Dtjytyk

Was credit for the black hole image misappropriated?

Does Parliament need to approve the new Brexit delay to 31 October 2019?

What is the padding with red substance inside of steak packaging?

One-dimensional Japanese puzzle

Mortgage adviser recommends a longer term than necessary combined with overpayments

How do I design a circuit to convert a 100 mV and 50 Hz sine wave to a square wave?

Are there continuous functions who are the same in an interval but differ in at least one other point?

What happens to a Warlock's expended Spell Slots when they gain a Level?

Single author papers against my advisor's will?

Accepted by European university, rejected by all American ones I applied to? Possible reasons?

Deal with toxic manager when you can't quit

Can the DM override racial traits?

1960s short story making fun of James Bond-style spy fiction

should truth entail possible truth

For what reasons would an animal species NOT cross a *horizontal* land bridge?

Am I ethically obligated to go into work on an off day if the reason is sudden?

Do working physicists consider Newtonian mechanics to be "falsified"?

Example of compact Riemannian manifold with only one geodesic.

Word to describe a time interval

How to read αἱμύλιος or when to aspirate

Keeping a retro style to sci-fi spaceships?

Huge performance difference of the command find with and without using %M option to show permissions

Why did Peik Lin say, "I'm not an animal"?

Would an alien lifeform be able to achieve space travel if lacking in vision?

Bound on log integral

0

$begingroup$

I am looking for an explanation of the bound
$$frac{1}{2pi}left(-frac{T log T}{1+(t-T)^2} - 2 int_T^infty frac{x log x (t-x)}{(1+(t-x)^2)^2} dx right)ll left( frac{1}{t+1} + frac{1}{T-t+1}right) log T,$$

where $t in [0,T].$

This is not for homework. Instead, this is a question from my senior thesis reading, so full answers are appreciated. I can bound the first term by $frac{log T}{T - t +1}$, but I am having trouble bounding the second.

real-analysis integration analytic-number-theory

share|cite|improve this question

edited Mar 23 at 2:06

William Chang

asked Mar 22 at 12:35

William ChangWilliam Chang

1,81111227

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  just bumping this.
  $endgroup$
  – William Chang
  Mar 23 at 2:05
  

  
  
  
  

add a comment | 

0

$begingroup$

I am looking for an explanation of the bound
$$frac{1}{2pi}left(-frac{T log T}{1+(t-T)^2} - 2 int_T^infty frac{x log x (t-x)}{(1+(t-x)^2)^2} dx right)ll left( frac{1}{t+1} + frac{1}{T-t+1}right) log T,$$

where $t in [0,T].$

This is not for homework. Instead, this is a question from my senior thesis reading, so full answers are appreciated. I can bound the first term by $frac{log T}{T - t +1}$, but I am having trouble bounding the second.

real-analysis integration analytic-number-theory

share|cite|improve this question

edited Mar 23 at 2:06

William Chang

asked Mar 22 at 12:35

William ChangWilliam Chang

1,81111227

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  just bumping this.
  $endgroup$
  – William Chang
  Mar 23 at 2:05
  

  
  
  
  

add a comment | 

$begingroup$

I am looking for an explanation of the bound
$$frac{1}{2pi}left(-frac{T log T}{1+(t-T)^2} - 2 int_T^infty frac{x log x (t-x)}{(1+(t-x)^2)^2} dx right)ll left( frac{1}{t+1} + frac{1}{T-t+1}right) log T,$$

where $t in [0,T].$

This is not for homework. Instead, this is a question from my senior thesis reading, so full answers are appreciated. I can bound the first term by $frac{log T}{T - t +1}$, but I am having trouble bounding the second.

real-analysis integration analytic-number-theory

share|cite|improve this question

edited Mar 23 at 2:06

William Chang

asked Mar 22 at 12:35

William ChangWilliam Chang

1,81111227

$endgroup$

share|cite|improve this question

edited Mar 23 at 2:06

William Chang

asked Mar 22 at 12:35

William ChangWilliam Chang

1,81111227

share|cite|improve this question

edited Mar 23 at 2:06

William Chang

asked Mar 22 at 12:35

William ChangWilliam Chang

1,81111227

asked Mar 22 at 12:35

William ChangWilliam Chang

1,81111227

asked Mar 22 at 12:35

William ChangWilliam Chang

1,81111227