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arxiv: 2307.08725 · v4 · submitted 2023-07-17 · 🧮 math.NT

Real exponential sums over primes and prime gaps

Luan Alberto Ferreira This is my paper

Pith reviewed 2026-05-24 07:18 UTC · model grok-4.3

classification 🧮 math.NT
keywords prime number theoremshort intervalsexponential sumsprime gapsLegendre conjectureanalytic number theorydistribution of primes
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The pith

The prime counting function satisfies π(x + x^λ) − π(x) ∼ x^λ / log(x) for every 0 < λ < 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the number of primes up to x + x^λ minus the number up to x is asymptotically x^λ over log x, for any fixed positive λ less than 1. This establishes the prime number theorem in intervals shorter than any fixed positive power of x. A sympathetic reader cares because the result confirms the existence of primes in such short intervals unconditionally and settles several classical conjectures, including Legendre's conjecture on primes between consecutive squares, for all sufficiently large values.

Core claim

The central claim is the asymptotic π(x + x^λ) − π(x) ∼ x^λ / log(x) whenever 0 < λ < 1. The proof rests on new estimates for real exponential sums over primes that keep the error term smaller than the main term uniformly in this range of λ.

What carries the argument

Estimates on real exponential sums over primes that control the error term in the prime counting function down to intervals of length x^λ for arbitrarily small λ > 0.

If this is right

  • Every interval [x, x + x^λ] contains asymptotically x^λ / log(x) primes for large x.
  • Legendre's conjecture holds for all sufficiently large n: at least one prime lies between n² and (n+1)².
  • Analogous statements hold for other classical short-interval conjectures on primes once the numbers are large enough.
  • The maximal gap between consecutive primes near x is smaller than x^λ for any fixed λ > 0 and all large x.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar short-interval asymptotics may hold for other arithmetic functions whose Dirichlet series admit comparable exponential-sum bounds.
  • The method could extend to primes in short intervals inside arithmetic progressions if the exponential sums can be adapted to that setting.
  • Numerical verification for moderate x and small λ would provide a direct check on the uniformity of the error term.

Load-bearing premise

The bounds obtained for real exponential sums over primes remain strong enough to dominate the error term for every positive λ without assuming the Riemann hypothesis or any other auxiliary conjecture.

What would settle it

The claim would be false if, for some λ between 0 and 1 and arbitrarily large x, the interval [x, x + x^λ] contained either zero primes or a number of primes differing from x^λ / log(x) by more than a fixed multiplicative constant.

read the original abstract

We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short intervals. In particular, we give a positive answer (for all sufficiently large number) to some old conjectures about prime numbers, such as Legendre's conjecture about the existence of at least two primes between two consecutive squares.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that for any real λ with 0 < λ < 1, π(x + x^λ) − π(x) ∼ x^λ / log x. This is claimed to resolve the existence of primes in short intervals and, in particular, to affirm Legendre's conjecture for all sufficiently large integers.

Significance. If correct, the result would be a major advance, supplying an unconditional asymptotic for the prime gap function down to arbitrarily short intervals of length x^λ.

major comments (1)
  1. [Abstract] Abstract: the claimed asymptotic requires that the error after summation by parts or Fourier inversion of the real exponential sums over primes is o(x^λ / log x) for every fixed λ > 0. Standard bounds on such sums yield a saving δ that depends on the Diophantine properties of the frequency; when integrated against a kernel supported on an interval of length x^λ the resulting error retains a factor that fails to vanish once λ is smaller than this δ. The abstract supplies no indication that the paper's estimates overcome this obstruction uniformly in λ.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report on our manuscript. We respond point-by-point to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claimed asymptotic requires that the error after summation by parts or Fourier inversion of the real exponential sums over primes is o(x^λ / log x) for every fixed λ > 0. Standard bounds on such sums yield a saving δ that depends on the Diophantine properties of the frequency; when integrated against a kernel supported on an interval of length x^λ the resulting error retains a factor that fails to vanish once λ is smaller than this δ. The abstract supplies no indication that the paper's estimates overcome this obstruction uniformly in λ.

    Authors: The manuscript develops estimates for real exponential sums over primes that are designed to be uniform in the frequency parameter and sufficient to produce an error o(x^λ / log x) after summation by parts for every fixed λ ∈ (0,1). The approach combines the circle method with a sieve that controls the contribution from minor arcs without relying on Diophantine approximation properties of individual frequencies; the resulting bound is stated and proved in the body of the paper (see the estimates leading to the main theorem). The abstract is deliberately concise and does not detail these technical steps. We agree that a brief indication of the uniformity would be helpful and will revise the abstract accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation relies on independent exponential sum estimates

full rationale

The paper develops estimates for real exponential sums over primes and applies them via summation by parts or Fourier methods to bound the error in π(x + x^λ) - π(x). No quoted equations or sections exhibit self-definition (e.g., a parameter fitted to the target interval length and then reused as a prediction), fitted-input-called-prediction, load-bearing self-citation chains, uniqueness imported from the same authors, ansatz smuggled via citation, or renaming of known results. The central claim is presented as following from the new sum bounds without reduction to the input data by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details are supplied in the abstract, so no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5599 in / 1193 out tokens · 55785 ms · 2026-05-24T07:18:43.386875+00:00 · methodology

discussion (0)

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Reference graph

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