A result concluded by Dirichlet's theorem
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We know from the Prime Number Theorem (PNT) that
$$frac{1}{N}sum_{n=1}^N Lambda(n)= 1+ o(1),$$
where $Lambda$ is von Mangoldt function. Now consider $ W in mathbb{N}$ and define
$$tilde{Lambda} (n) :=
frac{Phi(W)}{W} ln(Wn+1) $$
if $Wn+1$ is prime and $0$ otherwise.$Phi$ is the Euler function. I saw somewhere that by Dirichlet's famous theorem about primes in arithmetic progressions and the PNT, it can be proved that
$$frac{1}{N}sum_{n=1}^N tildeLambda(n)= 1+ o(1).$$
Would anyone please introduce me some references to read the proof?
number-theory prime-numbers fourier-analysis analytic-number-theory
edited Nov 21 at 12:09
amWhy
191k27223439
asked Jul 11 at 20:16
user115608
1,253926
add a comment |
up vote
3
down vote
favorite
We know from the Prime Number Theorem (PNT) that
$$frac{1}{N}sum_{n=1}^N Lambda(n)= 1+ o(1),$$
where $Lambda$ is von Mangoldt function. Now consider $ W in mathbb{N}$ and define
$$tilde{Lambda} (n) :=
frac{Phi(W)}{W} ln(Wn+1) $$
if $Wn+1$ is prime and $0$ otherwise.$Phi$ is the Euler function. I saw somewhere that by Dirichlet's famous theorem about primes in arithmetic progressions and the PNT, it can be proved that
$$frac{1}{N}sum_{n=1}^N tildeLambda(n)= 1+ o(1).$$
Would anyone please introduce me some references to read the proof?
number-theory prime-numbers fourier-analysis analytic-number-theory
edited Nov 21 at 12:09
amWhy
191k27223439
asked Jul 11 at 20:16
user115608
1,253926
It is not so much a consequence of Dirichlet's theorem and PNT, but the proof comes out of combining the proof of PNT with the $L$-functions and characters that Dirichlet had already used in his proof about 60 years earlier. The technical details of this combination were worked out by de la Vallée-Poussin shortly after the proof of PNT. You'll want to search for "Prime Number Theorem for Arithmetic Progressions".
– Erick Wong
Jul 12 at 23:27
@ErickWong Thanks.I searched but I found nothing. would you please help?
– user115608
Jul 13 at 6:36
Did you sincerely find nothing? Literally searching for that exact phrase in Google results in at least 3 PDF proofs on the first page alone. One of them by Soprounov is particularly simple at only 3 pages long.
– Erick Wong
Jul 13 at 7:04
@ErickWong yes I did. None of them was exactly the proof I want.
– user115608
Jul 13 at 7:13
That is completely different from “I found nothing”, and completely different from your question which merely asks for references. If you can’t specify exactly what proof you want, no one can provide any references.
– Erick Wong
Jul 13 at 20:07
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
We know from the Prime Number Theorem (PNT) that
$$frac{1}{N}sum_{n=1}^N Lambda(n)= 1+ o(1),$$
where $Lambda$ is von Mangoldt function. Now consider $ W in mathbb{N}$ and define
$$tilde{Lambda} (n) :=
frac{Phi(W)}{W} ln(Wn+1) $$
if $Wn+1$ is prime and $0$ otherwise.$Phi$ is the Euler function. I saw somewhere that by Dirichlet's famous theorem about primes in arithmetic progressions and the PNT, it can be proved that
$$frac{1}{N}sum_{n=1}^N tildeLambda(n)= 1+ o(1).$$
Would anyone please introduce me some references to read the proof?
number-theory prime-numbers fourier-analysis analytic-number-theory
edited Nov 21 at 12:09
amWhy
191k27223439
asked Jul 11 at 20:16
user115608
1,253926
We know from the Prime Number Theorem (PNT) that
$$frac{1}{N}sum_{n=1}^N Lambda(n)= 1+ o(1),$$
where $Lambda$ is von Mangoldt function. Now consider $ W in mathbb{N}$ and define
$$tilde{Lambda} (n) :=
frac{Phi(W)}{W} ln(Wn+1) $$
if $Wn+1$ is prime and $0$ otherwise.$Phi$ is the Euler function. I saw somewhere that by Dirichlet's famous theorem about primes in arithmetic progressions and the PNT, it can be proved that
$$frac{1}{N}sum_{n=1}^N tildeLambda(n)= 1+ o(1).$$
Would anyone please introduce me some references to read the proof?
number-theory prime-numbers fourier-analysis analytic-number-theory
number-theory prime-numbers fourier-analysis analytic-number-theory
edited Nov 21 at 12:09
amWhy
191k27223439
asked Jul 11 at 20:16
user115608
1,253926
edited Nov 21 at 12:09
amWhy
191k27223439
asked Jul 11 at 20:16
user115608
1,253926
edited Nov 21 at 12:09
amWhy
191k27223439
edited Nov 21 at 12:09
amWhy
191k27223439
edited Nov 21 at 12:09
amWhy
191k27223439
191k27223439
asked Jul 11 at 20:16
user115608
1,253926
asked Jul 11 at 20:16
user115608
1,253926
asked Jul 11 at 20:16
user115608
1,253926
1,253926
It is not so much a consequence of Dirichlet's theorem and PNT, but the proof comes out of combining the proof of PNT with the $L$-functions and characters that Dirichlet had already used in his proof about 60 years earlier. The technical details of this combination were worked out by de la Vallée-Poussin shortly after the proof of PNT. You'll want to search for "Prime Number Theorem for Arithmetic Progressions".
