On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function

Harald Grobner

REVIEW

On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function

Harald Grobner

Number Theory · Submitted 2 Sep 2026

Pith's verdict: A clean, credible proof that the conventional zero-indexed Riemann–von Mangoldt formula diverges, with one imported estimate worth checking before publication.

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arXiv ID2609.02713 (version 2)

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CitationHarald Grobner. "On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function." arXiv:2609.02713 (2026). https://pith.science/paper/KH3JUSLC

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Editor's note

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Dear colleague,

The paper does what it says: it proves that the conventional zero-indexed sum Σ_ρ R(x^ρ) in the Riemann–von Mangoldt formula diverges for each fixed x>1, with limsup of the partial sums infinite, and similarly for the trivial zeros. That is genuinely new relative to Zagier and Elliott, who flagged the interchange-of-limits issue but did not settle it. The main mechanism is clean: a truncated Mellin transform whose continuation has poles at nontrivial zeros forces an auxiliary function P_A to grow faster than any T^θ below the real part of such a zero, and a contour decomposition transfers that growth to the partial sums. The residue, Mellin, and Fourier steps are worked out in detail. I found no circularity and no fitted parameters; the cited inputs are standard or at least external.

The soft spot is exactly the one you flagged. Lemma 2.3, hence Lemma 2.5, uses Elliott's Prop. 4.5.2: ERi decays faster than any power of log|z| uniformly in fixed vertical strips. That estimate is imported without proof, and it is load-bearing. If it failed in the relevant strip, the horizontal contribution would not be o(1), and the connection between ΣR_T(x) and P_A would collapse. I do not think this is a reason to reject: the estimate is plausible, it is explicitly cited to a recent monograph, and the rest of the paper shows the author can prove such things. But a referee should ask for the statement to be verified, ideally a proof or a precise pointer in [Ell25]. A smaller point is the Fourier-limit step in Prop. 2.6: it is delicate because Q_c has infinitely many jumps accumulating at 0. The argument handles this by working on compact subintervals and using bounded variation there, and I think that is adequate.

Overall, the central claim holds up on my reading. The result matters for textbook presentations and for any formalization of explicit formulas; it does not change the analytic content of prime number theory. The paper deserves a serious referee. I would send it out and expect it to come back publishable once the Elliott estimate has been double-checked.

Referee report

Major objections (4)

[§2.2 (Lemma 2.3 and Lemma 2.5)] The entire proof that the horizontal contour contribution H_{a,c}(T')=o(1) depends critically on the estimate (2.4) from [Ell25, Prop. 4.5.2], which asserts that ERi(z) decays faster than any power of log|z| uniformly in any fixed vertical strip. This estimate is not proved in the manuscript; it is cited. If this estimate fails in the strip Re(z) between -a log x and c log x (the strip relevant for the contour), the bound on the horizontal integrals would not hold, and the key identity (3.2) connecting ΣR_T(x) and P_A(log x T) would not follow. Consequently the main theorem would not be reachable. The paper should either provide a self-contained proof of the needed uniform decay, or at least give a detailed justification of why the estimate can be trusted, given its specialized and recent nature.
[§1.3 (Proposition 1.7)] The meromorphic continuation of H_A(z) uses the series expansion (1.8) and relies on the claim that the series Σ_{k≥0} (-1)^k M_A(2k)/((2k)! (2k+1-z)) converges absolutely and locally uniformly on C\{positive odd integers}. The justification is only sketched (a boundedness of M_A(2k) and factorial decay). A more detailed argument should be provided, for instance showing that (2k+1-z) remains bounded away from zero for k large on any compact set, or at least stating explicitly that the series converges normally.
[§2.3 (Proposition 2.6)] The Fourier analysis relies on the claim that Q_c(u) is of bounded variation on every compact subinterval of (0,1]. The function Q_c(u) = e^{log x c u} Q(u) with Q(u)=u^{-1}B(u^{-1}) and B(y)=Σ_{n>y} μ(n)/n. The Möbius sum introduces jumps at u=1/n. The statement that Q_c is of bounded variation should be justified or a reference given. Without this justification the application of Fourier's single-integral theorem (Titchmarsh, Thm.12) is not fully supported.
[§4.1 (Proposition 4.1)] In the proof for the trivial zeros, the condition v0=2 log(x) δ < 2π is imposed. Since x>1 is fixed, one can indeed choose δ small enough. This should be stated explicitly: 'choose δ>0 so small that 2 log(x) δ < 2π'. The current text simply says 'choose 0<δ<1 so small that v0:=2 log(x)δ<2π', which is acceptable if one mentions that such a δ exists because 2 log(x) is finite. The sentence should be clarified.

