What is the simplest way of explaining the Riemann Hypothesis to a layman?
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Almost all people from number theory and many more from other branches of mathematics know the Riemann Hypothesis and its importance. However I am having trouble explaining why the Riemann Hypothesis is considered to one of the most important open problems in mathematics today.
What is the best way to explain the Riemann Hypothesis to a layman?
complex-analysis number-theory analysis prime-numbers analytic-number-theory
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
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up vote
2
down vote
favorite
Almost all people from number theory and many more from other branches of mathematics know the Riemann Hypothesis and its importance. However I am having trouble explaining why the Riemann Hypothesis is considered to one of the most important open problems in mathematics today.
What is the best way to explain the Riemann Hypothesis to a layman?
complex-analysis number-theory analysis prime-numbers analytic-number-theory
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
2
My professor explained it by defining the $p-series$. Then he extends the $p-series$ to the complex plane, defining it as $zeta-$function. Next, he shows how Analytic Continuation works, gave some examples like $1+x+x^2+...$ and $frac11-x$ are the analytic continuation of each other. Next, he mixes them up and tells us the "Riemann Hypothesis", Zeros of Zeta function, etc. He next shows the relation between Euler's Prime product and Zeta function. I personally find some links that help me a lot. youtube.com/watch?v=NaL_Cb42WyY & youtube.com/watch?v=sD0NjbwqlYw
– Sujit Bhattacharyya
Sep 1 at 5:16
This question has been asked many times; have a look at the web, e.g. here.
– Dietrich Burde
Sep 1 at 11:35
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Almost all people from number theory and many more from other branches of mathematics know the Riemann Hypothesis and its importance. However I am having trouble explaining why the Riemann Hypothesis is considered to one of the most important open problems in mathematics today.
What is the best way to explain the Riemann Hypothesis to a layman?
complex-analysis number-theory analysis prime-numbers analytic-number-theory
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
Almost all people from number theory and many more from other branches of mathematics know the Riemann Hypothesis and its importance. However I am having trouble explaining why the Riemann Hypothesis is considered to one of the most important open problems in mathematics today.
What is the best way to explain the Riemann Hypothesis to a layman?
complex-analysis number-theory analysis prime-numbers analytic-number-theory
complex-analysis number-theory analysis prime-numbers analytic-number-theory
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
asked Sep 1 at 5:02
Nilotpal Kanti Sinha
3,26321232
3,26321232
2
My professor explained it by defining the $p-series$. Then he extends the $p-series$ to the complex plane, defining it as $zeta-$function. Next, he shows how Analytic Continuation works, gave some examples like $1+x+x^2+...$ and $frac11-x$ are the analytic continuation of each other. Next, he mixes them up and tells us the "Riemann Hypothesis", Zeros of Zeta function, etc. He next shows the relation between Euler's Prime product and Zeta function. I personally find some links that help me a lot. youtube.com/watch?v=NaL_Cb42WyY & youtube.com/watch?v=sD0NjbwqlYw
– Sujit Bhattacharyya
Sep 1 at 5:16
This question has been asked many times; have a look at the web, e.g. here.
– Dietrich Burde
Sep 1 at 11:35
add a comment |Â
2
My professor explained it by defining the $p-series$. Then he extends the $p-series$ to the complex plane, defining it as $zeta-$function. Next, he shows how Analytic Continuation works, gave some examples like $1+x+x^2+...$ and $frac11-x$ are the analytic continuation of each other. Next, he mixes them up and tells us the "Riemann Hypothesis", Zeros of Zeta function, etc. He next shows the relation between Euler's Prime product and Zeta function. I personally find some links that help me a lot. youtube.com/watch?v=NaL_Cb42WyY & youtube.com/watch?v=sD0NjbwqlYw
– Sujit Bhattacharyya
Sep 1 at 5:16
This question has been asked many times; have a look at the web, e.g. here.
