The Music of the Primes: Why an unsolved problem in mathematics matters. Marcus Sautoy du

every teenager doing calculations in the schoolroom. With the advent of pocket calculators, they lost their place as an essential tool in everyday life, but several hundred years ago every navigator, banker and merchant would have been exploiting these tables to turn difficult multiplication into simple addition. Included at the back of Gauss’s new book was a table of prime numbers. It was uncanny that primes and logarithms should appear together, because Gauss noticed after extensive calculations that there seemed to be a connection between these two seemingly unrelated topics.

      The first table of logarithms was conceived in 1614 in an age when sorcery and science were bedfellows. Their creator, the Scottish Baron John Napier, was regarded by local residents as a magician who dealt in the dark arts. He skulked around his castle dressed in black, a jet-black cock perched on his shoulder, muttering that his apocalyptic algebra foretold that the Last Judgement would fall between 1688 and 1700. But as well as applying his mathematical skills to the practice of the occult, he also uncovered the magic of the logarithm function.

      If you feed a number, say 100, into your calculator and then press the ‘log’ button, the calculator spits out a second number, the logarithm of 100. What your calculator has done is to solve a little puzzle: it has looked for the number x that makes the equation 10^x = 100 correct. In this case the calculator outputs the answer 2. If we input 1,000, a number ten times larger than 100, then the new answer output by your calculator is 3. The logarithm goes up by 1. Here is the essential character of the logarithm: it turns multiplication into addition. Each time we multiply the input by 10, we get the new output by adding 1 to the previous answer.

      It was a fairly major step for mathematicians to realise that they could talk about logarithms of numbers which weren’t whole-number powers of 10. For example, Gauss would have been able to look up in his tables the logarithm of 128 and find that raising 10 to the power 2.107 21 would get him pretty close to 128. These calculations are what Napier had collected together in the tables that he had produced in 1614.

      Tables of logarithms helped to accelerate the world of commerce and navigation that was blossoming in the seventeenth century. Because of the dialogue that logarithms created between multiplication and addition, the tables helped to convert a complicated problem of multiplying together two large numbers into the simpler task of adding their logarithms. To multiply together large numbers, merchants would add together the logarithms of the numbers and then use the log tables in reverse to find the result of the original multiplication. The increase in speed that the sailor or seller would gain via these tables might save the wrecking of a ship or the collapse of a deal.

      But it was the supplementary table of prime numbers at the back of his book of logarithms that fascinated the young Gauss. In contrast to the logarithms, these tables of primes were nothing more than a curiosity to those interested in the practical application of mathematics. (Tables of primes constructed in 1776 by Antonio Felkel were considered so useless that they ended up being used for cartridges in Austria’s war with Turkey!) The logarithms were very predictable; the primes were completely random. There seemed no way to predict when to expect the first prime after 1,000, for example.

      The important step Gauss took was to ask a different question. Rather than attempting to predict the precise location of the next prime, he tried instead to see whether he could at least predict how many primes there were in the first 100 numbers, the first 1,000 numbers, and so on. If one took any number N, was there a way to estimate how many primes one would expect to find amongst the numbers from 1 to N? For example, there are twenty-five prime numbers up to 100. So you have a one in four chance of getting a prime if you choose a number at random between 1 and 100. How does this proportion change if we look at the primes from 1 to 1000, or 1 to 1,000,000? Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist.

      If we look at the table overleaf of values of the number of primes up to various powers of ten, based on more modern calculations, this regularity becomes apparent.

      This table, which contains much more information than was available to Gauss, shows us more clearly the regularity that Gauss discovered. It is in the last column that the pattern manifests itself. This column represents the proportion of prime numbers amongst all the numbers being considered. For example, 1 in 4 numbers are prime counting up to 100, so in that interval you will need to count, on average, 4 to get from one prime to the next. Of the numbers up to 10 million, 1 in 15 are prime. (So, for example, there is a 1 in 15 chance that a seven-digit telephone number is a prime.) For N greater than 10,000, this last column seems to be just increasing by about 2.3 each time.

      So every time Gauss multiplied by 10, he had to add about 2.3 to the ratio of the primes to all the numbers. This link between multiplication and addition is precisely the relationship embodied in a logarithm. Gauss, with his book of logarithms, would have found this connection staring him in the face.

      The reason why the proportion of primes was increasing by 2.3 rather than 1 every time Gauss multiplied by 10 is because primes favour logarithms based not on powers of 10 but on powers of a different number. Pressing the ‘log’ button on your calculator when you input 100 produced the answer 2, which is the solution to the equation 10^x = 100. But there is nothing that says we have to have 10 as the number to raise to the power x. It is our obsession with our ten fingers which makes 10 so appealing. The choice of the number 10 is called the base of the logarithm. We can talk about the logarithm of a number to a base other than 10. For example, the logarithm of 128 to base 2 rather than base 10 requires us to solve a different puzzle, to find a number x so that 2^x = 128. If we had a ‘log to base 2’ button on our calculator, we could press it and get the answer 7, because we need to raise 2 to the power of 7 to get up to 128: 2⁷ = 128.

      What Gauss discovered is that primes can be counted using logarithms to the base of a special number, called e, which to twelve decimal places is 2.718 281 828459 … (like π, it has an infinite decimal expansion with no repeating patterns), e turns out to be as important in mathematics as the number π, and occurs all over the mathematical world. This is why logarithms to the base e are called ‘natural’ logarithms.

      The table that Gauss had made at the age of fifteen led him to the following guess. For the numbers from 1 to N roughly 1 out of every log(N) numbers will be prime (where log(N) denotes the logarithm of N to the base e). He could then estimate the number of primes from 1 to N as roughly N/log(N). Gauss was not claiming that this magically gave him an exact formula for the number of primes up to N – it just seemed to provide a very good ballpark estimate.

      It was a similar philosophy that he would later apply in his rediscovery of Ceres. His astronomical method made a good prediction for a small region of space to look at, given the data that had been recorded. Gauss had taken the same approach for the primes. Generations had become obsessed with trying to predict the precise location of the next prime, with producing formulas that would generate prime numbers. By not getting hung up on the minutiae of which number was prime or not, Gauss had hit on some sort of pattern. By stepping back and asking the broader question of how many primes there are up to a million rather than precisely which numbers are prime, a strong regularity seemed to emerge.

      Gauss had made an important psychological shift in looking at the primes. It was as if previous generations had listened to the music of the primes note by note, unable to hear the whole composition. By concentrating instead on counting how many primes there were as one counted higher, Gauss found a new way to hear the dominant theme.

      Following Gauss, it has become customary to denote the number of primes we find in the numbers from 1 to N by the symbol π(N) (this is just a name for this count and has nothing to do with the number π). It is perhaps unfortunate that Gauss used a symbol that makes one think of circles and the number 3.1415 … Think of this instead as a new button on your calculator. You input a number