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On the analytic continuation of the Riemann zeta function

The paper analyzes the analytic continuation of the Riemann zeta function in detail to derive several new identities with potential applications in physics and mathematics.

Original authors: Paolo Valtancoli

Published 2026-05-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Paolo Valtancoli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Riemann zeta function, $\zeta(s)$ , as a very picky musician. In its original form (defined by a simple sum of numbers), this musician can only play a song perfectly when the "temperature" of the room (a mathematical value called the real part of $s$ ) is above 1. If the room gets any cooler (below 1), the music stops, the notes become chaotic, and the formula breaks down.

The paper by P. Valtancoli is essentially a guide on how to teach this musician to play the song in colder rooms, all the way down to freezing temperatures, without the music falling apart.

Here is how the author does it, explained through simple analogies:

1. The Problem: The Music Stops at Room Temperature 1

The standard definition of the zeta function is like a line of people passing a bucket of water. As long as the line is short enough (Re $s > 1$ ), the bucket gets filled perfectly. But if you try to make the line infinitely long in a cold room, the bucket overflows, and the math breaks.

2. The Solution: "Integration by Parts" as a Moving Ladder

The author uses a mathematical technique called "integration by parts." Think of this as a ladder.

The original formula is the bottom rung of the ladder.
By applying the technique once, the author builds a new rung that allows the formula to work in a slightly colder room (Re $s > 0$ ).
By doing it again, they build another rung, allowing it to work even colder (Re $s > -1$ ).
They keep climbing this ladder, rung by rung, extending the "validity" of the music into colder and colder territories.

3. The Secret Ingredient: The "Periodic" Mask

To climb each rung of the ladder, the author has to deal with a "step function." Imagine a staircase where the height jumps up suddenly at every whole number. This staircase is messy and hard to work with in the cold.

To fix this, the author subtracts a specific, smooth polynomial curve (a fancy algebraic shape) from the staircase.

The Analogy: Imagine you have a jagged, saw-toothed fence. You want to make it smooth. You take a perfectly shaped piece of wood (the polynomial) and cut it out of the jagged fence.
The Result: What's left isn't a jagged mess anymore; it's a repeating pattern (a periodic function). In math, repeating patterns are much easier to handle and predict.

The author identifies these "wooden pieces" to remove as related to Bernoulli polynomials. For every new rung of the ladder (labeled by a number $p$ ), they have to find a new, slightly more complex wooden piece to cut out.

4. The New Identities: A Chain of Clues

Once the jagged fence is smoothed out into a repeating pattern, the author translates the whole problem into a new set of rules, or identities.

These new rules look like a chain reaction. They say:
"The value of the zeta function at this cold temperature is equal to a few simple numbers, plus a long list of other zeta values (specifically $\zeta(k) - 1$ ) multiplied by some coefficients."

Why is this cool? The values in this long list ( $\zeta(k) - 1$ ) get smaller and smaller as you go further down the list, eventually vanishing. This makes the new formulas very stable and useful for calculation.

5. The Journey from $p=1$ to $p=12$

The paper walks us through this process step-by-step:

Case $p=1$ : The simplest version. It reveals a famous result: if you add up all the differences between the zeta function and 1 (starting from $k=2$ ), you get exactly 1.
Case $p=2$ : The author finds a way to calculate the value of the zeta function at 0 (which is $-1/2$ ) and its slope at 0.
Cases $p=3$ through $p=12$ : As they go higher, the "wooden pieces" (polynomials) they have to subtract get more and more complicated. The formulas become huge, with long strings of numbers and powers.
- The author admits that for the higher cases (like $p=9$ to $p=12$ ), the math is so messy that they had to use a computer (Mathematica) to do the heavy lifting.
- However, the pattern holds true: even with these complex formulas, they successfully extend the function's reach and confirm known "trivial zeros" (specific points where the function equals zero, like at -2, -4, -6, etc.).

Summary

In short, this paper is a construction manual. It shows how to take a mathematical formula that only works in a warm room and, by systematically smoothing out its jagged edges and climbing a ladder of integration, extend its reach into the freezing cold.

For every step up the ladder (increasing $p$ ), the author generates a new, unique "identity"—a new equation that connects the zeta function to a series of other numbers. While the equations get messy and complex at the top of the ladder, they prove that the function can be understood and calculated in many new ways, revealing hidden structures in the numbers.

