc1dc955000dbfd2c97adcbe864b241af6.png)
which is just possible by making 
.
be a number such that 
isn't a 
power modulo 
(which is just CRT). Now we look at this 
table![\[
\begin{bmatrix}
1+n+a & 2^t+n+a & \dots & N^t+n+a \\
1+2n+a & 2^t+2n+a & \dots & N^t+2n+a\\
\vdots & \vdots & \ddots & \vdots \\
1+N^{100t}n+a & 2^t+N^{100t}n+a & \dots & N^t+N^{100t}n+a \\
\end{bmatrix}
\]](http://latex.artofproblemsolving.com/2/8/f/28fde65e00fc124236899bebd59e31082e90be7f.png)
Now by Prime Number Theorem on AP, we get that the number of primes in 
column is 
And hence the expected number of primes in each row is atleast ![\[\frac{C N^{100t}}{ \log N} \cdot \frac{N}{N^{100t}}=\frac{CN}{\log N}\]](http://latex.artofproblemsolving.com/b/1/3/b132cf47a94862e3a75801e0b10194017d742385.png)
As required.
