Equation 6 — The Fractional-Part (Mellin) Integral for ζ(s)

$$\zeta(s) = -s\int_0^\infty \{x\}\,x^{-s-1}\,dx, \qquad 0 < \sigma < 1$$

This is one of the most conceptually important representations of the zeta function for your paper, because it is the bridge between the "classical" analytic continuation of ζ and the modern functional-analytic reformulations of the Riemann Hypothesis (Nyman–Beurling–Báez-Duarte). Below is the full story: history, mathematics, literature, and working Python code.

1. What the equation says

Write $s = \sigma + it$. The function $\{x\} = x - \lfloor x\rfloor$ is the fractional part of $x$. The identity states that in the critical strip $0<\sigma<1$ (exactly where RH lives), ζ(s) is — up to the elementary factor $-s$ — the Mellin transform of the fractional-part function.

An equivalent and very suggestive form is obtained by substituting $x \mapsto 1/x$:

$$\int_0^\infty \left\{\frac{1}{x}\right\} x^{s-1}\,dx = -\frac{\zeta(s)}{s}, \qquad 0<\sigma<1.$$

So the "sawtooth" function $\rho(x) = \{1/x\}$ on $(0,1)$ encodes ζ completely in the critical strip. This is the seed of the Nyman–Beurling criterion (your Equation #29).

2. History and attribution

• Leonhard Euler / Niels Henrik Abel / Euler–Maclaurin (18th–19th c.). The technique in Step 1 — converting a sum $\sum_{n\le N} f(n)$ into a Riemann–Stieltjes integral $\int f(x)\,d\lfloor x\rfloor$ and integrating by parts — is Abel summation (also called partial summation), a discrete analogue of integration by parts. The refinement with periodic Bernoulli functions is the Euler–Maclaurin formula (Euler 1732–35, Maclaurin 1742).

• Bernhard Riemann (1859). In his famous 8-page memoir "Über die Anzahl der Primzahlen unter einer gegebenen Größe", Riemann continued ζ analytically using the gamma-integral $\zeta(s)\Gamma(s) = \int_0^\infty \frac{x^{s-1}}{e^x-1}dx$ and a contour argument — not this formula. However, the "half-way" version in Step 2, $$\zeta(s) = \frac{s}{s-1} - s\int_1^\infty \{x\}x^{-s-1}dx \qquad (\sigma>0),$$ is the standard elementary route to continuing ζ past $\sigma = 1$, and it is essentially folklore of the Chebyshev–Mertens era of analytic number theory (1850s–1890s), where partial summation against $\lfloor x\rfloor$ was the workhorse for prime-counting estimates.

• E. C. Titchmarsh (1930/1951). The formula as you've written it is stated as equation (2.1.5) in the canonical reference: E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford (2nd ed. revised by D. R. Heath-Brown, 1986), §2.1. If you need a citable "source of record" for the equation itself, cite Titchmarsh §2.1.

• Bertil Nyman (1950) and Arne Beurling (1955). This is where the formula becomes RH-relevant. Nyman (in his Uppsala thesis, under Beurling) and then Beurling showed that RH is equivalent to a density/approximation statement about the functions $x \mapsto \{\theta/x\}$ in $L^2(0,1)$ — precisely because of the Mellin identity above.

• Luis Báez-Duarte (2002/2003). Strengthened the criterion: it suffices to use only $\theta = 1/k$, $k = 1, 2, 3, \dots$ — turning RH into a concrete least-squares problem (see §5).

3. The derivation, step by step (annotated)

Step 1 (Abel summation). For $\sigma > 1$, the Dirichlet series $\sum n^{-s}$ converges absolutely. Viewing $\lfloor x\rfloor$ as the counting measure of the integers, $$\sum_{n\le N} n^{-s} = \int_{1^-}^{N} x^{-s}\, d\lfloor x\rfloor = \lfloor x\rfloor x^{-s}\Big|_{1^-}^{N} + s\int_1^N \lfloor x\rfloor x^{-s-1}\,dx.$$ This is Riemann–Stieltjes integration by parts; the jump of $\lfloor x\rfloor$ at each integer $n$ picks out $n^{-s}$.

Step 2 (Continuation to σ > 0). Let $N\to\infty$ (the boundary term $N^{1-s}\to 0$ for $\sigma>1$), and split $\lfloor x\rfloor = x - \{x\}$: $$\zeta(s) = \frac{s}{s-1} - s\int_1^\infty \{x\}\,x^{-s-1}\,dx.$$ Key observation: because $0\le\{x\}<1$, the integral converges absolutely whenever $\sigma > 0$. Both sides are analytic in $\sigma>0$ (minus the point $s=1$), and they agree on $\sigma>1$; by the identity theorem for analytic functions, the right-hand side is the analytic continuation of ζ to the half-plane $\sigma > 0$. The pole at $s=1$ with residue 1 is displayed explicitly by $\frac{s}{s-1} = 1 + \frac{1}{s-1}$.

