Euler's Product for the Zeta function via Boxes II

1,911 views Jul 3, 2024

We continue exploring elementary connections between Box Arithmetic and Number theory, centered on Euler's product formula for the Zeta function. The Fundamental Identity of Arithmetic that we introduced in our last video plays a big role. But now we introduce some interesting variants: when we replace the numbers occurring in this Fundamental Identity with suitable powers of them, including also negative powers, and then apply the Sum operator as before. What makes this works is a crucial multiplicative property of taking powers. We make contact with another of Euler's famous formulas, for the sum of the reciprocal squares of natural numbers. In this talk, we necessarily extend our usual finite careful mathematics with unbounded analogs. But that means that we need to be mindful of the exact meaning of the statements we make. Our general orientation is that the meanings of expression can be reduced to finite expressions by using the natural ordering of the Natural numbers to apply truncation consistently. Video Contents Introduction Powers And The Caret Operation Applying f To The FIA A Variant ( replacing with cubes Another Variant (replacing with reciprocals Suggests An Identity Squared Reciprocals Examples A calculation *********************** My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at Patreon, / njwildberger Your support would be much appreciated. Here are the Insights into Mathematics Playlists: • MathHistory: A course in the History ... • WildTrig: Intro to Rational Trigonometry • Math Foundations • Wild Linear Algebra • Famous Math Problems • Box Arithmetic • Elementary Mathematics (K-6) Explained • Year9Maths • Ancient Mathematics • Wild West Banking • Sociology and Pure Mathematics • Sociology and Pure Physics • Old Babylonian Mathematics • Probability and Statistics: an introd... • Boole's Logic and Circuit Analysis • Universal Hyperbolic Geometry • Differential Geometry • Algebraic Topology • MathSeminars • Playing Go • Diffusion Symmetry: A bridge between ...Here are the Wild Egg Maths Playlists (some available only to Members!) • Algebraic Calculus One • Classical to Quantum (Members only)[ • Algebraic Calculus Two](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-) • Advice for mathematics exploration (M... • Solving Polynomial Equations (Members...[ • The Hexagrammum Mysticum (Members only)](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_) • Algebraic Calculus and dCB curves (Me... • Dynamics on Graphs (Members only) • Playlist • Triangle Geometry • Explorations with q-series (Members o... • Six: An elementary course in Pure Mat... • Maxel Inverses and Orthogonal Polynom...************************

0:08

[Music] Hello everyone. I'm Norman Wildberger. This is our second video on Euler product and zeta functions. We are looking at this from a box arithmetic point of view. So,

0:20

we've identified something that we might call the fundamental identity of arithmetic,

0:25

which is largely equivalent to the more usual fundamental theorem of arithmetic, and we express it here in terms of this series of boxes which are being carried together to form

0:37

this beautiful M set or box with just the natural numbers starting from one in it. So here we have

0:44

powers of the prime two, here we have powers of the prime three, powers of the prime five,

0:50

of the prime seven, and so on. So we form the carrot over all primes in this fashion,

0:55

and I remind you that to form the carrot of two boxes we take the box consisting of all possible

1:02

products of elements, one from here and one from here. Now, because these are unbounded objects,

1:11

we require a different kind of theory, so we need to go beyond what we've already set up.

1:17

I'm not going to do that today, so we are just going to think about this sort of informally. But generally, our orientation is that we're going to try ultimately to assign meaning to

1:26

something like this in a finite way. We want to always avoid the assumption that we can do

1:33

an infinite number of things because we can't. So we have to try to state what we're talking about

1:38

here, and we have to bring out the meaning of such an equation in a purely finite way. Okay, so today

1:47

what we're going to do is we're going to take this fundamental identity and we're going to augment it

1:53

in certain ways, moving towards the Euler product for the zeta function. But the zeta function is,

2:00

from my point of view, not really a proper mathematical object, and neither is the Euler

2:05

product. Nevertheless, we're going to be moving in that direction, and we'll see that we are touching base with a wide variety of interesting things. In fact, we'll be able to, even from

2:16

just this elementary point of view, any prior understanding of the zeta function or any of the

2:22

associated number theory, we're going to be able to see how the zeta function story somehow has a

