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.
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 ∞
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 prove that the RZF satisﬁes the functional equation ζ(s) = 2s π s−1 sin πs Γ(1 − s)ζ(1 − s) 2 (1.4)
where Γ(s) is the usual gamma function deﬁned as
e−t ts−1 dt
The functional equation or the gamma function will not be studied in this paper but it shows the zeros of the RZF in the plane where σ < 0. These zeros are all the negative even integers i.e. -2, -4, -6, etc. These are the trivial zeros of the RZF. The rest of the zeros, the non trivial zeros, must lie in the plane 0 < σ < 1 which is denoted the critical strip(picture blow on the left). After calculating some zeros in the critical strip, Riemann made the observation that the zeros all had a real part of one half. This brings rise to
the Riemann Hypothesis(RH).
Figure 4: The critical strip: all of those complex number with real part greater than 0 and The RH states that all non trivial zeros of the RZF have a real less than 1. From . 1
part of one half. The line of σ = 2 is denoted the critical line where all the non trivials zeros lie (picture above on the right). This hypothesis has boggled the minds of mathematicians for 150 years and still remains unproven today. In Hilbert’s list of 23 math problems to solve in the 20th century, there are 3 problems in particular that he pointed out. The ﬁrst was the irrationality of the The critical line with zeros Figure 5: square root of 2, the next was Fermat’s Last theorem, and ﬁnally function. As in predicted that the RH would be solved in a of the zeta the RH. Hethis ﬁgure, few years, zeta function in the all zeros of the Fermat’s Last Theorem in decades and the irrationality critical strip calculated to of 2 line on of the square root date possibly never. His prediction was wrong. the critical line. From . The irrationality of the square root of 2 was proved in a few years, Fermat’s Last Theorem a century later, and the RH still remains. Computers have calculated that the ﬁrst 10 trillion zeros have a 1 real...
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