Minor objections (5)

[(Title page)] The reference list spells 'Fourier Integrals' as 'Fourier Integrals' in [Tit48]; the correct title is 'Introduction to the Theory of Fourier Integrals' (actually 'Integrals' is correct, but the type-setting has a typographical error in the citation).
[Proof of Lemma 2.5] The authors choose B>3 to obtain o(1); the choice should be stated earlier in the lemma statement or in the proof. It is clear but could be made explicit.
[§2.3, after (2.7)] The notation A_0(y) is introduced but not used later; it might be omitted or integrated directly.
[§2.4, line after (2.14)] The constant Ca,x is said to be independent of T, but its construction absorbs several terms. It would be helpful to state that Ca,x = (constant from compact interval) + (constant from i(a+1/2)/t term) + (constant from Dirichlet series) + C′_a,x.
[General] The paper uses the term 'Riemann's constant' for Θ; while this is a recent convention from [Ell25], it might be better to specify 'the supremum of the real parts of non-trivial zeros' the first time it is used.

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The paper

The paper claims

The paper proves that the partial sums over non-trivial zeros in the Riemann–von Mangoldt explicit formula are unbounded, so the series diverges and the formula as usually stated is not valid.

The reading

The paper challenges the widely used Riemann–von Mangoldt explicit formula for the prime-counting function. It proves that, for every fixed x>1 and any exponent θ less than Riemann's constant Θ, the partial sum ΣR_T(x) over non-trivial zeros ordered by increasing imaginary part is not O(T^θ). Consequently, the limit superior of |ΣR_T(x)| is infinite and the series diverges. This means that the expression ∑_ρ R(x^ρ) in the conventional ordering is not a well-defined complex number and the identity π₀(x)=R(x)−∑_ρ R(x^ρ) cannot hold with that summation. The argument also shows that the contribution of the trivial zeros diverges, further undermining the standard form of the formula.

What carries the argument

The key machinery is the auxiliary function P_A(t) = ∫_1^t log(y/(2π log x)) Re(ERi(−A+iy)) dy and a contour integral decomposition that relates ΣR_T(x) to P_A(log x T). The growth of P_A is controlled by poles of a truncated Mellin transform at the non-trivial zeros of ζ(s); Proposition 1.13 shows P_A(t) cannot be O(t^θ) for θ < Re(ρ). The contour integral, using a choice of good ordinates where horizontal contributions vanish (Lemma 2.5), yields ΣR_T(x) = (1/(π log x)) P_A(log x T) + constant + o(1), transferring the growth obstruction to ΣR_T(x).

Extended reading notes

Core claim

The central discovery is that for each fixed x>1 and any θ<Θ (where Θ is the supremum of real parts of non-trivial zeros, called Riemann’s constant), the partial sum ΣR_T(x) = ∑_{0<|Im(ρ)|≤T} m_ρ ERi(ρ log x) is not O(T^θ). Hence limsup_{T→∞}|ΣR_T(x)|=∞, so the series diverges. This implies that the expression ∑_ρ R(x^ρ) in the conventional ordering is not a well-defined complex number and the identity π₀(x)=R(x)−∑_ρ R(x^ρ) cannot hold with that summation. The argument also extends to show the divergence of the contribution of the trivial zeros.

Load-bearing premise

The proof that the horizontal contour sides vanish (Lemma 2.5) depends on the estimate that ERi(z) decays faster than any power of log|z| uniformly in every fixed vertical strip as |Im z|→∞ (cited from [Ell25, Prop. 4.5.2]). If this estimate fails in the strip needed for the argument, the relation between ΣR_T(x) and P_A would not hold, and the contradiction would collapse.

