028 029 The Riemann-Zeta Function Agastya Kumar Application in Number Theory Here is the beginning of a list of integers beginning at one. I will use this string of numbers to firstly explain what led mathematicians to attempt solving the mystery of the primes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17… Some amongst you may have noticed that the highlighted numbers are prime. Now, can any of you immediately spot a pattern of how these prime numbers are distributed? If so, you are a genius or probably wrong. This then begs the question. Is there a function that represents this distribution? This is the question that the mathematicians began to ask in the 18th century. The search for the profound answer to this intimidating problem was initiated by young (at the time) mathematician, Carl Friedrich Gauss. His student, Bernhard Riemann, inherited the problem and later developed a hypothesis partially using mathematics established by Leonhard Euler. The Zeta-Function To begin with, I will take you through the mathematical lore of Euler’s contribution to the Riemann Hypothesis and to Prime Numbers. Infinite series – what is it Convergent and Divergent Series What Euler did Zeta Function Euler’s primary influence on the Riemann Hypothesis was the Zeta Function. Euler rose to fame when he was ‘playing around’ with infinite series. You may be asking what an infinite series is. An infinite series is the sum of infinitely many numbers that are related in a given way and listed in a given order. Here is a famous infinite series, namely the Basel problem. So, this series takes the sum of infinitely many numbers. However, can you think of how these numbers are related? If we can spot a simple pattern, we are able to see that the denominators are all square numbers. Therefore, we can simplify the notation to this. It may look complicated. However: The ‘Σ’ just represents the operation of ‘the sum of’. The ‘n=1’ means that we will begin inputting numbers into the function, which is the sum of the reciprocals of all square numbers, at one. Finally, the ∞ means that the series is infinite and will continue to sum infinitely many numbers. At this point we must ask what the question mark represents. What value does the series ‘converge’ to, or does it even have such a limit? This brings us to convergent and divergent series. A convergent series has a limit that the series ‘tends to’. Euler famously proved that the Basel series converges onto π2/6. If a series does not tend towards a limit, it is a divergent series. For example, if we sum all ordinary integers infinitely, we cannot place a limit on that series. Let’s now return to the infinite series we saw before. This time it has a slightly different notation. We have converted the notation for a particular infinite series to a general function. This function is the zeta function. Now, Euler had lots of fun with this function. He was riding high off his achievements and kept finding the limits of all these infinite series. He eventually proved that the Zeta Function will converge for all integer values of s that are greater than 1. However, one of the most significant discoveries he made, if not the most remarkable, was him expressing the Zeta Function as the infinite product of infinite series. That might be a mouthful, so let’s visualise this. If we let each infinite series be the summation of all ‘one over the powers of a prime number raised to s’ and then multiply that series by the same series for the next prime The Riemann-Zeta Function Application in Number Theory Here is the beginning of a list of integers beginning at one. I will use this string of numbers to firstly explain what led mathematicians to attempt solving the mystery of the primes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17… Some amongst you may have noticed that the highlighted numbers are prime. Now, can any of you immediately spot a pattern of how these prime numbers are distributed? If so, you are a genius or probably wrong. This t en begs th question. Is there a function that represents this distribution? This i he question t at the mathematician began to ask in the 18th century. The search f r the profound answ r to this intimid ting problem was initiated by young (at the time) athematician, Carl Friedrich G uss. His student, Bernhard Riemann, inherited the problem and later developed a hypothesis partially using mathematics established by Leonhard Euler. The Zeta-Function To begin with, I will take you through the mathematical lore of Euler’s contribution to the Riemann Hypothesis and to Prime Numbers. • Infinite series – what is it • Convergent and Divergent Series • What Euler did • Zeta Function Euler’s primary influence on the Riemann Hypothesis was the zeta function. Euler rose to fame when he was ‘playing around’ with infinite series. You may be asking w at an infinite series is. An infinite series is • the sum of infinitely many numbers that are • related in a given way and listed in a given order. Here is a famous infinite series, namely the Basel problem. 1 + 1/ 4 + 1/ 9 + 1/ 16 +1/ 25 + … = ? So, this series takes the sum of infinitely many numbers. However, can you think of how these numbers are related? If we can spot a simple pattern, we are able to see that the denominators are all square numbers. Therefore, we can simplify the notation to this. ! 1 " ! #$% = ? Here is a famous infinite series, namely the Basel problem. 1 + 1/ 4 + 1/ 9 + 1/ 16 +1/ 25 + … = ? So, this series takes the sum of infinitely many numbers. However, can you think of how these numbers are related? If we can spot a simple pattern, we are able to see that the denominators are all square numbers. Therefore, we can simplify the notation to this. ! 1 " ! #$% = ? It may look complicated. However: • The ‘S’ just represents the operation of ‘the sum of’. • The ‘n=1’ means that we will begin inputting numbers into the function, which is the sum of the reciprocals of all square numbers, at one. • Finally, the ∞ means that the series is infinite and will continue to sum infinitely many numbers. At this point we must ask what the question mark represents. What value does the series ‘converge’ to, or does it even have such a limit? This brings us to convergent and divergent series. • A convergent series has a limit that the series ‘tends to’. Euler famously proved that the Basel series converges onto &! ' . • If a e i oes not tend towards a limit, it is a divergent seri s. For example, if we sum all ordinary integers infinitely, we can o p ace a limit on that s ries. Let’s now re ur to the infinite seri s we saw before. This time it has a slightly difer nt no ation. We h ve converted the otation for a particular infinite series to a general function. This function is the zeta function. z ( ) = ! 1 " ( #$) Now, Euler had lots of fun with this function. He was riding high of his achievements and kept finding the limits of all these infinite series. He eventually proved that the zeta function will converge for all integer values of s that are greater than 1. How ver, one f the most significant discoveries h made, if not the most remarkable, was him expressing the zeta function as the infinite p oduct of infini e series. That might be a mouthful, so let’s visualise this. If we let each infinite series be the summation of all ‘one over the powers of a prime number raised to s’ and then multiply that series by the same series for the next prime number, we can derive the zeta function. Alas, another mouthful, let’s visualise this. z ( ) = 2 1 % ( + 2 1 !" + 2 1 * ( + 2 1 +" … This is the first of the infinite series that I mentioned. 2, which is the first prime number is the base. Then we are taking incremental powers of the prime number as we progress further in the series. Lastly, that is all raised to
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