1. Noname manuscript No.
(will be inserted by the editor)
The Riemann Zeta Function for Beginners
Douglas Leadenham
Received: date / Accepted: date
Abstract The Riemann zeta function is not usually oered as teaching material. Still,
questions arise from Internet videos and posted graphics that have been produced by
investigators of the Riemann hypothesis that all nontrivial zeros of the function lie
on the critical line. Herein is an outline of a possible class lecture on the elementary
algebra of the zeta function.
Keywords Riemann zeta function
Mathematics Subject Classication (2010) 11Mxx
1 Introduction
A student had watched the YouTube video cited below and asked how this could be
possible.
ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12, Jan 9, 2014
https://www.youtube.com/watch?v=w-I6XTVZXww
He knew, of course, that
n
k=1
k =
1
2
n(n + 1)
which tends to innity with n2
. The narrator in the video, a mathematician, com-
mented that without getting our knickers in a twist over the Riemann zeta function
one could show how with only real algebra that the innite sum of natural numbers
really does converge to − 1
12 .
The student did ask about the Riemann zeta function
D. Leadenham
675 Sharon Park Drive, No. 247, Menlo Park, CA 94025
Tel.: 650-233-9859
E-mail: douglasleadenham@gmail.com
2. 2 Douglas Leadenham
ζ(s) =
∞
n=1
n−s
; s = σ + ıt
and I said only that the variable s is a complex number that must be taken into
account in the calculation. This math subject happened to be o point of the class at
the time.
2 Why the astounding result is true
In this case only natural numbers are summed, so the complex variable s = −1 with
the imaginary part t = 0. The notation is from Edwards, who followed Riemann's 1859
notation.[Edwards]
The Internet abounds with graphics on the Riemann hypothesis that the nontrivial
zeros of the zeta function all lie on the line in the complex plane s = 1
2 +ıt. The million
dollar challenge by the Clay Mathematics Institute to prove this has got professional
and amateur mathematicians quite interested. Articles appear nearly daily that com-
municate such work.[Fanelli] Two of the Internet graphics are shown in Figures 1 and
2.
In Figure 2 the intersection of the dark trajectory line of ζ(1
2 +ιt) with the real axis
is at ζ(1
2 + ı · 0) = −1.46035 . . .. This point is shown as a gray dot, in contrast with
the nontrivial zeros, ζ(1
2 + ιt) = 0, indicated by black dots. What one needs to do to
show the convergence of the innite sum of natural numbers to − 1
12 is plot ζ(−1 + ıt)
for a sequence of t−values as they decrease to zero exactly. This is shown in Figure 3.
Here the upper branch begins with t = −1 and increases to 0. The lower branch
begins with t = +1 and decreases to 0. Thus, the innite sum of natural numbers as
developed in the complex plane has a cusp at ζ(−1) = − 1
12 . The astounding result
seen on YouTube Numberphile is now only amazing along with being mathematically
true.
Data for Figure 3 were obtained with the help of WolframAlpha, a free style front
end to Mathematica
®. To make it work right, it was necessary to format the queries in
Mathematica
® syntax. Table 1 shows the data for Figure 3, for all of which σ = −1.
3 How the complex variable s makes the zeta function spiral around
Go back to the denition of the zeta function.
ζ(s) =
∞
n=1
n−s
The complex variable s = σ + ıt in the exponent makes ζ(s) an innite sum of the
product n−σ
n−ıt
. With σ = −1, rewrite the base n in the second factor as n = eln n
.
Now
ζ(s) =
∞
n=1
n+1
e−ıt ln n
=
∞
n=1
n [cos (−t ln n) + ı sin (−t ln n)]
3. Easy Zeta 3
Fig. 1 Some Zeros of Zeta Viewed from t 0 toward 0.
This is equivalent to
ζ(s) =
∞
n=1
n [cos (ln n · t) − ı sin (ln n · t)]
which is a sum of products of circles in the complex plane. Consider the case of
small n and t = 0. Then
4. 4 Douglas Leadenham
Fig. 2 Some Zeros of Zeta Viewed along the t−axis
ζ(s) =
∞
n=1
n (cos 0 − ı sin (0)) =
∞
n=1
n
and this is the special case featured on Numberphile, but it ignores the complex
variables ζ and s. Moreover, n is the product of its prime factors. n = Πjpj. That
makes
eln n
= eln(p1p2...pj ) = eln p1+ln p2+···+ln pj
e−ıt ln n
= e−ıt ln p1−ıt ln p2−···−ıt ln pj
This is an innite product of circles in the complex plane. Each such factor applies
a dierent amplitude, depending on pj, to the spiraling curve, so the curve, being an
innite sum, never has a constant radius. Figures 1 and 2 are thus understood.
6. 6 Douglas Leadenham
References
[Fanelli] Fanelli, M. and Fanelli, A. The Riemann hypothesis about the non-trivial zeroes of
the Zeta function, arXiv:1509.01554, 5 pages (2015)
[Edwards] Edwards, H. M., Riemann's Zeta Function, pp. 12,96-135. Academic Press, New
York (1974)
![2 Douglas Leadenham
ζ(s) =
∞
n=1
n−s
; s = σ + ıt
and I said only that the variable s is a complex number that must be taken into
account in the calculation. This math subject happened to be o point of the class at
the time.
2 Why the astounding result is true
In this case only natural numbers are summed, so the complex variable s = −1 with
the imaginary part t = 0. The notation is from Edwards, who followed Riemann's 1859
notation.[Edwards]
The Internet abounds with graphics on the Riemann hypothesis that the nontrivial
zeros of the zeta function all lie on the line in the complex plane s = 1
2 +ıt. The million
dollar challenge by the Clay Mathematics Institute to prove this has got professional
and amateur mathematicians quite interested. Articles appear nearly daily that com-
municate such work.[Fanelli] Two of the Internet graphics are shown in Figures 1 and
2.
In Figure 2 the intersection of the dark trajectory line of ζ(1
2 +ιt) with the real axis
is at ζ(1
2 + ı · 0) = −1.46035 . . .. This point is shown as a gray dot, in contrast with
the nontrivial zeros, ζ(1
2 + ιt) = 0, indicated by black dots. What one needs to do to
show the convergence of the innite sum of natural numbers to − 1
12 is plot ζ(−1 + ıt)
for a sequence of t−values as they decrease to zero exactly. This is shown in Figure 3.
Here the upper branch begins with t = −1 and increases to 0. The lower branch
begins with t = +1 and decreases to 0. Thus, the innite sum of natural numbers as
developed in the complex plane has a cusp at ζ(−1) = − 1
12 . The astounding result
seen on YouTube Numberphile is now only amazing along with being mathematically
true.
Data for Figure 3 were obtained with the help of WolframAlpha, a free style front
end to Mathematica
®. To make it work right, it was necessary to format the queries in
Mathematica
® syntax. Table 1 shows the data for Figure 3, for all of which σ = −1.
3 How the complex variable s makes the zeta function spiral around
Go back to the denition of the zeta function.
ζ(s) =
∞
n=1
n−s
The complex variable s = σ + ıt in the exponent makes ζ(s) an innite sum of the
product n−σ
n−ıt
. With σ = −1, rewrite the base n in the second factor as n = eln n
.
Now
ζ(s) =
∞
n=1
n+1
e−ıt ln n
=
∞
n=1
n [cos (−t ln n) + ı sin (−t ln n)]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)