Technical Summary: On the Analytic Continuation of the Riemann Zeta Function

Problem Statement
The Riemann zeta function, $\zeta(s)$ , is initially defined by the Dirichlet series $\sum_{n=1}^{\infty} n^{-s}$ , which converges only in the half-plane $\text{Re}(s) > 1$ . While the analytic continuation of $\zeta(s)$ to the entire complex plane is a well-established result (typically derived via the functional equation or contour integration), this paper focuses on deriving the continuation through a specific iterative method: repeated integration by parts. The objective is to generate a family of new identities for $\zeta(s)$ that extend the domain of validity step-by-step and express the function in terms of series involving $(\zeta(k) - 1)$ for integers $k \geq 2$ .

Methodology
The author employs a systematic procedure based on integration by parts applied to an integral representation of the zeta function involving the Heaviside step function $\Theta(x-n)$ . The core methodology proceeds as follows:

Integral Representation: The starting point is the integral form derived from the series definition, where the summation is converted into an integral involving $\Theta(x-n)$ .
Iterative Integration by Parts: By repeatedly applying integration by parts, the author shifts the validity of the series representation from $\text{Re}(s) > 1$ to $\text{Re}(s) > -(p-1)$ for an integer parameter $p$ .
Periodization via Polynomial Subtraction: To handle the divergent behavior at the lower bounds of integration, a specific polynomial $f_p(x)$ is subtracted from the step function summation. This polynomial is constructed by taking the sum of powers formula (related to Faulhaber's formula) for the first $n$ integers, $P_{p-1}(n) = \sum_{i=1}^n i^{p-1}$ , and substituting $n \to (x-1)$ .
Conversion to Periodic Functions: The subtraction transforms the integrand into a periodic function (involving fractional parts or Bernoulli polynomials).
Series Expansion: The resulting periodic integrals are expanded into infinite series involving terms of the form $\int_0^1 \frac{x^m}{(x+n)^{s+k}} dx$ , which are subsequently related to $\zeta(s+k) - 1$ .

Key Contributions and Results
The paper derives explicit identities for $\zeta(s)$ for values of $p$ ranging from 1 to 12. Each value of $p$ corresponds to a specific identity that extends the region of convergence and provides a new representation of the zeta function.

Case $p=1$ : The author derives the identity $\sum_{k=0}^{\infty} \frac{\Gamma(s+k)}{\Gamma(k+2)\Gamma(s)}(\zeta(s+k)-1) = \frac{1}{s-1}$ . This recovers the simple pole at $s=1$ with residue 1. A specific corollary is presented: $\sum_{k=2}^{\infty} (\zeta(k)-1) = 1$ .
Case $p=2$ : The derivation incorporates the polynomial $f_2(x) = \frac{x(x-1)}{2}$ . This yields an identity valid for $\text{Re}(s) > -1$ , allowing for the calculation of $\zeta(0) = -1/2$ and an expression for the derivative $\zeta'(0)$ as a series of $(\zeta(k)-1)$ .
Cases $p=3$ to $p=12$ : The paper presents explicit formulas for higher orders. As $p$ $p$ increases, the polynomial subtractions become more complex (involving higher powers of $x$ $x$ ), and the resulting identities for $\zeta(s)$ $ζ (s)$ contain increasingly complex rational functions of $s$ $s$ and higher-order infinite series.
- For even $p$ , the identities are shown to naturally contain the trivial zeros of the zeta function at negative even integers ( $s = -2, -4, \dots$ ).
- The author explicitly calculates the polynomial coefficients for $p$ up to 12, noting that for $p \geq 9$ , computer algebra systems (Mathematica) were utilized to manage the complexity of the polynomial decomposition.
- The general pattern suggests that for any $p$ , a general formula can be obtained where $\zeta(s)$ is expressed as a rational function of $s$ plus a series involving $\Gamma$ functions and $(\zeta(s+k)-1)$ .

Significance and Claims
The paper claims that these derived identities offer "new identities which may be useful for applications in physics and mathematics." Specifically, the author highlights that:

The method provides a constructive way to extend the analytic continuation of $\zeta(s)$ without relying solely on the functional equation.
The identities express $\zeta(s)$ and its derivatives entirely in terms of the convergent series $(\zeta(k)-1)$ for $k \geq 2$ . Since these terms tend to zero rapidly as $k \to \infty$ , the series are numerically well-behaved.
The approach unifies the calculation of values at negative integers (including trivial zeros) and derivatives at $s=0$ within a single framework of integration by parts.

The author maintains a modest tone regarding the broader implications, stating that the hope is for these identities to "shed light on new properties of $\zeta(s)$ ," without asserting the resolution of the Riemann Hypothesis or specific new physical applications beyond the general utility of the identities.