– Erick Wong
Jul 12 at 23:27
@ErickWong Thanks.I searched but I found nothing. would you please help?
– user115608
Jul 13 at 6:36
Did you sincerely find nothing? Literally searching for that exact phrase in Google results in at least 3 PDF proofs on the first page alone. One of them by Soprounov is particularly simple at only 3 pages long.
– Erick Wong
Jul 13 at 7:04
@ErickWong yes I did. None of them was exactly the proof I want.
– user115608
Jul 13 at 7:13
That is completely different from “I found nothing”, and completely different from your question which merely asks for references. If you can’t specify exactly what proof you want, no one can provide any references.
– Erick Wong
Jul 13 at 20:07
add a comment |
It is not so much a consequence of Dirichlet's theorem and PNT, but the proof comes out of combining the proof of PNT with the $L$-functions and characters that Dirichlet had already used in his proof about 60 years earlier. The technical details of this combination were worked out by de la Vallée-Poussin shortly after the proof of PNT. You'll want to search for "Prime Number Theorem for Arithmetic Progressions".
– Erick Wong
Jul 12 at 23:27
@ErickWong Thanks.I searched but I found nothing. would you please help?
– user115608
Jul 13 at 6:36
Did you sincerely find nothing? Literally searching for that exact phrase in Google results in at least 3 PDF proofs on the first page alone. One of them by Soprounov is particularly simple at only 3 pages long.
– Erick Wong
Jul 13 at 7:04
@ErickWong yes I did. None of them was exactly the proof I want.
– user115608
Jul 13 at 7:13
That is completely different from “I found nothing”, and completely different from your question which merely asks for references. If you can’t specify exactly what proof you want, no one can provide any references.
– Erick Wong
Jul 13 at 20:07
It is not so much a consequence of Dirichlet's theorem and PNT, but the proof comes out of combining the proof of PNT with the $L$-functions and characters that Dirichlet had already used in his proof about 60 years earlier. The technical details of this combination were worked out by de la Vallée-Poussin shortly after the proof of PNT. You'll want to search for "Prime Number Theorem for Arithmetic Progressions".
– Erick Wong
Jul 12 at 23:27
It is not so much a consequence of Dirichlet's theorem and PNT, but the proof comes out of combining the proof of PNT with the $L$-functions and characters that Dirichlet had already used in his proof about 60 years earlier. The technical details of this combination were worked out by de la Vallée-Poussin shortly after the proof of PNT. You'll want to search for "Prime Number Theorem for Arithmetic Progressions".
– Erick Wong
Jul 12 at 23:27
@ErickWong Thanks.I searched but I found nothing. would you please help?
– user115608
Jul 13 at 6:36
@ErickWong Thanks.I searched but I found nothing. would you please help?
– user115608
Jul 13 at 6:36
Did you sincerely find nothing? Literally searching for that exact phrase in Google results in at least 3 PDF proofs on the first page alone. One of them by Soprounov is particularly simple at only 3 pages long.
– Erick Wong
Jul 13 at 7:04
Did you sincerely find nothing? Literally searching for that exact phrase in Google results in at least 3 PDF proofs on the first page alone. One of them by Soprounov is particularly simple at only 3 pages long.
– Erick Wong
Jul 13 at 7:04
@ErickWong yes I did. None of them was exactly the proof I want.
– user115608
Jul 13 at 7:13
@ErickWong yes I did. None of them was exactly the proof I want.
– user115608
Jul 13 at 7:13
That is completely different from “I found nothing”, and completely different from your question which merely asks for references. If you can’t specify exactly what proof you want, no one can provide any references.
– Erick Wong
Jul 13 at 20:07
That is completely different from “I found nothing”, and completely different from your question which merely asks for references. If you can’t specify exactly what proof you want, no one can provide any references.
– Erick Wong
Jul 13 at 20:07
add a comment |
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It is not so much a consequence of Dirichlet's theorem and PNT, but the proof comes out of combining the proof of PNT with the $L$-functions and characters that Dirichlet had already used in his proof about 60 years earlier. The technical details of this combination were worked out by de la Vallée-Poussin shortly after the proof of PNT. You'll want to search for "Prime Number Theorem for Arithmetic Progressions".
– Erick Wong
Jul 12 at 23:27
@ErickWong Thanks.I searched but I found nothing. would you please help?
– user115608
Jul 13 at 6:36
Did you sincerely find nothing? Literally searching for that exact phrase in Google results in at least 3 PDF proofs on the first page alone. One of them by Soprounov is particularly simple at only 3 pages long.
– Erick Wong
Jul 13 at 7:04
@ErickWong yes I did. None of them was exactly the proof I want.
– user115608
Jul 13 at 7:13
That is completely different from “I found nothing”, and completely different from your question which merely asks for references. If you can’t specify exactly what proof you want, no one can provide any references.
– Erick Wong
Jul 13 at 20:07