2:30

connection with prime numbers and the general growth of prime numbers. So we'll be able to

2:37

access some of that in a very elementary way here. Okay, so just make sure that we understand how to

Powers And The Caret Operation

2:42

form the carrot of two boxes. Here's a box with elements a, b, another one with elements c, d. The

2:49

carrot of these two boxes is the box consisting of the elements a * c, a * d, a * e, b * c, b * d,

2:57

and b * e. So now in the world of boxes, you can always square any box just by multiplying the box

3:06

by itself, and this property pleasantly follows from the associative law that the square of a * b

3:17

is equal to the square of a * the square of b. So when we're talking about squares of elements,

3:25

there's, in the box world, various levels at which we can kind of apply an operation like that. So

3:32

instead of just squaring a box, we could also do something sort of one level down, by which I mean

3:39

square the elements of a box. Alright, so given the box a, we could look at the box consisting

3:45

of the squares a² as little a runs through the box capital A. So for example, here's a box with

3:55

two elements, the number three and alpha. If we apply this F to it, we square three to get nine,

4:03

we square alpha to get alpha squared. So that's what F does to this box. It's not really squaring

4:08

the box itself, it's squaring the elements in the box, and then we get an interesting connection

4:21

with the carrot operation because F of a carrot b is F of a carrot F of b. So for example, here if

4:31

we take this identity here and we consider forming this F to all the ingredients, so F of this thing

4:39

would just be the box a² b². F of this is the box c² d² e². So if we carrot those, we have to form

4:47

all possible products a² * c², a² * d², etc., but a² * c² by this rule here is the same as (a * c)².

5:00

So what we see is that as we run through all six possible combinations, what we're getting are just the squares of all the elements over here in the carrot of these two original boxes. Okay,

5:11

so if this is box a and this is box b, this is box a carrot b, then F of a carrot b is this,

5:18

and F of a is here and F of b is here, and we see the identity is valid. So now let's take the

5:24

fundamental identity of arithmetic and let's apply this F operation to both sides. So with this M set

Applying f To The FIA

5:36

consisting of the powers of two, we're going to square all of those elements. So there's 1², 2²,

5:41

2²², 2³², etc. Similarly, we're going to square all the elements in the M set consisting of powers

5:50

of three, and of powers of five, and of powers of seven, etc., and according to the identity that

5:56

we've just established, on the right-hand side what we should be getting is the M set consisting

6:03

of the squares of the natural numbers. And I hope you see this is not surprising at all. I mean if

6:09

you take some element here, for example, 4², and ask, well, how can you get 4² by taking products

6:17

of things like this? Well, you could do it here. You could take 2² * 1², and that's the only way

6:26

you can get 4², just in the same way that the only way of getting 4 is to take 2² * 1. Okay,

6:35

so now let's just rewrite this a little bit. So instead of having the squares of all of these things, let's put the squares right above the two. So, for example, this next one over here would

6:47

be 2³². Let's rewrite that as 2²³. Then what we have here is an M set consisting of powers of 2²,

7:01

and over here we'll have an M set consisting of powers of 3², and so on. On the right-hand side,

7:09

we still have the same thing over here, and now we can try to sort of connect with the

7:16

sums that are appearing in the zeta function and in the Euler product formula by applying what we

7:23

call the summation or the sum operator, capital S. I remind you what that does:

7:29

it takes an M set and just takes the sum of all the elements of that M set. And we saw

7:37

that S of a carrot b equals S of a times S of b. So this summation operation works well

7:48

on carrots of things. So let's apply S to both sides of this thing that we've just obtained,

7:54

this thing here. So S of this one here is just the sum of all the elements. That's the

8:00

series in quotes: 1 + 2² + 2²² + 2²³, etc. And S of this thing is the sum of all the powers of 3²,

8:12

so it's 1 + 3² + 3²², etc., and then a similar one for five and seven, etc., and we're going to

8:19

multiply all of these sums. That's what's being described here. Now this sum here

8:30

is like a geometric series. Okay, it starts with one and then the common ratio is just 2². At each

8:38

stage, we're just multiplying by 2². So if we liberally apply the standard formula for the

8:46

sum of an unbounded geometric series, it's 1 over 1 minus this common ratio 2². Okay, so this is all