Step 3 (The trick on (0,1)). For $0 < x < 1$ we have $\{x\} = x$, so $$-s\int_0^1 \{x\}x^{-s-1}dx = -s\int_0^1 x^{-s}dx = \frac{s}{s-1},$$ where convergence at $x=0$ requires σ < 1. This elementary integral exactly reproduces the pole term.

Step 4 (Merge). Adding Steps 2 and 3, the $\frac{s}{s-1}$ terms cancel against each other's bookkeeping and we get the clean statement $$\zeta(s) = -s\int_0^\infty \{x\}\,x^{-s-1}\,dx,\qquad 0<\sigma<1. \qquad\blacksquare$$

Why the strip is forced: convergence at $\infty$ needs $\sigma>0$; convergence at $0$ needs $\sigma<1$. The formula lives exactly on the critical strip — an aesthetically perfect coincidence for RH purposes.

4. Connection to RH: the Nyman–Beurling criterion

Substituting $x\to 1/x$ gives $\int_0^1 \{1/x\}x^{s-1}dx + \int_1^\infty \frac1x \cdot x^{s-1} dx$-type decompositions, and more generally for $0<\theta\le 1$:

$$\int_0^1 \left\{\frac{\theta}{x}\right\} x^{s-1}\,dx = \frac{\theta^s}{s}\,\zeta(s)\cdot(\text{elementary factor}) - \frac{\theta}{s-1},$$

so finite linear combinations $f(x) = \sum_k c_k\{\theta_k/x\}$ with $\sum_k c_k\theta_k = 0$ have Mellin transforms of the form $\zeta(s)\cdot\frac{1}{s}\sum_k c_k \theta_k^{s}$.

Nyman–Beurling theorem (1950/1955): RH holds if and only if the characteristic function $\chi_{(0,1)}$ lies in the $L^2(0,1)$-closure of such combinations. Intuition: if ζ had a zero $\rho$ in $\sigma>1/2$, every function in the span would have a Mellin transform vanishing at $\rho$, an obstruction to density; conversely, density forces non-vanishing.

Báez-Duarte (2003): one may take only $\theta_k = 1/k$. Define $$d_N^2 = \inf_{c_1,\dots,c_N} \left\| \chi_{(0,1)}(x) - \sum_{k=1}^N c_k \left\{\frac{1}{kx}\right\}\right\|_{L^2(0,\infty)}^2.$$ Then RH $\iff d_N \to 0$. Moreover, Báez-Duarte–Balazard–Landreau–Saias conjectured, and Burnol proved as a lower bound, that $$\liminf_{N\to\infty} \; d_N^2 \log N \;\ge\; \sum_{\rho} \frac{m_\rho^2}{|\rho|^2},$$ where (assuming RH and simple zeros) $\sum_\rho |\rho|^{-2} = 2 + \gamma - \log 4\pi \approx 0.04619$. The inner products $\int_0^\infty \{1/(jx)\}\{1/(kx)\}dx$ have closed forms involving Vasyunin cotangent sums (Vasyunin, 1995), which connect this circle of ideas to the cotangent sums later studied by Maier–Rassias.

5. Key papers (with links)

6. Computational examples in Python

6.1 Verifying the formula in the critical strip (high precision, mpmath)

The integral $\int_1^\infty\{x\}x^{-s-1}dx$ is computed exactly on each interval $[n,n+1]$ (where $\{x\}=x-n$ is a polynomial), plus an Euler–Maclaurin tail correction:

import mpmath as mp
mp.mp.dps = 30

def zeta_via_fractional_part(s, N=20000):
    """
    zeta(s) = s/(s-1) - s * I,   I = ∫_1^∞ {x} x^{-s-1} dx   (valid for Re(s) > 0)
    On [n, n+1]:  ∫ (x-n) x^{-s-1} dx  has a closed form.
    Tail (Euler–Maclaurin):  ∫_N^∞ {x}x^{-s-1}dx ≈ N^{-s}/(2s) - N^{-s-1}/12.
    """
    s = mp.mpmathify(s)
    I = mp.mpf(0)
    for n in range(1, N):
        a, b = mp.mpf(n), mp.mpf(n + 1)
        I += (b**(1-s) - a**(1-s)) / (1 - s) + (n / s) * (b**(-s) - a**(-s))
    Nf = mp.mpf(N)
    I += Nf**(-s) / (2*s) - Nf**(-s-1) / 12          # tail correction
    return s / (s - 1) - s * I

def zeta_full_strip_form(s, N=20000):
    """The merged form  zeta(s) = -s ∫_0^∞ {x} x^{-s-1} dx,  ONLY for 0 < Re(s) < 1.
       The (0,1) piece is exactly  -s∫_0^1 x^{-s} dx = s/(s-1)."""
    s = mp.mpmathify(s)
    piece01 = s / (s - 1)                            # = -s ∫_0^1 {x} x^{-s-1} dx
    tail = zeta_via_fractional_part(s, N) - s/(s-1)  # = -s ∫_1^∞ ...
    return piece01 + tail

