Partial Sums of Primes
Partial Sums of the Riemann Zeta Function
Carlos Villeda December 4th, 2010
Chapter 1 Introduction
1.1 Riemann Zeta Function
In 1859, Bernard Riemann published his paper “On The Number of Primes Less Than a Given Magnitude”, in which he defined a complex variable function which is now called the Riemann Zeta Function(RZF). The function is defined as: ζ(s) = Σ 1 ns (1.1)
Where n ranges over the positive integers from 1 to infinity and where s is a complex number. To get an understanding of the importance of this function, one needs to know some history about it. Dating back to the times of Euclid, the prime numbers have been studied in great deal. What makes them the most interesting over the other numbers is that they hold a special property of not being able to be decomposed into to separate integers. The only numbers that the primes are divisible by is one and themselves. Another, more interesting topic about the primes is their distribution. The question that is asked is “Are the primes distributed in a regular pattern or just randomly placed throughout the integers?” This question is answered by the Riemann Hypothesis to an extent.
1
In 1737, Leonard Euler was one of the first to work with the RZF given above; but it did not have the name it has today. It wasn’t until a century later when Riemann’s name was attached to it for his work. Euler had showed that the RZF was equivalent to Euler’s Product Formula. ζ(s) =
∞ 1 1 = s −s n=1 n p=prime 1 − p ∞
(1.2)
By proving this, Euler showed that there is some relation between the prime numbers and the RZF. The relation remained unknown until when Riemann expanded the range of s. Euler had only worked with real values of s and Riemann was the first to open up the function to the complex plane where s = σ + ıt (1.3)
Upon expanding the range of s to the complex plane, Riemann then studied the zeros of the RZF. It was shown that there were no zeros the lie beyond σ > 1. Moreover he was able to
Carlos Villeda December 4th, 2010
Chapter 1 Introduction
1.1 Riemann Zeta Function
In 1859, Bernard Riemann published his paper “On The Number of Primes Less Than a Given Magnitude”, in which he defined a complex variable function which is now called the Riemann Zeta Function(RZF). The function is defined as: ζ(s) = Σ 1 ns (1.1)
Where n ranges over the positive integers from 1 to infinity and where s is a complex number. To get an understanding of the importance of this function, one needs to know some history about it. Dating back to the times of Euclid, the prime numbers have been studied in great deal. What makes them the most interesting over the other numbers is that they hold a special property of not being able to be decomposed into to separate integers. The only numbers that the primes are divisible by is one and themselves. Another, more interesting topic about the primes is their distribution. The question that is asked is “Are the primes distributed in a regular pattern or just randomly placed throughout the integers?” This question is answered by the Riemann Hypothesis to an extent.
1
In 1737, Leonard Euler was one of the first to work with the RZF given above; but it did not have the name it has today. It wasn’t until a century later when Riemann’s name was attached to it for his work. Euler had showed that the RZF was equivalent to Euler’s Product Formula. ζ(s) =
∞ 1 1 = s −s n=1 n p=prime 1 − p ∞
(1.2)
By proving this, Euler showed that there is some relation between the prime numbers and the RZF. The relation remained unknown until when Riemann expanded the range of s. Euler had only worked with real values of s and Riemann was the first to open up the function to the complex plane where s = σ + ıt (1.3)
Upon expanding the range of s to the complex plane, Riemann then studied the zeros of the RZF. It was shown that there were no zeros the lie beyond σ > 1. Moreover he was able to