– Dietrich Burde
Sep 1 at 11:35
2
2
My professor explained it by defining the $p-series$. Then he extends the $p-series$ to the complex plane, defining it as $zeta-$function. Next, he shows how Analytic Continuation works, gave some examples like $1+x+x^2+...$ and $frac11-x$ are the analytic continuation of each other. Next, he mixes them up and tells us the "Riemann Hypothesis", Zeros of Zeta function, etc. He next shows the relation between Euler's Prime product and Zeta function. I personally find some links that help me a lot. youtube.com/watch?v=NaL_Cb42WyY & youtube.com/watch?v=sD0NjbwqlYw
– Sujit Bhattacharyya
Sep 1 at 5:16
My professor explained it by defining the $p-series$. Then he extends the $p-series$ to the complex plane, defining it as $zeta-$function. Next, he shows how Analytic Continuation works, gave some examples like $1+x+x^2+...$ and $frac11-x$ are the analytic continuation of each other. Next, he mixes them up and tells us the "Riemann Hypothesis", Zeros of Zeta function, etc. He next shows the relation between Euler's Prime product and Zeta function. I personally find some links that help me a lot. youtube.com/watch?v=NaL_Cb42WyY & youtube.com/watch?v=sD0NjbwqlYw
– Sujit Bhattacharyya
Sep 1 at 5:16
This question has been asked many times; have a look at the web, e.g. here.
– Dietrich Burde
Sep 1 at 11:35
This question has been asked many times; have a look at the web, e.g. here.
– Dietrich Burde
Sep 1 at 11:35
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
We all know what prime numbers are. Euclid has proven that there are infinitely many of them. Experience has taught us that they get more rare when we come to ever higher numbers. Of course mathematicians want to describe the "statistics" of the primes in more precise terms. The prime number theorem (proven at the end of the $19^rm th$ century) tells us that there are about $n/log n$ primes $leq n$ when $n$ is large. The next question then is: How much can the true number $pi(n)$ deviate from this estimate? Working on this problem already Riemann has remarked that there is a huge technical stumbling block, nowadays called the "Riemann Hypothesis". So far nobody has managed to move this block away. Therefore mathematicians go around it: Many papers have a proviso in their introduction: "Assuming that the Riemann Hypothesis is true, we prove the following: $> ldots> $".
answered Sep 1 at 9:10
Christian Blatter
166k7109311
add a comment |Â
up vote
1
down vote
One may interpret the Riemann Hypothesis by saying that the primes are distributed
as regularly as possible: for any real number $x$ the number of prime numbers less than $x$ is
approximately $Li(x)$ and this approximation is essentially square root
accurate. More precisely,
$$
pi(x)=Li(x)+O(sqrtxlog(x)).
$$
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
We all know what prime numbers are. Euclid has proven that there are infinitely many of them. Experience has taught us that they get more rare when we come to ever higher numbers. Of course mathematicians want to describe the "statistics" of the primes in more precise terms. The prime number theorem (proven at the end of the $19^rm th$ century) tells us that there are about $n/log n$ primes $leq n$ when $n$ is large. The next question then is: How much can the true number $pi(n)$ deviate from this estimate? Working on this problem already Riemann has remarked that there is a huge technical stumbling block, nowadays called the "Riemann Hypothesis". So far nobody has managed to move this block away. Therefore mathematicians go around it: Many papers have a proviso in their introduction: "Assuming that the Riemann Hypothesis is true, we prove the following: $> ldots> $".
answered Sep 1 at 9:10
Christian Blatter
166k7109311
add a comment |Â
up vote
2
down vote
We all know what prime numbers are. Euclid has proven that there are infinitely many of them. Experience has taught us that they get more rare when we come to ever higher numbers. Of course mathematicians want to describe the "statistics" of the primes in more precise terms. The prime number theorem (proven at the end of the $19^rm th$ century) tells us that there are about $n/log n$ primes $leq n$ when $n$ is large. The next question then is: How much can the true number $pi(n)$ deviate from this estimate? Working on this problem already Riemann has remarked that there is a huge technical stumbling block, nowadays called the "Riemann Hypothesis". So far nobody has managed to move this block away. Therefore mathematicians go around it: Many papers have a proviso in their introduction: "Assuming that the Riemann Hypothesis is true, we prove the following: $> ldots> $".
