A Small Taste from My New Book: Season 2 Episode 1

 

I am delighted to launch the first episode of Season 2, drawing directly from three paragraphs of my book, The Riemann Hypothesis Revealed. 

This new season continues our journey through the elegant landscape of complex analysis and analytic number theory, offering readers both foundational insights and advanced perspectives.

In my book, we explore how entire (complex analytic) functions with prescribed zeros can be represented as infinite products. This leads us naturally to the celebrated Weierstrass factorization theorem, which reveals how the structure of an entire function is intimately tied to the distribution of its zeros. Along the way, we discuss the convergence of infinite products, the role of canonical products, and the deep connections to analytic number theory.

Briefly, the reader will discover the following sections:

Infinite Products

This section discusses how entire (complex analytic) functions with prescribed zeros can be represented as infinite products. If a function  is analytic everywhere (entire) and has simple zeros at points , then locally around these zeros,  can be written as , where  is analytic and nonzero near the zeros. As the neighborhood expands to include all zeros, the logarithmic derivative  can be represented as a sum over the zeros, and, under suitable convergence conditions,  itself can be written as an infinite product involving its zeros and certain correction factors (to ensure convergence). This leads to the Weierstrass factorization theorem, which states that an entire function can be written as a product over its zeros, possibly multiplied by an exponential of an entire function.

On the Convergence of Infinite Products

This section explains the conditions under which infinite products of complex numbers (or functions) converge. An infinite product  converges absolutely if  and the series  converges absolutely. For analytic functions, if each factor is close to 1 in a certain region, then the product converges uniformly and absolutely in that region. The section also discusses Blaschke products (infinite products over the unit disk) and the convergence criteria for such products. The logarithmic derivative of an infinite product is the sum of the logarithmic derivatives of its factors, provided the product converges uniformly. The section further introduces canonical products—infinite products with specific correction factors (elementary factors) chosen to ensure convergence, especially when the zeros do not grow fast enough. The Weierstrass factorization theorem is revisited, stating that any entire function can be written as a product over its zeros, with suitable correction factors and an exponential of an entire function.

Hadamard’s Theorem

This section builds on the previous discussions to introduce Hadamard’s factorization theorem. It states that if an entire function  has zeros  and is of finite order , then  can be written as

where  is a polynomial (of degree at most the genus),  is the multiplicity of zero at the origin, and  are elementary factors chosen to ensure convergence. The order  of an entire function describes its growth rate at infinity, and the genus  is related to the minimal correction needed for convergence of the product over zeros. Hadamard’s theorem provides bounds on the order and genus: . If the order is not an integer, the function must have infinitely many zeros. The theorem is crucial in analytic number theory, especially in the study of the Riemann zeta function and its zeros.

In summary:

• Infinite products allow entire functions to be expressed in terms of their zeros, with convergence ensured by suitable correction factors.
• The convergence of such products is tied to the growth of the zeros and the function itself.
• Hadamard’s theorem gives a precise factorization for entire functions of finite order, relating the function’s growth, its zeros, and the structure of the infinite product representation.

The following exercises are designed to reinforce the concepts introduced in the book, such as the convergence of infinite products, the relationship between products and sums, and the subtleties of analytic continuation.

Let’s turn to the exercises and see these ideas in action.

In the classic text “A Collection of Problems on Complex Analysis” (Dover Publications, 1965) by L.I. Volkovyskii, G.L. Lunts, and I.G. Aramanovich, I came across the following exercise:

Exercise 1

Prove that, as is customary, if we restrict the arguments of the sequence  to , then the convergence or divergence of the infinite product  is equivalent to the convergence or divergence of the series .

Let’s do this carefully.

We assume:

 for all n,

we use the principal branch of the logarithm, so ,

and (as is standard for infinite products) we are interested in whether

converges (to a nonzero limit) or diverges, and we compare this to

Partial products and partial sums

Define the partial products

and the partial sums

where log is the principal branch.

Whenever the principal branch is consistently defined along the sequence , we have

This argument is too vague.

The problem is that

is false in general for the principal logarithm. It is true when no branch cut is crossed.

