Deep within the complex plane lies a hidden axis of symmetry where the secrets of the primes reside. Across the line
every point reflects into a twin, binding the zeta function in a
perfect mathematical duality.
Welcome to another post where arithmetic, geometry and chaos meet.
In “The Riemann Hypothesis Revealed” I derive Euler’s Summation
formula
for real numbers 
and 
directly from the
Euler-Maclaurin formula. The derivation emerges naturally from the theory of
Bernoulli numbers and provides an illuminating bridge between discrete sums and
continuous integrals. Take a look. I think you'll find it surprisingly
enjoyable.
We can apply the formula for 
positive integers and 
, 
because 
and 
.
Now write
Thus,
Take 
and make 
But the left-hand side is
Therefore,
or
It can be shown (see Analytic Continuation of the Zeta function) that this representation is valid for 
. Furthermore, as explained at the
end of Ramanujan’s Master Theorem, the quantity 
arises naturally in the
analytic continuation of the zeta function. In fact, one obtains the integral
representation
This formula (see Analytic
Continuation of the Zeta Function for details) leads to the identity
which is the celebrated functional equation of the
Riemann zeta function. The functional equation implies that the nontrivial
zeros are symmetric with respect to the critical line 
. More precisely since 
, the nontrivial zeros are symmetric with respect to complex
conjugation:
Combining the functional-equation symmetry 
with complex conjunction
gives the particularly useful symmetry
Indeed, for a zero 
we have
Thus, a zero-free region near 
has a mirror-image
zero-free region near 
with respect to the
critical line .
Restrict 
to the rectangle 
. Since 
has only finitely many zeros in this compact
region, and none of them lie on the line (see this post), the
minimum distance from any zero to the line is positive. Hence there
exists a 
such that the sub
rectangle 
contains no zeros of 
. Normally 
depends on 
and 
as 
; but we can obtain a positive lower
bound of 
for each value of .
Now consider the sum
Each term of 
satisfies 
. As 
grows, we choose 
such that 
. For any 
, 
and for any positive constant 
we have 
.
If 
then 
. That is, 
. From the asymptotic formula for
harmonic series, we know: 
(see: Euler’s constant and an asymptotic formula for Harmonic series),
hence
Additionally,
Indeed, we have
so
because
Thus, for sufficiently large ,
(apply the binomial expression: 
)
so
as claimed.
Moreover,
because for 
, 
and
so
Finally, it’s obvious that
Putting everything together
Now take 
That is,
Thus,
if we pick a constant 
and sufficiently large 
such that 
.
Note: For two complex functions 
we define
as 
, if there exist constants 
and a neighborhood of 
such that
for all 
in that neighborhood
(excluding possibly itself).
Important nuance: if 
we need to consider the
direction. Behavior may differ depending on direction in 
so typically we specify uniformly in all
directions unless stated otherwise. Of course, direction may be constrained if
we work inside a sector or the half-plane. (See also this post)
Thus, 
uniformly in the region that depends on 
: 
and 
For some absolute constant 
if
(the number 
is for convenience).
For any fixed constant 
we can choose sufficiently large so that
Consequently,
for 
, 
.
Take 
for 
, and has no zeros in and 
.
■
One is naturally tempted to ask:
what if
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