8:59

in we're just trying to get an idea, get our heads around this and see if anything comes up that

9:05

might be interesting and worthwhile. So at least formally, this geometric series is represented

9:12

in some fashion or another by this term here. Similarly, there's another geometric series with

9:19

first term one and constant ratio 3², so its sum in some sense is 1/1 - 3². The next one will be

9:28

1/1 - 5², etc., and on the right-hand side is the sum of these guys, just the sum of the squares:

9:37

1 + 2² + 3² + 4², etc. So a literal application of this rule in this context seems to give us

9:48

something which doesn't have any immediate sense, okay, although it's possible that perhaps there's

9:57

some interpretation of these unbounded objects for which some kind of equality like this has

10:06

some kind of meaning. But certainly, currently, this doesn't look like it's very promising. I

10:12

mean these things here are, you know, one, these are negative numbers, and they're getting bigger

10:20

in the denominators. This whole thing is getting smaller and, you know, and this one is obviously

10:25

getting bigger, at least if we sort of proceed in terms of partial sums. So the validity and

10:31

the meaning in fact of something like this is very much in question, but nevertheless, this is in the

10:36

spirit in the direction of the Euler product for the zeta function. So let's keep going a

10:42

little bit further with it. What made that work is that the squaring operation has the property

A Variant ( replacing with cubes)

10:50

that mn² = m² * n². But that property, that sort of multiplicative property, is shared by lots of

10:58

other things. For example, cubing. It's also true that if you cube m * n, you get m³ * n³. So in

11:07

fact the same reasoning allows us to replace all the numbers in the fundamental identity with their

11:15

cubes. So here are powers of two and we're cubing all of them. Here are powers of three and we're

11:21

cubing all of them, etc., and we're always going over all primes. So the first one is twos and

11:26

threes, then fives and sevens, then elevens, etc., and on the right-hand side we just get the M set

11:32

consisting of the cubes of the natural numbers. And again the meaning is the same as before. If

11:39

we take any one element in here, okay, then it's obtainable in exactly one way as a product of

11:46

something in this first box times something in the next box times something in the next box, etc.,

11:52

etc., where most of those product constituents are ones and only a finite number of them are not.

12:00

Okay, so again we can perform the same little twist where we interchange the powers here,

12:06

so instead of having 2²³, we could write 2³². Then we see that the entries of this thing here

12:14

are just given by this geometric sequence. Each term is 2³ times the previous one,

12:22

and similarly here, a geometric sequence: each one is 3³ times the previous one. And then when we

12:29

apply our S operator, well, again we have a geometric series: first term one, constant

12:38

ratio the prime cubed. So in general it would be 1 over 1 minus P³, and we're taking the product P

12:45

over all primes. It's a quick way of saying what we get when we perform the sum of all of these

12:52

guys and multiply them all together. And over here, well, we're just getting the sum of all the

12:59

cubes of the natural numbers. And again, this is a highly questionable thing for exactly the same

13:06

reason as the previous one was. This seems to us to be unbounded and not even vaguely approaching

13:14

anything, and these things here are all giving us something completely different from what appears to be happening on the right-hand side. Nevertheless, it's still kind of interesting.

13:23

So here's another variant now, which is not, we're sort of stepping away from box arithmetic

Another Variant (replacing with reciprocals)

13:30

generally and just focusing on something that we can do with these boxes of natural numbers.

13:36

So in our world, where we accept rational numbers also, we can take reciprocals of natural numbers,

13:42

and those are fractions or rational numbers. So that's perhaps something else that we could do, is to take reciprocals of everything in the fundamental identity of arithmetic, and the reason

13:53

why that works is because taking reciprocals still obeys this fundamental multiplicative property

14:00

that the reciprocal of n * m is n inverse * m inverse. So here are the powers of two and we're

14:09

taking reciprocals of all of them. Here are powers of three, we're taking reciprocals of all of them, and carrot everything together, and on the right-hand side just the reciprocals of the

14:18

natural numbers. So now let's apply the S operator. Okay, maybe we could rewrite this

14:27

first of all as one, here's 1/2, here's 1 over 2², here's 1 over 2³. It's another way of writing it.

