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Open research questions in Analytic Number Theory Research

118 unresolved questions extracted from the limitations and future-work sections of 536 Analytic Number Theory Research papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • The need for new proofs using different Ramsey-theoretic results - The gap in the existing literature deriving the infinitude of primes from Ramsey-type theorems

    Folkman's theorem and the primes · 2026 · DOI

  • The existing large sieve inequality is not explicit. There is a need for an explicit version of Heath-Brown's large sieve inequality for quadratic characters.

    Explicit quadratic large sieve inequality · 2026 · DOI

  • Further study of the sequence of ratios f(2n)/f(n) is needed. Further study of the properties of Mersenne numbers is needed. Development of new algorithms for computing the number of divisors is needed.

    On the Number of Divisors of Mersenne Numbers · 2026 · DOI

  • The current state-of-the-art cannot handle sums of the classical arithmetic functions τ, σ, φ. There is a lack of understanding of the behavior of the sequence of ratios f(2n)/f(n). There is a need for new insights into the properties of Mersenne numbers.

    On the Number of Divisors of Mersenne Numbers · 2026 · DOI

  • The length of the Dirichlet polynomial (X) is much larger than the range of the parameter t. The multiplicative structure of the coefficients of the Dirichlet polynomial makes it challenging to compute its mean value. The need to bypass strong twin-prime like conjectures.

    Conditional mean values of long Dirichlet polynomials · 2026 · DOI

  • There is a gap in the current understanding of the distribution of cubic character sums. Prior work has not provided a satisfactory estimate for the proportion of characters for which the maximum is greater than a given value.

    The distribution of the maximum of cubic character sums · 2026 · DOI

  • The conjecture is unproven. The author has conducted numerical verifications on a restricted domain which have proven successful, but this does not constitute a proof. The validity of the conjecture for all integers remains to be established.

    A Conjecture on the Structure of Indices Generating Prime Numbers via a Conditional Arithmetic Sequence · 2026 · DOI

  • Future steps consist of exhaustive numerical validation over large domains and, ideally, an attempt at a formal proof. The approach could be explored for other arithmetic progressions, for example for moduli 4, 6, 30 or larger moduli.

    A Conjecture on the Structure of Indices Generating Prime Numbers via a Conditional Arithmetic Sequence · 2026 · DOI

  • The central error in previous versions (V1.0-V1.3) was the assumption that Δ(H,A) ≤ K(P_A)•H^{D-1} + O(H^{D-2}).- The correct order is Δ(h,A) = Θ(h^d) for non-unimodular polytopes.

    Ehrhart Theory and Discrete Gaps in Minkowski Sumsets: Asymptotic Lattice Density and Refutation of O(h^{d-1}) Bounds (Version 1.4) · 2026 · DOI

  • Further study of the distribution of values of the newform Dedekind sums is needed. The paper's methods may be applicable to other problems in mathematics.

    Large values of newform Dedekind sums · 2026 · DOI

  • The distribution of values of the newform Dedekind sums remains open. There is a lack of understanding of the properties of the generalized Dedekind sum.

    Large values of newform Dedekind sums · 2026 · DOI

  • The paper assumes the 3x+1 conjecture holds and the Weak Residue Conjecture is valid. The results are limited to the case where N is a positive integer. The paper does not provide a complete proof of the 3x+1 conjecture.

    Iteration Steps of 3x + 1 Problem · 2026 · DOI

  • Future research could focus on providing a complete proof of the 3x+1 conjecture. The authors suggest that further study of the qx+1 problem could be fruitful. The paper suggests that the results could be used to improve our understanding of dynamical systems and number theory.

    Iteration Steps of 3x + 1 Problem · 2026 · DOI

  • The gap in prior work is the limited range of values for which the infinitude of Piatetski-Shapiro primes has been established. The paper identifies the need for a more general result on Diophantine approximation with restricted denominators.

    Diophantine approximation with Piatetski-Shapiro primes · 2026 · DOI

  • Future research directions include exploring the implications of the results for cryptography and coding theory. Investigating the potential applications of the spectral projection mechanism in other areas of mathematics and physics.

    PRIME NUMBERS AS SPECTRAL ARTIFACTS OF QUANTUM GEOMETRIC SYSTEMS · 2026 · DOI

  • The traditional treatment of integers and their prime constituents as fundamental mathematical objects is challenged. The discrete foundation of number theory may represent an emergent phenomenon rather than a fundamental reality.

    PRIME NUMBERS AS SPECTRAL ARTIFACTS OF QUANTUM GEOMETRIC SYSTEMS · 2026 · DOI

  • The lack of a general method for evaluating definite integrals of floor and ceiling functions. The lack of a simple formula for evaluating the accumulation function for the floor function.

    Integration of Floor and Ceiling · 2026 · DOI

  • Further study of the prime k-tuple conjecture and the second Hardy-Littlewood conjecture. Development of new methods for studying the distribution of prime numbers. Exploration of the implications of the result for other areas of number theory.

    Compatibility Analysis of Hardy– Littlewood Dual Conjectures Based on CRT Primitive Product Periodic Structure · 2026 · DOI

  • The Hensley-Richards incompatibility argument has a gap in its proof. The gap is due to the invalid assumption that remote intervals can break the leftmost density benchmark.

    Compatibility Analysis of Hardy– Littlewood Dual Conjectures Based on CRT Primitive Product Periodic Structure · 2026 · DOI

  • The need for an extra condition to specify the form of the limit measure. The complexity of the theorem. The need for a discrete result.

    A Discrete Limit Theorem for a Collection of Epstein Zeta-Functions · 2026 · DOI

  • The paper assumes the Generalized Riemann Hypothesis. The authors note that their results are limited to the case of cubic characters.

    The distribution of the maximum of cubic character sums · 2026 · DOI

  • The transcendency of Mills' constant remains unknown. The study of Mills' constant and its variants is ongoing.

    Transcendency of variants of Mills’ constant · 2026 · DOI

  • The lack of understanding of the 1-to-1 correspondence between even and odd zeros of the Riemann zeta function.

    Odd Values of the Riemann Zeta Function · 2026 · DOI

  • Extending the results to other types of definite integrals. Applying the results to real-world phenomena that involve floor and ceiling functions.

    Integration of Floor and Ceiling · 2026 · DOI

  • The covering radius bound derivation applies specifically to n≥5 with n≢2 (mod 4) per Lemma 10; the paper does not address whether tighter or alternative bounds can be established for the excluded cases (small n or n≡2 (mod 4)) or provide explicit calculations for these boundary conditions.

    An improved upper bound on the covering radius of the logarithmic lattice of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mi mathvariant="double-struck">Q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>ζ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> · 2026 · DOI

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118 open questions have been extracted from the limitations and future-work passages of 536 Analytic Number Theory Research papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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