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Open research questions in Analytic Number Theory Research

186 gap statements mined from Analytic Number Theory Research papers in our 4.5M-paper local library, which holds 940 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

What the literature leaves open

  • The Riemann Hypothesis has remained unproven for over 160 years. The distribution of prime numbers is complex and not fully understood. The Riemann zeta function is a complex and intricate mathematical object.

    Riemann Hypothesis: The Most Definitive and Comprehensive Disproof Ever Constructed · 2026 · DOI

  • His work addresses major open problems, including the P = NP conjecture, the Birch and Swinnerton-Dyer conjecture, and other fundamental conjectures at the interface of algebra, logic, and graph theory.

    Study and Consequences of the Ⓢ Function on the Riemann Hypothesis · 2026 · DOI

  • The paper does not require q to be squarefree or cubefree, but does not discuss other potential limitations. The results are limited to certain degree 3 and degree 6 L-functions.

    Strong hybrid subconvexity for twisted selfdual $${\text {GL}}_3$$ L-functions · 2026 · DOI

  • Future research may involve extending the results to other types of L-functions. Future research may also involve applying the paper's approach to other problems in number theory.

    Strong hybrid subconvexity for twisted selfdual $${\text {GL}}_3$$ L-functions · 2026 · DOI

  • The study of gaps of cyclotomic polynomials is a complex problem. The development of a new approach to studying the gaps of binary cyclotomic polynomials is challenging. The analysis of the structure of binary cyclotomic polynomials is difficult.

    Gaps in Binary Cyclotomic Polynomials · 2026 · DOI

  • Further study of the gaps of cyclotomic polynomials is needed. The development of new approaches to studying the gaps of binary cyclotomic polynomials is necessary. The application of the results of the paper to cryptographic systems should be explored.

    Gaps in Binary Cyclotomic Polynomials · 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

  • Future research can focus on obtaining an explicit formula for the constant H_a. Future research can also focus on studying the properties of integers represented by the quadratic form x^2+3y^2 and the numbers represented as a sum of two squares.

    GAPS BETWEEN HEXAGONAL AND SQUARE LATTICES · 2026 · DOI

  • There is a gap in the existing literature regarding the asymptotic formula for the number of elements of the set S(2, 2, a) in short intervals. The existing literature does not provide a explicit formula for the constant H_a.

    GAPS BETWEEN HEXAGONAL AND SQUARE LATTICES · 2026 · DOI

  • Its spectrum remains to be determined. Whether these polynomials converge to 1/zeta(s) is a precise mathematical conjecture — equivalent to RH — that is offered to the mathematical community as an open problem.

    The No-Vacuum Principle and the Riemann Hypothesis: A Proved Algebraic Theorem, the Seifert Matrix Construction from Prime Numbers, and the Hermitian Operator H₅₉ · 2026 · DOI

  • The paper aims to address this gap by obtaining improved bounds on Romanov's constant and refining the numerical bounds for the Linnik-Goldbach problem.

    An update on the Linnik--Goldbach and Romanov problems · 2026

  • The previous bound for the q-adic valuation of σ_k(n) is not asymptotically optimal when n is large and k ≥ q-2. There is a need for an improved upper bound for the q-adic valuation of σ_k(n). The paper aims to fill this gap using a new approach based on the LTE lemma and cyclotomic polynomials.

    A note on the $q$-adic valuation of $σ_k(n)$ · 2026

  • The study of prime values in sparse integer sequences is a challenging problem. The authors need to improve upon the previous range of γ obtained by Li and Pan. The use of nilsequence estimates and pseudorandom majorants requires careful analysis.

    Linear equations in Piatetski-Shapiro primes · 2026

  • Future research can aim to improve upon the current estimate by using a better zero-density estimate for Dirichlet L-functions. Future research can aim to prove an unconditional asymptotic formula for U = log^(4+ε)N. Future research can aim to study the properties of prime numbers and their distribution using the result.

    On the Goldbach problem with restricted primes · 2026

  • The paper aims to fill this gap by proving an asymptotic formula for the number of representations of an odd integer as the sum of three primes, one of which is smaller than a given U.

    On the Goldbach problem with restricted primes · 2026

  • The asymptotic behavior and density of the solution set were not well understood. The existence of solutions for the equation σ(n+1) = kσ(n) was not fully investigated.

    A generalization of the Erdős-Sierpiński conjecture · 2026

  • The paper seeks to improve a result of de Araujo and provide a new bound on the covering radius.

    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

  • 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

  • Further study of the distribution of prime numbers. Exploration of the implications of the disproof of the Riemann Hypothesis for number theory and related fields.

    Riemann Hypothesis: The Most Definitive and Comprehensive Disproof Ever Constructed · 2026 · DOI

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

    Folkman's theorem and the primes · 2026 · DOI

  • Further study of the port construction and its applications. Investigation of the uniqueness of the nine-prime-factor example. Exploration of the implications of the five-splitting Hardy-Littlewood-Bateman-Horn prime-points hypothesis.

    Port Fillings for Primary Pseudoperfect Numbers · 2026

  • Further study of the Piatetski-Shapiro sequence and its properties. Application of the authors' methods to other problems in analytic number theory. Investigation of the range of γ for other types of sequences.

    Linear equations in Piatetski-Shapiro primes · 2026

  • The paper identifies a gap in the understanding of the Riemann zeta function and its relation to the Riemann Hypothesis. The authors seek to fill this gap with a new explicit product formula.

    An Explicit Product Formula for 1/\zeta(s) and its Equivalence to the Riemann Hypothesis · 2026 · DOI

Most-cited papers in Analytic Number Theory Research

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Open questions raised by papers listed on this page include: (1) The distribution of prime numbers is complex and not fully understood (“Riemann Hypothesis: The Most Definitive and Comprehensive Disproof Ever…”, 2026); (2) Future research may involve extending the results to other types of L-functions (“Strong hybrid subconvexity for twisted selfdual $${\text {GL}}_3$$ L-functions”, 2026); (3) Further study of the gaps of cyclotomic polynomials is needed (“Gaps in Binary Cyclotomic Polynomials”, 2026); (4) There is a need for an explicit version of Heath-Brown's large sieve inequality for quadratic characters (“Explicit quadratic large sieve inequality”, 2026); (5) Further study of the sequence of ratios f(2n)/f(n) is needed (“On the Number of Divisors of Mersenne Numbers”, 2026). 186 gap statements have been mined from Analytic Number Theory Research papers in our 4.5M-paper local library, which holds 940 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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