More precisely, for the principal branch one has

for some integer k.

So to be allowed to write

we must assume something like:

for every N, the straight multiplicative accumulation of the arguments does not cross the negative real axis.

A standard sufficient hypothesis is:

This is exactly what “consistently defined” means.

Hence if we assume that for every N, the partial product  avoids  then

So after removing the ambiguity:

This identity is the core link between the product and the sum.

If  converges, then  converges

Assume the series

converges in . Then the sequence of partial sums  converges to some limit .

Since the exponential function is continuous,

So the infinite product  converges (to the nonzero limit P).

If   converges (to a nonzero limit), then  converges

Now assume the infinite product

converges to some nonzero limit . Then the partial products  form a Cauchy sequence and

Because , for all sufficiently large N,  stays in some compact subset of  that does not cross the branch cut of the principal logarithm. Thus, for all large N,  is well-defined and continuous in N, and we can write

Strictly speaking we need:

for all sufficiently large N, , so that the principal logarithm is single-valued and continuous in a neighborhood of the tail of the sequence.

Having established the right condition, since  and log is continuous on , we may conclude

Hence the series   converges (its partial sums converge to )

Divergence

if  diverges in , then the sequence  does not converge, so  cannot converge to a nonzero limit. The product either diverges or converges to 0 (in which case ).

The reason is not simply “because  does not converge”. In general, one can have a sequence  which does not converge in  but for which  does converge (for example by adding integer multiples of ).

What saves the argument is that: we are fixing the principal logarithms of the individual factors, and we require the identity .

Conversely, if the product does not converge to a nonzero limit, then  does not converge in , so  cannot converge in .

Thus, under the principal-branch restriction  (so that is consistently defined), we have:

 converges (to a nonzero limit)  
 converges

assuming that the partial products

avoid the branch cut .

And in the extended sense (allowing convergence of the product to ), divergence of one corresponds to divergence of the other.

One more missing classical hypothesis

There is a standard extra assumption which is usually stated explicitly in textbooks:

 converges to a non-zero limit if and only if  converges and .

In fact, from convergence of  we automatically get , but in the reverse direction authors often state it to avoid pathological formulations.

Exercise 2

Now we have to check whether the assertion of the preceding problem is still true if we assume that:

·       or

·      ,

This is exactly where the branch-cut geometry starts to bite.

We’re now changing the way we choose arguments of , hence changing the branch of log. The previous result relied crucially on using a single continuous branch of log along the sequence of partial products.

Let’s look at the two new cases.

Case 1:

Here we’re using arguments in . That’s the principal argument shifted: instead of , we’ve rotated the cut so that branch cut is now along the positive real axis.

The problem: as n grows the partial products

may cross the positive real axis infinitely often. Each time you cross the branch cut, the value of  jumps by . So:

·         We can still define  with ,

·         but  is no longer guaranteed to equal  in a way that is continuous in N.

·     The identity itself can fail, not only continuity because with this branch,  for some integers , and these integers can change infinitely often if the partial products cross the cut infinitely often. This is the real obstruction.

So the key identity

can fail to behave nicely as N increases, because of branch-cut jumps. That breaks the equivalence:

·      It is possible for  while  on the chosen path differs from  by varying multiples of ,

·         or for  to converge in this branch while the product behaves differently under another branch.

Conclusion for this case: The clean equivalence “product converges  sum of logs converges” is no longer guaranteed if you use .

 Case 2: ,

Same story, different rotation.

We’re now using a branch of log whose cut lies along the ray . Again, the crucial question is:

Do the partial products  stay in a region that does not cross this branch cut for all large N?

If they do, then for all sufficiently large N,  is a continuous function of N on that branch, and you recover

in a stable way. Then the same equivalence as before holds.

If they don’t, i.e. if  crosses the branch cut infinitely often, you again get jumps of  in , and the neat equivalence between convergence of   and  can fail.

So:

The assertion from the previous problem is not automatically true for arbitrary choices of argument intervals like  or .