14:35

Here's again sort of a geometric series, but now the ratio is 1/3. The next one the ratio will be

14:42

1/5, and so on. And on the right-hand side, well, we just get the M set of the reciprocal natural

14:50

numbers. Okay, what happens when we apply S now? Well, now this is a geometric series of

14:57

sort of a different kind. It's something that's more familiar to us. We're more comfortable with

15:03

assigning a meaning to this unbounded ongoing sum. Now this is the kind of infinite sum where there

15:13

actually is a way of assigning a meaning to it, okay, and the reason is because the end result

15:21

is a number that we already have obtained. It's already a known rational number. So if you add up

15:30

this thing here, 1 plus a half plus a quarter plus an eighth, etc., the total is two, which according

15:37

to the formula is 1 over 1 minus a half. So that's a half in the bottom there, we get two. Okay,

15:42

now that's different from the usual thing that I complain about because now we have an infinite series which is summing up to a rational number. And the point is that we can make sense of that

15:53

statement without requiring us to do an infinite amount of work. We don't have to do an infinite

15:59

amount of work to get the thing on the right-hand side because it's already a known rational number,

16:05

okay, and then we can state the meaning of such a statement in terms of approximations. We can say,

16:11

well, if we take the first k terms of the left-hand side, then we're within such and

16:17

such of the end result, two. So that's in dire contrast with the usual story where we have some

16:26

sequence of rational numbers or some series which supposedly converges to an irrational number. The

16:34

problem there is that the irrational number has not been brought into being except by the actual

16:40

process of creating this infinite series. You have to run the infinite series to the end before the

16:47

irrational number, which is the limit, is even obtained before it comes into existence. Right,

16:53

there's a huge difference between these two scenarios, one of the reasons why we really

17:00

want to stick to rational number arithmetic if we want to do mathematics properly. This is such a profound understanding. Okay, and I know it's going to be hard for many of you to

17:11

accept because you've been taught from an early age something different, but this is the way the world really is. Okay, so we actually can have legitimate meanings to these various sums in each

17:25

one of these things. So here we're going to get 1 over 1 - 1/2, 1 over 1 - 1/3, then 1 over 1 - 1/5,

17:31

etc., on the right-hand side, and on the right-hand side, well, we get this series,

17:36

okay, which is more problematic because here there's not an object that's known to us which

17:42

is at the end of the rainbow. And of course, all students of mathematics will know that this is a

17:47

series which diverges in the usual sense. So let's have another look at this sort of strange identity

Suggests An Identity

17:53

that seems to be appearing here. So we have 1 over 1 - 1/2 * 1 over 1 - 1/3 * 1 over 1 - 1/5, etc.,

18:01

and over here the harmonic series. Well, this is two, and this is 1 over 2/3, that's three halves,

18:07

and this is 1 over 4/5, which is the same as 5/4. And then the next one will be 7/6,

18:13

the next prime is 11, so it'll be 11/10, and the next one is 13, so it'll be 13/12. Now all of

18:20

these quotients that we're getting are pretty close to one because they're a prime divided

18:26

by one less than the prime, and as the prime gets big, that quotient will start to approach

18:32

one from on top. So what we're doing is we're multiplying a series of things here together,

18:37

and we're always multiplying by something that's a little bit bigger than one. Okay,

18:43

but as we proceed, the separation from one is decreasing, so we're getting closer and closer to

18:52

multiplying by one, although we never do. So it's a natural question maybe to ask, well, you know, if you just started computing these things, these partial things, and just recorded it and looked at

19:02

the totals, what's happening to those totals? And at this stage here, we see that this is probably

19:09

telling us something. This is suggesting to us, because let's say we know that this diverges, it's

19:14

suggesting to us that this is going to diverge as well, right? It would be rather strange if

19:19

this thing here converged to something like 13.5 and this diverged. So that's telling us something

19:28

about the prime numbers already. Actually, it's telling us something about how sparse they are

19:33

or how not sparse they are. The fact that this product presumably diverges is giving us some

19:42

kind of information about the large scale growth of prime numbers, so in a large scale kind of

19:49

way. So let's do one more variant which moves us quite close to the Euler product story, but also

