Perturbed cubic Gauss sums

Demonstrates the approximation (red lines) to the perturbed cubic Gauss sum (blue wiggles) occuring in the Bombieri-Iwaniec method (1986) for exponential sums for various parameter values.
This is typical of many of my programs: Check a (complicated to prove but quicker to use) approximation to something by superimposing two versions of a graph (or printing two sets of numbers) - one calculated directly, the other calculated using the sophisticated approximation.
The approximation uses "Poisson summation with a rational quadratic". Great - now what you do is a spot of "gardening" (tidying things up with some partial summation lemmata) to simplify the summation range (which depends on parameters in an intractable way), remove a smooth weight function, and approximate the phase. Then you evaluate the resulting quadratic Gauss sums and average (in mean 4th or maybe 5th power) the resulting bounds for a bunch of these sums using a few Holder inequalities and a kind of multidimensional large sieve (jargon descended from "sieve of Eratosthenes"). This results in two "spacing" problems (one looking a bit like Vinogradov's MVT, the other being a tricky problem to do with pairs of rational numbers - attacked by messing around with the modular group). When all that's done, you minimise the final resulting bound (for an exponential sum) by choosing a couple of free parameters. Bombieri and Iwaniec showed that mu(1/2) is no larger than 9/56 by using this method (which Huxley said only they could've thought of). Huxley and Watt (and others) have polished up the method, leading to the 9/56 being replaced by a series of exponents - the latest being 32/205. Exponent 1/6 was known almost a century ago. The Lindelof Hypothesis (which is implied by the Riemann Hypothesis) says the true answer is that exponent ZERO is allowable.
All that is just for bounding a "simple" 1D exponential sum. In 1988 Bombieri, Iwaniec and Mozzochi adapted the method for a 2D sum - leading (in particular) to exponent 7/22 in the Gauss circle and Dirichlet divisor problems (exponent 1/3 were obtained for both more than a century ago- Sierpinski and Voronoi). With a little help from Heath-Brown, Huxley adapted these ideas to obtain the corresponding result in the problem of the mean square modulus of the zeta function on the critical line. His latest exponent (with a bit more help from Watt for the E(T) problem) is 131/416 for all three of these problems. We know the best possible exponent would be 1/4 - and this is conjectured to be allowable. In many ways this optimal result would correspond to a bound of 1/8 for mu(1/2) - which doesn't even give the Lindelof Hypothesis. And yet Ivic tells me that he thinks exponent 1/4 in the divisor problem may be harder than the Riemann Hypothesis. Confused yet? Yeah - me too!
9 years ago, my official mission was to use these ideas to estimate the "mean square of the 2D exponential sum for the lattice point problem". I barely had (or still have) a clue where to begin and all the while was being told it may be impossible anyway.
The soundtrack is "07612543" (the numbers giving you the guitar frets for the chorus) by best mate Dougal. I do backing vocals on the chorus as well as the quacking noises.

Category:

Education

Tags:

• exponential sum
• Bombieri
• Iwaniec
• method
• zeta
• function
• Riemann
• Lindelof
• Hypothesis
• Huxley
• Watt
• Gauss sums
• large sieve
• Poisson summation
• Mozzochi
• Gauss circle
• Dirichlet divisor
• problem
• 07612543
• Contraband
• Dougal
• Chris Jones
• solo effort
• Tim Jameson
• tpjameson
• ortega24024
• Lancaster
• Cardiff
• university
• Freebeats
• The band that time forgot
• Beef Sharkey
• Borland
• C++
• bgi
• BC4.52
• graphical
• interface
• approximation

[ortega24024]

$\int_0^T|\zeta(1/2+it)|^4dt=T­P_4(\log T)+O(T^{2/3}\log^8T)$ (Yoichi Motohashi).

Just don't any of you 6 billion MAGGOTS even DREAM of sucking in a single stolen

BREATH in the same multiverse as me until you've got your tiny little minds around

that one (and preferably reduced the exponent to $1/2$ as well)!!!!

[ortega24024]

My fucking God...these TWATS

actually think Hitler was the bad guy...

