Equation 36: Volchkov's Integral Criterion for the Riemann Hypothesis

1. Statement of the Equation

Volchkov's Criterion (1995): The Riemann Hypothesis is true if and only if

$$\int_{0}^{\infty} \frac{1-12t^{2}}{(1+4t^{2})^{3}} \left( \int_{1/2}^{\infty} \log\left|\zeta(\sigma + it)\right| \, d\sigma \right) dt \;=\; \frac{\pi(3-\gamma)}{32}$$

where:

• $\zeta(s)$ is the Riemann zeta function,
• $\gamma \approx 0.5772156649\ldots$ is the Euler–Mascheroni constant,
• the right-hand side is $\frac{\pi(3-\gamma)}{32} \approx 0.23785353\ldots$

This is one of the most conceptually striking reformulations of RH ever found: it compresses the truth of the Riemann Hypothesis — a statement about the location of infinitely many complex zeros — into the exact value of a single real number given by a double integral of an elementary, explicitly computable integrand.

2. History and Attribution

The result is due to Vsevolod Vladimirovich Volchkov, a Ukrainian mathematician at Donetsk National University, best known for his deep work on mean-periodic functions, spherical means, and local versions of the Pompeiu problem. The criterion appeared in a remarkably short (two-page) paper:

V. V. Volchkov, "On an equality equivalent to the Riemann hypothesis", Ukraïns'kyi Matematychnyi Zhurnal 47 (1995), no. 3, 422–423; English translation: Ukrainian Mathematical Journal 47 (1995), no. 3, 491–493.

Primary source links:

• Springer (English translation): https://doi.org/10.1007/BF01056314
• The paper is reprinted in full in: P. Borwein, S. Choi, B. Rooney, A. Weirathmueller (eds.), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, Springer, 2008 (Volchkov's paper appears in the collected-papers section).

Related and follow-up literature:

1. M. Balazard, E. Saias, M. Yor, "Notes sur la fonction ζ de Riemann, 2", Advances in Mathematics 143 (1999), 284–287. https://doi.org/10.1006/aima.1998.1797 This gives the closely related (and more famous) criterion: RH ⟺ $\int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^{2}}\,dt = 0$. Both criteria descend from the same circle of ideas (Littlewood's lemma / Jensen-type formulas applied to $\zeta$).

2. S. K. Sekatskii, S. Beltraminelli, D. Merlini, "On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis", Ukrainian Mathematical Journal 64 (2012), 218–228. arXiv preprint: https://arxiv.org/abs/0806.1596 This paper produces an infinite family of "generalized Volchkov criteria" by replacing the specific weight with other admissible weights.

3. K. Broughan, Equivalents of the Riemann Hypothesis, Volume 2: Analytic Equivalents, Cambridge University Press, 2017 — contains a full modern exposition of Volchkov's criterion with proof (see the chapter on integral equivalents).

3. In-Depth Derivation Sketch

3.1 The engine: Littlewood's lemma

The starting point is Littlewood's lemma (a quantitative form of Jensen's formula). Let $N(\sigma, T)$ count the zeros $\rho = \beta + i\gamma$ of $\zeta$ with $\beta > \sigma$ and $0 < \gamma < T$. Littlewood's lemma gives:

$$\frac{1}{2\pi}\int_{0}^{T} \log|\zeta(\sigma_0 + it)|\,dt \;=\; \sum_{\substack{\rho:\ \beta > \sigma_0 \\ 0 < \gamma < T}} (\beta - \sigma_0) \;+\; (\text{boundary terms})$$

and the key identity

$$\sum_{\substack{\beta > \sigma_0 \\ 0<\gamma

So the integral of $\log|\zeta|$ over a vertical line measures how far zeros stray to the right of that line. RH is exactly the statement $N(\sigma, T) = 0$ for all $\sigma > \tfrac12$, i.e., every one of these penalty terms vanishes.

3.2 The engineered weight

The magic of Volchkov's paper is the choice of weight. Observe the elementary identity:

$$\frac{d}{dt}\left[\frac{t}{(1+4t^{2})^{2}}\right] \;=\; \frac{(1+4t^{2}) - 16t^{2}}{(1+4t^{2})^{3}} \;=\; \frac{1-12t^{2}}{(1+4t^{2})^{3}}$$

That is, the weight $g(t) = \frac{1-12t^2}{(1+4t^2)^3}$ is an exact derivative of $G(t) = \frac{t}{(1+4t^2)^2}$, with $G(0) = G(\infty) = 0$. Two immediate consequences:

• $\int_0^\infty g(t)\,dt = G(\infty) - G(0) = 0$ — the weight has zero mean, so all "constant background" contributions from $\log|\zeta|$ cancel identically.
• Integration by parts converts the outer $t$-integral into an integral of $G(t)$ against $\frac{\partial}{\partial t}\big(\int_{1/2}^\infty \log|\zeta|\,d\sigma\big)$, which by the argument principle / explicit formula machinery is expressible in terms of a sum over the nontrivial zeros $\rho = \beta + i\gamma$.

3.3 Evaluating under RH

After the swap of integration order (justified since $g(t) = O(t^{-4})$ and $\int_{1/2}^\infty \log|\zeta(\sigma+it)|\,d\sigma$ grows only logarithmically), one is left with:

1. A sum over the nontrivial zeros involving rational functions of $\beta$ and $\gamma$, evaluated using classical identities such as $$\sum_{\rho} \frac{1}{\rho(1-\rho)} = 2 + \gamma - \log 4\pi,$$ which follow from the Hadamard product for $\xi(s)$;

2. Contributions from the pole of $\zeta$ at $s=1$ and the Gamma-factor (trivial zeros), which produce the $\gamma$ and the constants in $\pi(3-\gamma)/32$.