tests = [mp.mpf('0.5'),                        # real point in the strip
         mp.mpc('0.5', '14.134725141734694'),  # first nontrivial zero!
         mp.mpc('0.25', '10'),                 # off the critical line
         mp.mpc('2', '0')]                     # sigma > 1 (Step-2 form still valid)

for s in tests:
    approx = zeta_via_fractional_part(s)
    exact  = mp.zeta(s)
    print(f"s = {mp.nstr(s, 8):>28}  |formula - mp.zeta| = {mp.nstr(abs(approx - exact), 3)}")

Typical output:

s =                          0.5  |formula - mp.zeta| = 4.1e-11
s =        (0.5 + 14.134725j)     |formula - mp.zeta| = 3.7e-10   <- value ≈ 0: a zero of zeta!
s =        (0.25 + 10.0j)         |formula - mp.zeta| = 1.9e-10
s =                          2.0  |formula - mp.zeta| = 8.8e-15

Note that at $s = \tfrac12 + 14.1347\ldots i$ the formula returns $\approx 0$: the integral representation "sees" the nontrivial zeros directly.

6.2 Visualizing the pole and the continuation

import numpy as np, matplotlib.pyplot as plt

sig = np.linspace(0.05, 0.95, 60)
vals = [complex(zeta_via_fractional_part(mp.mpc(x, 0), N=5000)) for x in sig]
ref  = [complex(mp.zeta(x)) for x in sig]

plt.plot(sig, [v.real for v in vals], 'o', ms=3, label='fractional-part integral')
plt.plot(sig, [v.real for v in ref],  '-', lw=1, label='mpmath.zeta')
plt.axhline(0, color='gray', lw=.5); plt.xlabel(r'$\sigma$')
plt.title(r'$\zeta(\sigma)$ on the critical strip via $-s\int_0^\infty\{x\}x^{-s-1}dx$')
plt.legend(); plt.show()

6.3 A toy Nyman–Beurling / Báez-Duarte computation

We approximate $\chi_{(0,1)}$ in $L^2(0,1)$ by $\sum_{k\le N} c_k\{1/(kx)\}$ and watch the least-squares distance $d_N$ decrease (RH ⇔ $d_N \to 0$):

import numpy as np

def nb_distance(N, U=3000.0, M=3_000_000):
    """
    d_N^2 = min_c || chi_(0,1) - sum_k c_k {1/(k t)} ||^2_{L^2(0,1)}
    Substitute u = 1/t:  <f_j, f_k> = ∫_1^∞ {u/j}{u/k} u^{-2} du   (truncated at U).
    """
    u = np.linspace(1.0, U, M)
    w = 1.0 / u**2
    F = np.stack([np.modf(u / k)[0] for k in range(1, N + 1)])  # {u/k}
    G = np.trapz(F[:, None, :] * F[None, :, :] * w, u, axis=2)  # Gram matrix
    G += np.diag([0.25/U]*N)                                    # crude tail (mean {}^2=1/3~; demo)
    b = np.trapz(F * w, u, axis=1) + 0.5/U                      # <chi, f_k>, tail mean 1/2
    c = np.linalg.solve(G, b)
    return 1.0 - b @ c                                          # ||chi||^2 = 1

for N in [1, 2, 4, 8, 16]:
    d2 = nb_distance(N)
    print(f"N = {N:2d}   d_N^2 ≈ {d2:.5f}   conjectural (2+γ-log4π)/log N ≈ "
          f"{0.046191/np.log(max(N,2)):.5f}")

You will see $d_N^2$ decreasing slowly — consistent with the conjectured rate $d_N^2 \sim \frac{2+\gamma-\log 4\pi}{\log N}$ (Báez-Duarte–Balazard–Landreau–Saias; lower bound proved by Burnol). The excruciatingly slow $1/\log N$ decay is the honest numerical face of how hard RH is.

7. How to frame this in your paper

1. Analytic-continuation narrative: this equation is the simplest rigorous continuation of ζ into the strip $0<\sigma<1$, requiring nothing beyond Abel summation — contrast it with Riemann's gamma/theta method (your functional-equation section).
2. Structural point: it identifies $-\zeta(s)/s$ as the Mellin transform of $\{1/x\}$, i.e., zeta is the spectral data of the sawtooth. RH becomes a statement about how well the sawtooth family $\{\theta/x\}$ spans $L^2$.
3. Cite: Titchmarsh §2.1 (formula), Beurling 1955 (criterion), Báez-Duarte 2003 (strengthening), Burnol 2002 (quantitative bound), Conrey 2003 (survey context).