Squared Reciprocals

19:59

brings suddenly into being this idea that actually we're doing something non-trivial here,

20:05

that we're actually touching base with some subtle and actually rather delicate mathematics just

20:11

from this sort of elementary arithmetic with these boxes and carrots and forming sums. So

20:19

the variant is that we're going to replace all numbers in the fundamental identity with their squared reciprocals, not their squares, not their reciprocals, but the squares of their reciprocals,

20:30

or the reciprocals of their squares, same thing. So looking at the first box with powers of two,

20:37

we're taking all those powers of two and raising them to the minus 2 power,

20:42

and same thing with the box of powers of three, etc. And on the right-hand side, well, we're getting the box consisting of all the square reciprocals of the natural numbers. Now this first

20:54

box here, we can write it as one, this one here is 1 over 2², this one is 1 over 2²², this one is 1 over 2²³. We do the same thing as before, always interchange the order of the powers. Okay,

21:07

that makes this also into a geometric series. Same thing with this one, and also with the next one,

21:13

etc. Here the ratio is 1 over 2². Here the ratio is 1 over 3². The next one, the ratio 1 over 5²,

21:20

etc. And on the right-hand side, the M set of reciprocals of squares of natural numbers.

21:29

Now let's see what happens when we apply S, and suddenly something really interesting comes into

21:36

focus. So here I've copied what we've just obtained: powers of two reciprocal squares,

Examples

21:44

powers of three reciprocal squares, powers of five reciprocal squares, etc., carrotting them all together. And here is the M set of the reciprocal squares of the natural numbers. Okay,

21:54

each of these inside is a geometric series. When we apply our summation operator, so we take the

22:01

sum of all these guys: first term one, ratio 1 over 2². So we get 1/1 - 1 over 2². The next one,

22:11

first term one, ratio 1 over 3². So we get 1 over 1 - 1 over 3² * 1 over 1 - 1/5², etc. And on the

22:23

right-hand side, the sum of the reciprocals of the squares of the natural numbers. So

22:32

what does this look like in practice? Well, this is 3/4. Over here, when we take the reciprocal,

22:37

we get 4/3. Here we have 1 - 1/9, so that's 8/9. Reciprocal 9/8. Over here, 5² is 25 - 1 is 24,

22:51

so we get 25/24. Next one is 7, we get 49/48, and so on. So now we have the square

22:58

of the primes in the numerators and one less than each of those in the denominator. Now the

23:07

right-hand side, however, is very different from the previous one because there's a square there,

23:12

and Euler famously realized that this thing sums not to infinity, doesn't go on and on. It actually

23:21

sums to something that he was able to identify in terms of pi, famously pi²/6. So that's certainly

23:29

one of Euler's great remarkable formulas which made a big impression on everyone. It is still

23:37

a remarkable kind of thing, even though strictly speaking, you know, what is pi? Well, we don't

23:43

really know. And actually, this sum strictly speaking also, it doesn't really work, but there's something going on here. Okay, definitely there's something going on here, something like magic. But

23:51

now what we have is from this box arithmetic point of view, some kind of identity like this, that for

23:58

assuming Euler's result, and we suppose we have some approximate meaning to this, it means that the product of these squared primes divided by the one less than them, that that is very likely

24:10

heading towards something actually reasonably concrete. It's like a region on the number line,

24:17

maybe not exactly a point, but some kind of object on the number line. So this is now interesting and

24:23

it's decidedly different from the previous one that we had where we were just looking at product of primes over primes minus one. Great, so now we actually can make a calculation. Okay,

A calculation

24:36

let's choose the first primes up to 100. So 2², 3², 5², 7², and all the numbers less than those,

24:46

11², 13², 17², all the way up to 79², 83², 89², 97². The last prime is 100. Okay, so that's

24:58

an initial sequence of the left-hand side of what we were obtaining. And this thing here,

25:05

okay, it's a rational number and we can approximate what it is numerically. My

25:11

computer tells me it's approximately 1.64945. And what is pi²/6 approximately? Okay,

25:20

in quotes because pi really is not a properly defined number, but approximately it's 1.644934.