answered Sep 1 at 9:10
Christian Blatter
166k7109311
add a comment |Â
up vote
2
down vote
up vote
2
down vote
We all know what prime numbers are. Euclid has proven that there are infinitely many of them. Experience has taught us that they get more rare when we come to ever higher numbers. Of course mathematicians want to describe the "statistics" of the primes in more precise terms. The prime number theorem (proven at the end of the $19^rm th$ century) tells us that there are about $n/log n$ primes $leq n$ when $n$ is large. The next question then is: How much can the true number $pi(n)$ deviate from this estimate? Working on this problem already Riemann has remarked that there is a huge technical stumbling block, nowadays called the "Riemann Hypothesis". So far nobody has managed to move this block away. Therefore mathematicians go around it: Many papers have a proviso in their introduction: "Assuming that the Riemann Hypothesis is true, we prove the following: $> ldots> $".
answered Sep 1 at 9:10
Christian Blatter
166k7109311
We all know what prime numbers are. Euclid has proven that there are infinitely many of them. Experience has taught us that they get more rare when we come to ever higher numbers. Of course mathematicians want to describe the "statistics" of the primes in more precise terms. The prime number theorem (proven at the end of the $19^rm th$ century) tells us that there are about $n/log n$ primes $leq n$ when $n$ is large. The next question then is: How much can the true number $pi(n)$ deviate from this estimate? Working on this problem already Riemann has remarked that there is a huge technical stumbling block, nowadays called the "Riemann Hypothesis". So far nobody has managed to move this block away. Therefore mathematicians go around it: Many papers have a proviso in their introduction: "Assuming that the Riemann Hypothesis is true, we prove the following: $> ldots> $".
answered Sep 1 at 9:10
Christian Blatter
166k7109311
answered Sep 1 at 9:10
Christian Blatter
166k7109311
answered Sep 1 at 9:10
Christian Blatter
166k7109311
answered Sep 1 at 9:10
Christian Blatter
166k7109311
166k7109311
add a comment |Â
add a comment |Â
up vote
1
down vote
One may interpret the Riemann Hypothesis by saying that the primes are distributed
as regularly as possible: for any real number $x$ the number of prime numbers less than $x$ is
approximately $Li(x)$ and this approximation is essentially square root
accurate. More precisely,
$$
pi(x)=Li(x)+O(sqrtxlog(x)).
$$
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
add a comment |Â
up vote
1
down vote
One may interpret the Riemann Hypothesis by saying that the primes are distributed
as regularly as possible: for any real number $x$ the number of prime numbers less than $x$ is
approximately $Li(x)$ and this approximation is essentially square root
accurate. More precisely,
$$
pi(x)=Li(x)+O(sqrtxlog(x)).
$$
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
add a comment |Â
up vote
1
down vote
up vote
1
down vote
One may interpret the Riemann Hypothesis by saying that the primes are distributed
as regularly as possible: for any real number $x$ the number of prime numbers less than $x$ is
approximately $Li(x)$ and this approximation is essentially square root
accurate. More precisely,
$$
pi(x)=Li(x)+O(sqrtxlog(x)).
$$
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
One may interpret the Riemann Hypothesis by saying that the primes are distributed
as regularly as possible: for any real number $x$ the number of prime numbers less than $x$ is
approximately $Li(x)$ and this approximation is essentially square root
accurate. More precisely,
$$
pi(x)=Li(x)+O(sqrtxlog(x)).
$$
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
answered Sep 1 at 14:31
Dietrich Burde
75.1k64185
75.1k64185
add a comment |Â
add a comment |Â
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2
My professor explained it by defining the $p-series$. Then he extends the $p-series$ to the complex plane, defining it as $zeta-$function. Next, he shows how Analytic Continuation works, gave some examples like $1+x+x^2+...$ and $frac11-x$ are the analytic continuation of each other. Next, he mixes them up and tells us the "Riemann Hypothesis", Zeros of Zeta function, etc. He next shows the relation between Euler's Prime product and Zeta function. I personally find some links that help me a lot. youtube.com/watch?v=NaL_Cb42WyY & youtube.com/watch?v=sD0NjbwqlYw
– Sujit Bhattacharyya
Sep 1 at 5:16
This question has been asked many times; have a look at the web, e.g. here.
– Dietrich Burde
Sep 1 at 11:35