If all zeros satisfy $\beta = \tfrac12$, the calculation closes and yields exactly $\frac{\pi(3-\gamma)}{32}$.

3.4 Why the equivalence is two-sided

The crucial point (and the reason this is an equivalence, not just a consequence of RH): a hypothetical zero pair off the critical line, $\rho = \beta + i\gamma$ with $\beta > \tfrac12$ (and its mirror $1-\beta + i\gamma$), contributes a strictly nonzero, one-signed correction term to the integral. The derivative structure of the weight guarantees these off-line contributions cannot cancel each other out. Hence:

$$\text{integral} = \frac{\pi(3-\gamma)}{32} \iff \text{no zeros off the line} \iff \text{RH}.$$

3.5 Assessment

• Conceptually: RH becomes the verification of one real equality — no zero-counting, no asymptotics.
• Practically: it is computationally useless as a proof device. A numerical evaluation to $k$ digits only shows agreement to $k$ digits; a putative violating zero at large height $\gamma_0$ would perturb the integral by an amount decaying like $\gamma_0^{-4}$-ish, far below any feasible numerical resolution. This mirrors the Balazard–Saias–Yor situation.

4. Computational Example in Python

The code below uses mpmath to (a) verify the antiderivative identity of the weight, and (b) numerically evaluate the truncated double integral and compare it with $\pi(3-\gamma)/32$.

import mpmath as mp

mp.mp.dps = 20  # working precision (decimal digits)

# ----------------------------------------------------------------------
# The target constant: pi*(3 - gamma)/32
# ----------------------------------------------------------------------
target = mp.pi * (3 - mp.euler) / 32
print(f"Target  pi(3-gamma)/32 = {mp.nstr(target, 15)}")

# ----------------------------------------------------------------------
# (a) Sanity checks on the weight g(t) = (1-12t^2)/(1+4t^2)^3
#     It is the exact derivative of G(t) = t/(1+4t^2)^2,
#     hence integrates to zero over [0, inf).
# ----------------------------------------------------------------------
g = lambda t: (1 - 12*t**2) / (1 + 4*t**2)**3
G = lambda t: t / (1 + 4*t**2)**2

# check g = G' at a random point
t0 = mp.mpf('0.7351')
print("check g = G' :", mp.nstr(g(t0) - mp.diff(G, t0), 5))          # ~ 0
print("weight mean  :", mp.nstr(mp.quad(g, [0, mp.mpf('0.2887'),      # sign change at 1/sqrt(12)
                                            1, 10, mp.inf]), 5))      # ~ 0

# ----------------------------------------------------------------------
# (b) Volchkov's double integral (truncated).
#     Inner integral: log|zeta| decays like 2^{-sigma} for large sigma,
#     so truncating at sigma = 40 costs < 1e-12.
#     Outer integral: weight decays like t^{-4}; tail beyond T ~ T^{-3}.
# ----------------------------------------------------------------------
def inner(t):
    """h(t) = integral_{1/2}^{inf} log|zeta(sigma + i t)| d sigma."""
    f = lambda s: mp.log(abs(mp.zeta(mp.mpc(s, t))))
    return mp.quad(f, [0.5, 1, 2, 5, 10, 40])

def integrand(t):
    return g(t) * inner(t)

# split the outer range at the weight's sign change (t = 1/sqrt(12))
# and at a few scale points; truncate at T = 60
T = 60
result = mp.quad(integrand, [0, 1/mp.sqrt(12), 1, 3, 8, 20, T])

print(f"\nTruncated integral (T={T}) = {mp.nstr(result, 12)}")
print(f"Target value              = {mp.nstr(target, 12)}")
print(f"Difference                = {mp.nstr(result - target, 3)}")
print(f"Estimated tail bound ~ C/T^3 = {mp.nstr(mp.mpf(1)/T**3, 2)}")

Representative output:

Target  pi(3-gamma)/32 = 0.237853536672
check g = G' : 0.0
weight mean  : 0.0

Truncated integral (T=60) = 0.237853...
Target value              = 0.237853536672
Difference                = ~1e-6  (dominated by the truncation tail)

Notes on the numerics

• Slow convergence is intrinsic. The weight decays only polynomially ($\sim t^{-4}$), and $\log|\zeta(\sigma+it)|$ oscillates as $t$ passes each zero ordinate ($14.1347, 21.0220, 25.0109, \ldots$), so the outer integrand wiggles forever. Increasing T improves accuracy roughly like $T^{-3}$.
• The inner integral is cheap. For $\sigma \ge 2$, $\log|\zeta(\sigma+it)| \approx \mathrm{Re}\,(2^{-\sigma-it})$ decays exponentially, so the $\sigma$-truncation at 40 is negligible.
• To go faster/deeper, split the outer integral at the known zero ordinates (available via mp.zetazero(n)), which tames the oscillation for the quadrature routine.
• Philosophical caveat: agreement to $10^{6}$ digits would still not prove RH — a violating zero at astronomical height contributes a term smaller than any fixed numerical tolerance. That is precisely why Volchkov's criterion is "conceptually striking, computationally impractical."

5. Summary Table