PRIME NUMBER PATTERNS

After all the stir caused by Wolframs rule 110, here's another puzzle about our universe whose understanding might help us get things a little clearer: Riemann's Zeta function.

The Riemann hypothesis, first tossed off by Bernhard Riemann in 1859 in a paper about the distribution of prime numbers, is still widely considered to be one of the greatest unsolved problems in mathematics, sure to wreath its conqueror with glory — and, incidentally, lots of cash. Two years ago, to celebrate the millennium, the Clay Mathematics Institute announced an award of a million dollars for a proof (or refutation) of the hypothesis.

Whether in pursuit of glory, cash ("prizes attract cranks," one mathematician sniffed) or pure mental satisfaction, more than a hundred of the world's leading mathematicians came to New York City recently to attend an unusual conference at New York University's Courant Institute. While most math conferences are devoted to presenting completed work, this one was held for mathematicians to swap hunches, warn of dead ends and get new ideas that could ultimately lead to a solution......
As in all sports, it helps to know the rules of the game. Riemann made his hypothesis in the course of a 10-page paper he wrote on the distribution of prime numbers that is considered to be one of the most important papers in the history of number theory, a history that stretches back more than 2,500 years..............Despite the random occurrence of individual primes, the primes themselves were found to follow a remarkably simple distribution. In 1792, when he was 15, Karl Friedrich Gauss decided to examine the number of primes less than a given number. He discovered that the primes became, on average, sparser the further out he looked and that this dwindling obeyed a simple, logarithmic law. He had no idea why this was so, but it was intriguing.

In 1859, Riemann, who had been a student of Gauss, took up the question of the distribution of primes in his only paper on number theory. With that paper he revolutionized the field, as he had the fields of geometry (his math became the basis for Einstein's theory of gravitation) and several other branches of mathematics. What Riemann discovered was a way of using the properties of a relatively simple function to count the primes.

While the real numbers can be thought of as points on an infinite line, the complex numbers are points on a plane. One axis of this complex plane corresponds to the real numbers, and the other corresponds to the "imaginary" numbers — which were introduced so that negative numbers could have square roots, and are no more imaginary than real numbers. A function like Riemann's zeta function is simply a rule that takes a point on this plane and sends it to some other point.

Riemann showed that if he knew where the value of his zeta function went to zero he would be able to predict the distribution of the primes. He was able to prove that aside from some "trivial" zeros — located at -2, -4, -6, and so on and thus easily included in his equations — the zeros of the zeta function all lay within a strip one unit wide running along the imaginary axis.

Somehow the distribution of these zeros mirrored or encoded the distribution of the prime numbers. Riemann guessed that all of the zeros ran along the middle of the critical strip like the dotted line on a highway. Nobody is sure why he made this guess, but it has proven to be inspired. Over the past few decades billions of zeros of the zeta function have been calculated by computer, and every one of them obeys Riemann's hypothesis.

Source: New York Times
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The problem is to know whether this zeta function for primes has any significance, and if so, what is it?

PENSIONS AND DEFLATION RISK

The latest acknowledgement from the UK Office of National Statistics that private pension assets have been overvalued by a factor of a one third, only serves to highlight once again the precarious situation of pension prospects (and hence economic ones too , since pensioners are going to big a big factor in future consumption) in many G23 countries. The effect of the stock market collapses on pension plan values is well know, but the comments of the UK Chief Statistician on the impact of protracted low interest rates should sound alarm bells in the heads of those thinking about the problems we may face if we hit a serious wave of general deflation.

A day after apologizing for drastically overestimating the amount of money in pension savings accounts, Britain's government ordered an unusual review Tuesday of the data used to value the funds. After initially valuing private pension savings at $132 billion, the government said statistical errors may have led it to overestimate the amount in pension savings account by $46 billion. The disclosure battered public confidence in Britain's pension funds at a time of rising anxiety about whether the pensions are adequate to meet the demands of the country's aging population.

Two high-profile companies also reported pension woes. Cash-strapped insurer Equitable was struggling to limit losses in its employee pension fund, while British Telecommunications denied allegations of a large hole in its own portfolio.

