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 o�ered 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 Classi�cation (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

2n(n+ 1)

which tends to in�nity 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 in�nite sum of natural numbers

really does converge to − 112 .

The student did ask about the Riemann zeta function

D. Leadenham675 Sharon Park Drive, No. 247, Menlo Park, CA 94025Tel.: 650-233-9859E-mail: douglasleadenham@gmail.com

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 = 12 +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(12+ιt) with the real axis

is at ζ(12 + ı · 0) = −1.46035 . . .. This point is shown as a gray dot, in contrast with

the nontrivial zeros, ζ(12 + ιt) = 0, indicated by black dots. What one needs to do to

show the convergence of the in�nite sum of natural numbers to − 112 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 in�nite sum of natural numbers as

developed in the complex plane has a cusp at ζ(−1) = − 112 . 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 de�nition of the zeta function.

ζ(s) =

∞∑n=1

n−s

The complex variable s = σ+ ıt in the exponent makes ζ(s) an in�nite sum of the

product n−σn−ıt. With σ = −1, rewrite the base n in the second factor as n = elnn.Now

ζ(s) =

∞∑n=1

n+1e−ıt lnn =

∞∑n=1

n [cos (−t lnn) + ı sin (−t lnn)]

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 (lnn · t)− ı sin (lnn · t)]

which is a sum of products of circles in the complex plane. Consider the case of

small n and t = 0. Then

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 . Thatmakes

elnn = eln(p1p2...pj) = eln p1+ln p2+···+ln pj

e−ıt lnn = e−ıt ln p1−ıt ln p2−···−ıt ln pj

This is an in�nite product of circles in the complex plane. Each such factor applies

a di�erent amplitude, depending on pj , to the spiraling curve, so the curve, being an

in�nite sum, never has a constant radius. Figures 1 and 2 are thus understood.

Easy Zeta 5

�

���

���

���

���

�

���

���

���

���� ���� � ��� ��� �� ��� ��

��������

��������

������������������� ������

������ �����������

����������

Fig. 3 In�nite Sum of Natural Numbers Resulting from the Zeta Function

Table 1 Data for Figure 3.

t <(ζ) =(ζ)

-1 0.4166 1.499-0.9 0.321667 1.35-0.8 0.236667 1.20-0.7 0.161667 1.05-0.6 0.096667 0.9-0.5 0.041667 0.75-0.4 -0.003333 0.6-0.3 -0.038333 0.45-0.2 -0.06333 0.3-0.1 -0.078333 0.15-0.05 -0.0820833 0.075-0.03 -0.0828833 0.0450 -0.083333333 00.03 -0.0828833 0.0150.05 -0.0820833 0.0250.1 -0.078333 0.050.2 -0.06333 0.10.3 -0.038333 0.150.4 -0.003333 0.20.5 0.041667 0.250.6 0.096667 0.30.7 0.161667 0.350.8 0.236667 0.40.9 0.321667 0.451 0.4166 0.499

6 Douglas Leadenham

References

[Fanelli] Fanelli, M. and Fanelli, A. The Riemann hypothesis about the non-trivial zeroes ofthe Zeta function, arXiv:1509.01554, 5 pages (2015)

[Edwards] Edwards, H. M., Riemann's Zeta Function, pp. 12,96-135. Academic Press, NewYork (1974)