The central aim of my book, The Riemann Hypothesis Revealed,
was to guide readers to the heart of the Riemann Hypothesis as swiftly as
possible—without ever sacrificing mathematical rigor. Every essential theorem,
proof, and conceptual bridge needed to reach the Riemann Hypothesis is
included, ensuring that nothing is left to assumption or left unproven.
Achieving this clarity and completeness required a careful balance: while many
fascinating side topics beckon for attention, the book maintains a focused trajectory,
avoiding digressions that might disrupt the continuity of the journey toward
RH.
In contrast, this blog series offers us the freedom to
explore vertically—delving into supplementary topics, alternative approaches,
and elegant formulas that, while not strictly necessary for the main narrative,
enrich our understanding and appreciation of the broader mathematical
landscape. Here, we can pause to investigate intriguing byways, share
additional insights, and enjoy the full depth and beauty of complex analysis
and its connections to the Riemann Hypothesis.
Today I’ll demonstrate that the series
converges uniformly on every compact subset of the complex plane,
provided we exclude disks of arbitrarily small radius centered at the points
In addition, I will show that this series fails to converge
absolutely at any point in the complex plane.
Finally, I will reveal its connection to the digamma function—a
link that plays a key role in the derivation of the Stirling series and is
discussed in detail in my book.
We’re talking about two things:
Conditional
convergence that’s actually uniform on compacta away from the poles.
Absolute convergence failing everywhere.
1. Uniform convergence on finite
parts of the plane with small disks removed
“Finite part of the plane” = a
bounded (say, closed and bounded) set 
.
We remove disks of radius 
around the poles 
, so we’re looking at a compact set
where 
.
First key trick: regroup the series
in pairs.
Pick an integer 
and write
Define
Then our series is
where “
” here means equality of partial sums
up to grouping (we’re not reordering, only grouping, so convergence properties
are preserved).
Estimate 
uniformly on 
for large 
Fix a compact with 
for all 
.
For 
large enough (say 
with 
), we have, for any and 
Similarly
For finitely many small 
, the functions 
are continuous on 
except the poles and our hypothesis says avoids small disks around those poles, so each
is bounded on . That’s only finitely many terms,
hence they cause no convergence issues.
Thus, we have a majorant for the
tail:
and
because the tail 
converges and the head is finite.
By the Weierstrass M-test, 
converges absolutely and uniformly on .
Since the original series differs
from this regrouped one only by finite partial sums and grouping of consecutive
terms, it also converges uniformly on .
Because the choice of was any bounded region with disks around the
poles removed (radius as small as you like), this proves
The series
converges uniformly on any finite
part of the plane from which small disks around 
have been removed.
2. The series is nowhere absolutely
convergent
Now I show that
diverges for every 
that is not one of the
poles.
Fix any such and set 
. For all sufficiently large 
(say 
), we have
Hence
Therefore,
which means that the series dominates the harmonic series. So
the absolute-value series diverges.
Combined those two results form a nice classic example: a series
of meromorphic functions that is uniformly convergent on compacta away from
its poles but is nowhere absolutely convergent.
Let us consider the following formula,
which is discussed in detail in The Riemann Hypothesis Revealed under
the topics of the logarithmic derivative and the functional equation for the
Gamma function:
We define the function:
It is called the digamma function.
We have
Therefore,
I am about to demonstrate how the series we established as
uniformly convergent is intimately connected to the digamma function.
Observe that
Now consider the “standard” alternating sum
and notice that
Split 
into even/odd terms:
By comparing this with the earlier representation of the digamma
function, we obtain
Since
we have
Epilogue
The exploration in this episode
highlights the subtle beauty and depth of classical analysis. By examining the
convergence properties of our alternating series and uncovering its connection
to the digamma function, we see how seemingly simple expressions can reveal
profound links across different areas of mathematics. These results not only
enrich our understanding of special functions and series but also demonstrate
the power of rigorous reasoning and careful analysis. As we continue to venture
beyond the main path toward the Riemann Hypothesis, may these supplementary
journeys inspire further curiosity and appreciation for the intricate tapestry
of mathematical ideas.
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