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: Jerusalem stone may answer Jewish revolt questions

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

US state reaches deal to keep dinosaur mummy

59 minutes ago

North Dakota reached a $3 million deal to keep a rare fossil of a duckbilled dinosaur on display at the state's heritage center, where it will serve as a cornerstone for the facility's $51 million expansion, officials said ...

Jerusalem stone may answer Jewish revolt questions

3 hours ago

Israeli archaeologists said Tuesday they have discovered a large stone with Latin engravings that lends credence to the theory that the reason Jews revolted against Roman rule nearly 2,000 ago was because ...

Kung fu stegosaur

4 hours ago

Stegosaurs might be portrayed as lumbering plant eaters, but they were lethal fighters when necessary, according to paleontologists who have uncovered new evidence of a casualty of stegosaurian combat. The ...

User comments : 0