On Tuesday, National Statistician Len Cook announced a review of data, which the Department for Work and Pensions had called "urgent." The department had used official statistics in making the pensions estimate that it later called into question.Cook's Office of National Statistics, which supplied the department with the data, said the numbers could mislead if not interpreted correctly. In fact, Cook's office had already detected errors in its pensions data in January. It recalled the data, recalculated them, then issued a revised set of figures in May.

Demographic trends have increased the pressure on firms that manage their own employee pensions. Low interest rates combined with the increased longevity of Britain's working population added 45 percent to the costs of administering private pensions over the past 15 years, said KPMG partner David Fairs. With interest rates at their lowest in almost 40 years, companies must contribute more money to their pension funds to earn the same financial return for their employees. At the same time, an aging population strains the ability of pension funds to provide for all beneficiaries. Plunging stock markets have contributed to the problem. Because many firms have invested the bulk of their employee pension funds in shares, the collapse in stock values over the past year has eroded the value of those assets.
Source: Yahoo News
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EU REFORM STUCK IN A RUT

The sixth months of the Danish presidency were to have been a time of imaginative and far reaching changes, but if the recent reaction to one of the proposals for strengthening the EU's international role are anything to go by, then it's going to be all uphill work:

Proposals by Tony Blair and Jacques Chirac for a powerful new European Union council president have been dealt a blow by Denmark, which on Monday took over the rotating EU presidency.The British prime minister and French president are leading calls to create the new high-profile post to give the EU more political direction and to represent Europe on the world stage. But Denmark has added its voice to claims that the new position would be a means of strengthening the power of the bigger countries in the EU to the detriment of the smaller ones.
Anders Fogh Rasmussen, the Danish prime minister, said it was inevitable that the new council president would come from one of the larger EU members. "I'm not in favour of a president of the council, serving a term of five years, because such a proposal tends to favour big countries," he told the Financial Times."Equal treatment for big and small countries is essential for us."His views are significant, because Denmark has been charged with developing plans to reform the EU council before December's Copenhagen summit.


The final shape of the EU is being considered by Valery Giscard d'Estaing's convention on the future of Europe. The former French president, who meets Mr Fogh Rasmussen on Tuesday in Copenhagen, is understood to be an enthusiast for the idea of an EU council president.
Source: Financial Times
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So it seems likely that the process of achieving a consensus for change is going to be a long one, with plenty of obstacles along the way, and, from the coments above this is a process which is unlikely to bear fruit during the Danish presidency. Which brings us back to the problem of the timing of the expansion agenda. Anyone willing to place any bets?

ITALY'S DEGENERATE POLITICS FURTHER DEGENERATES

A top Italian politician called on Saturday for the castration of rapists in the wake of a series of rapes in northern Italy.

"It is an intolerable situation. To avoid these shameful crimes there is only one solution -- the physical castration of these delinquents," Roberto Calderoli, the vice president of Italy's Senate (upper house), said in a statement. "Once upon a time one spoke of chemical castration, but personally I tend more toward simpler methods: scissors, and ones that are not necessarily sterilized," Calderoli, a member of the rightist Northern League, added.
Source: Yahoo News LINK

No comment, but unfortunately today there's more news in a similar vein:

Italy's Interior Minister offered to resign on Sunday as a row flared over comments he reportedly made about a murdered government aide. But Prime Minister Silvio Berlusconi said he had not accepted Interior Minister Claudio Scajola's resignation.

Scajola tendered his resignation after two respected newspapers reported comments he had made about the assassinated aide in which he called the murdered man "a pain in the ass who wanted to have his contract renewed.............. Biagi's assassination became front-page news again last week when letters written by the economist to a number of government officials, were published, highlighting his security concerns over a decision to withdraw his police escort. The Interior Ministry has launched an internal investigation into the decision to withdraw Biagi's bodyguards, but Scajola has defended the decision saying not everyone could be escorted.
"If (Biagi) had been protected there would have been three dead," Il Corriere quoted Scajola as saying.
Source: Yahoo News LINK

The continuing question mark over the judgemental capacity of Italy's political class which is once again re-inforced by the above must also raise the serious doubts about this country's capacity to rise to the challenges which are undountedly waiting just around the corner.