Editorial extensions

If this is right

The standard explicit formula for π₀(x) as usually presented in textbooks and online sources would need to be revised; the sum over zeros in the conventional ordering is not convergent.
Numerical computations that rely on the truncated sum ΣR_T(x) as an approximation of π₀(x)−R(x) must be interpreted with caution; the limit does not exist.
The result underscores that the interchange of summation over n and over ρ in deriving the formula is invalid unless additional summability methods are specified.
The divergence of the trivial-zero contribution (Proposition 4.1) further shows that even the second sum in the formula does not converge in the usual sense.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the paper is correct, then any use of the formula π(x) = R(x) − ∑ R(x^ρ) in heuristic or asymptotic arguments must be understood as a formal identity only, not as a convergent series in the standard sense. This raises questions about many applications that implicitly assume convergence.
The proof isolates the imported estimate from Elliott (2025) as the most delicate external assumption; a verification or independent proof of that estimate would strengthen the conclusion.
The auxiliary function P_A and its growth obstruction may provide a new testable condition: if numerical experiments could compute P_A(t) to high t, they should see oscillations that do not decay like any power less than Θ.
The divergence result does not preclude the existence of a different summation order (e.g., summing over zeros grouped by real part) that yields convergence; that remains an open question.

Original abstract

An explicit formula for the prime-counting function $\pi(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $\pi(x)=R(x)-\sum_\rho R(x^\rho)$, where the sum runs over all zeros $\rho$ of the Riemann $\zeta$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, $\Sigma R_T(x):=\sum_{0<|\Im m(\rho)|\le T} R(x^\rho)$ converge as ${T\to\infty}$. Writing $\Theta:=\sup\{\Re e(\rho):\ \zeta(\rho)=0,\ 0<\Re e(\rho)<1\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $\theta<\Theta$, the sums $\Sigma R_T(x)$ are not $O(T^\theta)$. As a consequence, $\limsup_{{T\to\infty}}|\Sigma R_T(x)|=\infty$ and $\sum_\rho R(x^\rho)$ diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the analogous result -- and hence divergence -- for the contribution of the trivial zeros to $\sum_\rho R(x^\rho)$. In both cases the exponent $\Theta$ is proved to be sharp.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,702 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.73 · compares with prior results 0.27 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

An explicit formula for the prime-counting function $\pi(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $\pi(x)=R(x)-\sum_\rho R(x^\rho)$, where the sum runs over all zeros $\rho$ of the Riemann $\zeta$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, $\Sigma R_T(x):=\sum_{0<|\Im m(\rho)|\le T} R(x^\rho)$ converge as ${T\to\infty}$. Writing $\Theta:=\sup\{\Re e(\rho):\ \zeta(\rho)=0,\ 0<\Re e(\rho)<1\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $\theta<\Theta$, the sums $\Sigma R_T(x)$ are not $O(T^\theta)$. As a consequence, $\limsup_{{T\to\infty}}|\Sigma R_T(x)|=\infty$ and $\sum_\rho R(x^\rho)$ diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the divergence of the contribution of the trivial zeros to $\sum_\rho R(x^\rho)$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	An explicit formula for the prime-counting function $\pi(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $\pi(x)=R(x)-\sum_\rho R(x^\rho)$, where the sum runs over all zeros $\rho$ of the Riemann $\zeta$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity.	background 1.00	0.04	0.03	asserted 1.00
s1	This particularly entails the claim that the partial sums over the non-trivial zeros, $\Sigma R_T(x):=\sum_{0<\|\Im m(\rho)\|\le T} R(x^\rho)$ converge as ${T\to\infty}$.	background 0.95	0.06	0.05	asserted 0.99
s2	Writing $\Theta:=\sup\{\Re e(\rho):\ \zeta(\rho)=0,\ 0<\Re e(\rho)<1\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $\theta<\Theta$, the sums $\Sigma R_T(x)$ are not $O(T^\theta)$.	result 0.95	0.94	0.21	asserted 1.00
s3	As a consequence, $\limsup_{{T\to\infty}}\|\Sigma R_T(x)\|=\infty$ and $\sum_\rho R(x^\rho)$ diverges.	result 0.90	0.92	0.81	asserted 1.00
s4	We conclude the paper by showing that an adapted, but simpler strategy also gives the divergence of the contribution of the trivial zeros to $\sum_\rho R(x^\rho)$.	result 0.86	0.95	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v1

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