And don't think I'm going to thank you for "tidying" up my "mess" either...when

it was you (or more to the point the SCUM you let walk all over you) that "messed" it up in the 1st Maryfucking place!!!!

[ortega24024]

ou're not really one of these fucking fucking goddamn shitface braindead schizing,

TV/muzac-guzzling, tax-paying (genocide funding), psycho-babbling, clothes-wearing, ultra-hypocritical, soulless junkie hedonist murdering cowardly 65 million English bogus-shopkeeping SCUMBAGS really, are you by any chance?!

[ortega24024]

@SaulHirschberger Danke schoene, Herr Doktor Professor Hirschberger (PhD pending)...

....but who the HELL are you to presume to know or criticise the sound methods of

elite mathematicians, humanists (and their humble imitators such as me)?!!!

[bygota]

hey, i liked this soundtrack!!!!

[ortega24024]

@bygota Then if you e-mail me at tpjameson@gmail.com then maybe I can e-mail you the whole album that it's from. If you refer to the earlier comments you'll see me engaging in a discussion with 'eedahl': It was the music that sucked him in to... but I ended up visiting him in Oslo to give him some maths tuition. The song is '07612543' (the fret numbers on the guitar for the chorus). It is by my best mate Dougal - and I do backing vox and quacking noises on it.

[eedahl]

Sent you a PM :-) Yeah, I had no interest in mathematics before about a year ago. I randomly learned about complex numbers, and something just clicked: I wanted to know more stuff. So I lost my math angst, learnt the entire high school curriculum and got some formal papers so I could start studying math at uni, which is a real treat for me. I know only a little abstract algebra, but the fact that I know anything at all after a year is nice to me. Goes to show how random that kind of stuff can be

[eedahl]

BTW, I showed the song to my GF and she liked it as well ^^

[SaulHirschberger]

...until he gets sick of you (and all the other 6 billion cunts) again so gives you

(and everycunt else around him) yet another lovely gr8 big whopping concussion...ALWAYS GOOD TO BASH OUT THE COBWEBS - try "The brain that changes itself" by Norman Doidge for example).

[SaulHirschberger]

but there's GOT TO BE A BETTER WAY (e.g. go into Barclays (sort code 20-18-15 remember) and pulverise the sodding lot of them...hence earn a nice cig-detox

and some PEACE in jail with your good mate Paul di'Anno - then make some damn fine honest songs about it all...and marry the least bitchy chick that sucks you cos of it all...after doing a support-slot for the aforementioned BEAST..

[SaulHirschberger]

...Yes I KNOW you're making some damn good videos out of it all (WHEN YOU

CAN BE ARSED ENOUGH TO DO THE EDITING...getting the HELL out of your

SODDING parent's place might be a good start)..which WILL be very good

at cashing in from the SODDING sicko rock'n'roll industry...and that IS

a good realistic career plan (which has sort of been working for you for

15 years)...

[SaulHirschberger]

...BUT THEN EACH TIME THOSE cowardly murdering scheming fuckers THE Risk Management A.

(just GET IN THE RING muthafuckas) phone you...YOU JUST BATTER THE FUCK OUT

OF YOUR BEDROOM WALLS...AND IT'S BACK TO SQUARE 1 AGAIN IN A HEAP OF

100,000 even-more-messedup goddamn maths notes (and none of them

dizzy chicks in the pubs will screw you back to peace again when you're that wound up...TOO SCARY)...

[SaulHirschberger]

PS - I keep neatly sorting out some of your crummy (yet strangely exquisite) ash-covered, kitten-poo-covered notes (ready for scanning - that scanner you Wombled away from the Scarecrow Festival in Wray is fast enough for the job...the 1 you built for your GCSE Technology project is NOT...so DON'T bother going off on another tangent rebuilding it...YET)...

[SaulHirschberger]

...And when the HEAVEN are you going to digest Chapters2 and 3

properly and IN 3 DAYS FLAT rewrite the rulebook on

the RATIONAL POINTS CLOSE TO A CURVE (for the Serbian Journal

...LOL you've got your solution to Donalds's Gamma question

in the CROATIAN one now!...F knows what Martin's making of all this...the guy's been waiting for SEVEN years.)