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Prime Number Error Terms

Nathan Ng

TL;DR

The paper proposes a unifying conjectural framework, GELI, for the true size of a broad class of prime-number error terms, unifying Montgomery’s conjecture for \(\psi(x)\) and the Mertens-type behavior of \(M(x)\) under a single principle. It builds on Lamzouri’s approach, modeling explicit prime-error sums by a random cosine sum and proving that GELI implies corresponding large/small value bounds; it also derives a general L^2 bound for almost periodic functions that strengthens previous results. The work applies the framework to automorphic \(L\)-functions, primes in arithmetic progressions, and classical prime sums, producing omega-type lower/upper bounds for extremal errors and linking these to the sizes of terms involving \(\zeta'(\rho_n)\). A key technical achievement is a robust smoothing and moment-analytic method, including a new bound (MengProp) valid under a weakened theta-parameter constraint, enabling broader applicability. Overall, the paper provides a comprehensive, flexible approach to predicting and bounding extreme fluctuations in prime-counting related error terms, with implications for several families of L-functions and related arithmetic sums.

Abstract

In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In 2012 the author made an analogous conjecture for the true order of the sum of the Möbius function, $M(x)$. This refined an earlier conjecture of Gonek from the 1990's. In this article we speculate on the true size of a large class of prime number error terms and present a general conjecture. This general conjecture includes both Montgomery's conjecture and the conjecture for $M(x)$ as special cases. Recently, Lamzouri (Springer volume: Essays in Analytic Number Theory, In Honor of Helmut Maier's 70th birthday) showed that an effective linear independence conjecture (ELI) for the zeros of the zeta function implies one of the inequalities in Montgomery's conjecture. In this article we adapt Lamzouri's method to show that a generalized effective linear independence (GELI) conjecture implies a lower bound for general prime number error terms. Furthermore, of independent interest, we prove an $L^2$ bound for almost periodic functions. This allows us to weaken significantly one of the conditions in Lamzouri's main result and also give an improvement of the main theorem in an article of Akbary-Ng-Shahabi (Q. J. Math. 65 (2014), no. 3).

Prime Number Error Terms

TL;DR

The paper proposes a unifying conjectural framework, GELI, for the true size of a broad class of prime-number error terms, unifying Montgomery’s conjecture for \(\psi(x)\) and the Mertens-type behavior of \(M(x)\) under a single principle. It builds on Lamzouri’s approach, modeling explicit prime-error sums by a random cosine sum and proving that GELI implies corresponding large/small value bounds; it also derives a general L^2 bound for almost periodic functions that strengthens previous results. The work applies the framework to automorphic -functions, primes in arithmetic progressions, and classical prime sums, producing omega-type lower/upper bounds for extremal errors and linking these to the sizes of terms involving \(\zeta'(\rho_n)\). A key technical achievement is a robust smoothing and moment-analytic method, including a new bound (MengProp) valid under a weakened theta-parameter constraint, enabling broader applicability. Overall, the paper provides a comprehensive, flexible approach to predicting and bounding extreme fluctuations in prime-counting related error terms, with implications for several families of L-functions and related arithmetic sums.

Abstract

In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In 2012 the author made an analogous conjecture for the true order of the sum of the Möbius function, . This refined an earlier conjecture of Gonek from the 1990's. In this article we speculate on the true size of a large class of prime number error terms and present a general conjecture. This general conjecture includes both Montgomery's conjecture and the conjecture for as special cases. Recently, Lamzouri (Springer volume: Essays in Analytic Number Theory, In Honor of Helmut Maier's 70th birthday) showed that an effective linear independence conjecture (ELI) for the zeros of the zeta function implies one of the inequalities in Montgomery's conjecture. In this article we adapt Lamzouri's method to show that a generalized effective linear independence (GELI) conjecture implies a lower bound for general prime number error terms. Furthermore, of independent interest, we prove an bound for almost periodic functions. This allows us to weaken significantly one of the conditions in Lamzouri's main result and also give an improvement of the main theorem in an article of Akbary-Ng-Shahabi (Q. J. Math. 65 (2014), no. 3).
Paper Structure (11 sections, 17 theorems, 204 equations, 1 table)

This paper contains 11 sections, 17 theorems, 204 equations, 1 table.

Table of Contents

  1. Introduction
  2. Key ingredients and outline of the Proof of Theorem \ref{['generalsumthm']}
  3. Further examples
  4. Prime number theorem for automorphic $L$-functions
  5. Primes in arithmetic progressions
  6. Classical prime number sums
  7. Proof of the key Proposition \ref{['measureprop']}
  8. Smoothing the sum
  9. Proof of Theorem \ref{['generalsumthm']} and Corollaries \ref{['Mofxcorr']}, \ref{['Lofxcorr']}
  10. Proof of Proposition \ref{['MengProp']}
  11. Concluding remarks

Key Result

Theorem 1.5

Assume $\bm{\lambda}= (\lambda_n)_{n \in \mathbb{N}}$ satisfies the Generalized Effective LI conjecture (Conjecture GELI). Assume the sequences $\bm{\lambda}= (\lambda_n)_{n \in \mathbb{N}}$ and $\mathbf{r}=(r_n)_{n \in \mathbb{N}}$ satisfy Assumptions 1-5, ass1,ass2, ass3, ass4, ass5. Let $X$ be la and

Theorems & Definitions (42)

  • Conjecture 1.1: Montgomery, 1980
  • Conjecture 1.2: Ng, 2012
  • Conjecture 1.3: Effective LI conjecture: ELI
  • Conjecture 1.4: Generalized Effective LI conjecture: GELI
  • Theorem 1.5
  • Remark 1.6
  • Conjecture 1.7: Hughes-Keating-O'Connell
  • Corollary 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 32 more