25:29

So the claim of that formula that we had is that if we keep on doing this by multiplying by further

25:36

and further primes, we should be heading towards this number. So this number here is getting bigger

25:42

a little bit at a time, and I want you to think about this in terms of like perhaps an interest

25:48

payment. Okay, let's say we're at this stage here, we've computed this thing here. What does it mean to multiply by 121/120? Well, this is, you know, a little bit more than 1% more than one. So it's

26:01

something like 1.01. So this is like how much you owe if you borrowed this amount of money for one

26:09

year at a roughly 1% interest rate. Okay, just a little ratio here is giving us 1% interest.

26:17

So the amount that you owe after a year is a little bit more, roughly about 1% more. Okay,

26:23

now the next year, well, we're talking about 13²/13² - 1, that's a little bit closer to one,

26:30

so the interest rate has gone down a little bit, maybe it's 8% or something like that, or 0.9%. Okay, so your principal that you're going to owe after that year is still going to be bigger, but

26:40

it's going to be bigger by geometrically less than it was the previous year. And that's carrying on,

26:48

so the interest rate every year is decreasing. It would be nice if banks did this, but anyway,

26:55

the interest rates are decreasing and heading towards zero. But what we're saying is that the

27:02

total amount that you're going to owe as you keep on going is going to head towards this thing here.

27:11

Now on the other hand, on the right-hand side of the equation, we had the sum of the reciprocal

27:17

squares, and we can calculate those as well. That's just a very explicit calculation. So I

27:22

summed k = 1 to 300. My computer told me that it's about 1.641606, which is pretty close to

27:30

this thing here, right? So you see there's two different ways of calculating something,

27:37

right? This calculation here is in some sense calculating the sum of the squared reciprocals

27:46

of the natural numbers from k = 1 to 300, or reciprocally, this thing here is calculating

27:52

this product over primes of these things here. Now, which of those is more work? I don't know.

28:00

But you see that there are two quite different things here that are being compared, and we're

28:07

actually touching base with some real mathematics here, having to do with partial sums of this thing

28:13

and having to do with partial products of this thing, and some kind of closeness between these two things. Now at this stage here, we can also perhaps appreciate that this kind of information

28:29

is telling us something about the growth of primes. So just let's suppose that we sort of stop

28:35

at this level here and pause for a second and ask, okay, what's going to happen from here on? Okay,

28:40

so if the primes from here on get really far apart and they get big fast, okay, if the primes

28:48

are big, then we're going to be multiplying by numbers which are really close to one,

28:54

so the interest rate will be very close to zero. Then it's possible for us to make a computation

28:59

that shows that we're not going to be able to get from here to here by continued multiplications by

29:08

these further ratios if the primes are going too fast. Okay, so the fact that as we keep on going,

29:17

we're going from here to here, is telling us something about the growth rate of primes. If

29:23

the primes are growing too fast, we're not going to be able to get to here. On the other hand, if the primes are not growing fast enough, you know, if there's a lot of primes, say, between

29:32

100 and 200, then we'll be multiplying too often by interest rates which are still reasonably big,

29:39

and that's going to bump us over going from here to here. For example, if you have an interest rate

29:44

of 1% and you have that for 20 years, and compound interest, you know, you're going to probably pay

29:49

something like 22 or 23 cents on your original, so multiplying one by 1.23 will definitely bump this

30:00

over that. Right, so that's the kind of reasoning that allows you to see that this scenario is going

30:09

to allow you to extract some information about prime numbers. Now this is all very primitive,

30:15

of course, but you think about the Riemann zeta function and the associated number theory as encoded by the Riemann hypothesis as being some much more sophisticated analysis somehow in the

30:27

same direction. Okay, that the form of the zeta function and these sums tell you something about

30:36

primes. And what I'm trying to convince you today is that you can see some of this at least in an

30:41

elementary fashion without any complex numbers, without any supposition that there is such a

30:48

thing as the Riemann zeta function. You can see this simply as a consequence of formulas

30:54

that are obtainable from the fundamental identity of arithmetic by suitably applying multiplicative

31:04

functions to this identity in box arithmetic. I'm Norman Wildberger, thanks for listening. [Music]

31:22

Let me know if you need any further assistance!

English

AllFrom the seriesFrom Insights into MathematicsRelatedFor youRecently uploadedWatched