It remains true only if the chosen branch of log is such that the partial products eventually stay in a simply connected region avoiding the branch cut, so that is single–valued and continuous along the sequence . In fact, it is actually enough that (branch cut) for all large N. and that all those points lie in the same connected component of the cut plane.

In other words: the equivalence is a statement about choosing a consistent branch of logalong the path of partial products, not about any arbitrary way of assigning arguments.

______________________________

Moving beyond these pathological cases, we have the following result:

Exercise 3

Prove that for the absolute convergence of the infinite product

that is, for the absolute convergence of the series

it is necessary and sufficient that the series

should converge absolutely.

In my book, I prove that absolute convergence of the series  ensures the absolute convergence of the infinite product  . This result serves as a motivating principle for what follows. That’s why I named it The Riemann Hypothesis Revealed and not Complex Analysis. When writing about the Riemann Hypothesis and striving for minimal compromises in exposition, certain technical details are sometimes omitted for focus. My readers will know by now that I don’t follow academic conventions. The reason is simple: my priority is to make mathematics transparent and accessible, rather than adhering to traditional formalities that can obscure the essential ideas.

However, through this blog, we have the opportunity to revisit, validate, and enrich the story. Here, we can explore the technical nuances that are sometimes omitted in the book for the sake of clarity, offering readers a more comprehensive perspective. By expanding on these foundational principles, we not only reinforce the motivation behind key results, but also provide a deeper understanding of their role in analytic number theory and the journey toward the Riemann Hypothesis.

Let’s see both directions now.

 convergent  product absolutely convergent

Assume

Then . So there exists N such that for all ,

For , the Taylor series

converges absolutely and we have uniform estimate

for some constant  and all .

Apply this with  for  to get

The finite initial segment  is irrelevant for absolute convergence, so

converges absolutely. By definition the product  converges absolutely.

Product absolutely convergent   convergent

Now assume the product converges absolutely, i.e.

converges absolutely.

Then in particular , hence , so . For n large, , and again we can use Taylor expansion.

For small z, we have the reverse-type estimate

for some constant  and all . This follows from  and continuity near 0.

Thus, for large n,

Since  converges, the comparison test gives

So absolute convergence of the product is equivalent to absolute convergence of .

 

Exercise 4

The infinite products  and  converge. Investigate the convergence of the infinite products

         i.           

       ii.           

      iii.           

      iv.           

Let’s assume “converge” in the usual sense for infinite products of complex numbers:  converges means ,  and  converges (similarly for ).

So for large n .

Product

Here  also  so  a sum of two convergent series. Hence  converges.

Similar for .

Product

Now  so the product diverges (blows up in modulus).

Product

Here . Therefore, it is impossible to have  so the product always diverges.

 

Exercise 5

Investigate the convergence of the following products

         i.           

       ii.            ,

      iii.           

 

For (i), it is natural to write the factors as pairs.

Hence,

However, if we set

then  and .

Thus,

We have , i.e., converges (alternating harmonic) and  converges. So  converges, hence the product converges to a nonzero limit.

But  diverges, so the product is not absolutely convergent only conditionally.

Conclusion: convergent but not absolutely.

For (ii), we have convergence for  and divergence for . Reader’s exercise

For (iii), notice that

and

diverges like a harmonic series. Reader’s exercise.

Epilogue

Season 2 Episode 1 has taken us deep into the heart of complex analysis, exploring the elegant machinery of infinite products. These are not just technical results—they are the keys that unlock the hidden architecture of entire functions and illuminate the path toward some of mathematics’ most celebrated mysteries.

If you found this episode insightful, I invite you to take the next step and invest in your mathematical growth by purchasing my book, The Riemann Hypothesis Revealed. This book is more than just a collection of theorems—it’s a carefully crafted guide designed to make advanced concepts accessible, transparent, and inspiring. Whether you’re a student, researcher, or lifelong learner, you’ll find that this investment pays dividends in clarity, confidence, and a deeper understanding of the mathematical universe.

Don’t just read about mathematics—experience it fully. Make The Riemann Hypothesis Revealed your next investment in knowledge and discovery.