A zero sum game

Mar 21, 2011
Different colours in this image mark regions of different phase of an angular lattice sum divided by the basic two-dimensional sum. The light blue areas must contain exactly the same numbers of zeros of each function, proving important properties of each.

(PhysOrg.com) -- New light has been shed on the 150-year-old math puzzle known as the Riemann hypothesis, say mathematical physicists at the University of Sydney.

The Riemann relates to the location of the zeros of a one-dimensional sum, which is known as the Riemann zeta function.

Along with suitable generalisations, it is considered by some to be the most prominent challenge in pure mathematics.

"The celebrated Riemann hypothesis is generally recognised as being the most outstanding unsolved problem in ," says Professor Ross McPhedran from the School of Physics.

"Until now the full version of the hypothesis has remained unsolved, although modern computer calculations have shown that the first 10 trillion zeros lie on the critical line."

In a paper to be published in the March edition of the Proceedings of the Royal Society A, Professor Ross McPhedran and colleagues from the University of Sydney and the University of Technology, Sydney considered two-dimensional sums.

"We have identified a previously uninvestigated class of two-dimensional sums which depend on angles in the plane as well as distances.

"We have shown that these angular sums all satisfy the Riemann hypothesis if and only if the basic two-dimensional sum satisfies it.

"We have also proved that the distributions of zeros of these sums are all the same."

It is hoped that these new insights will give mathematicians new tools which will enable them to finally prove the Riemann hypothesis, thereby deepening our understanding of the distribution of .

Explore further: Researchers help Boston Marathon organizers plan for 2014 race

More information: rspa.royalsocietypublishing.org/

Provided by University of Sydney

5 /5 (9 votes)

Related Stories

New Pattern Found in Prime Numbers

May 08, 2009

(PhysOrg.com) -- Prime numbers have intrigued curious thinkers for centuries. On one hand, prime numbers seem to be randomly distributed among the natural numbers with no other law than that of chance. But ...

Putting math problems in proper order

Nov 17, 2009

Mathematics is driven by the quest to solve problems and today the American Institute of Mathematics (AIM) announces a new tool to help attack those questions. Research problems can take decades or centuries to answer, with ...

The sum of digits of prime numbers is evenly distributed

May 12, 2010

(PhysOrg.com) -- On average, there are as many prime numbers for which the sum of decimal digits is even as prime numbers for which it is odd. This hypothesis, first made in 1968, has recently been proven by French researchers ...

Set theory mathematician Paul Cohen dies

Apr 02, 2007

Trailblazing mathematician Paul Cohen, who developed a technique known as forcing to test a hypothesis, has died in Stanford, Calif., at the age of 72.

Congress Commends UM-Led Math Team's Breakthrough E8 Calculation

Mar 30, 2007

A major mathematical breakthrough by a team of 18 scientists, led by University of Maryland mathematician Jeffrey Adams, has been commended by Congress, one week after the work made international headlines when it was announced by the American Instit ...

Recommended for you

Egypt archaeologists find ancient writer's tomb

7 hours ago

Egypt's minister of antiquities says a team of Spanish archaeologists has discovered two tombs in the southern part of the country, one of them belonging to a writer and containing a trove of artifacts including reed pens ...

Study finds law dramatically curbing need for speed

Apr 18, 2014

Almost seven years have passed since Ontario's street-racing legislation hit the books and, according to one Western researcher, it has succeeded in putting the brakes on the number of convictions and, more importantly, injuries ...

User comments : 0

More news stories

Egypt archaeologists find ancient writer's tomb

Egypt's minister of antiquities says a team of Spanish archaeologists has discovered two tombs in the southern part of the country, one of them belonging to a writer and containing a trove of artifacts including reed pens ...

NASA's space station Robonaut finally getting legs

Robonaut, the first out-of-this-world humanoid, is finally getting its space legs. For three years, Robonaut has had to manage from the waist up. This new pair of legs means the experimental robot—now stuck ...

Ex-Apple chief plans mobile phone for India

Former Apple chief executive John Sculley, whose marketing skills helped bring the personal computer to desktops worldwide, says he plans to launch a mobile phone in India to exploit its still largely untapped ...

Filipino tests negative for Middle East virus

A Filipino nurse who tested positive for the Middle East virus has been found free of infection in a subsequent examination after he returned home, Philippine health officials said Saturday.

Airbnb rental site raises $450 mn

Online lodging listings website Airbnb inked a $450 million funding deal with investors led by TPG, a source close to the matter said Friday.