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Lower bounds for shifted moments of Dirichlet $L$-functions of fixed modulus

Peng Gao, Liangyi Zhao

TL;DR

The article proves sharp GRH-conditional lower bounds for shifted moments of Dirichlet L-functions of fixed modulus, focusing on the two-shift case. It combines an approximate functional equation, prime-sum estimates, and Dirichlet-polynomial techniques to construct a majorant for the moment and to control auxiliary terms via Hölder’s inequality. The authors establish that, for large q with φ^*(q)≥ηφ(q), the shifted moment M_{f{t},f{a}}(q) is bounded below by φ(q)(log q)^{a_1^2+...+a_k^2} times a product of shifted zeta-values, matching the expected main-term magnitude and aligning with conjectural predictions. The methods extend prior upper-bound frameworks to produce matching lower bounds for the shifted moments of Dirichlet L-functions.

Abstract

We establish sharp lower bounds for shifted (with two shifts) moments of Dirichlet $L$-function of fixed modulus under the generalized Riemann hypothesis.

Lower bounds for shifted moments of Dirichlet $L$-functions of fixed modulus

TL;DR

The article proves sharp GRH-conditional lower bounds for shifted moments of Dirichlet L-functions of fixed modulus, focusing on the two-shift case. It combines an approximate functional equation, prime-sum estimates, and Dirichlet-polynomial techniques to construct a majorant for the moment and to control auxiliary terms via Hölder’s inequality. The authors establish that, for large q with φ^*(q)≥ηφ(q), the shifted moment M_{f{t},f{a}}(q) is bounded below by φ(q)(log q)^{a_1^2+...+a_k^2} times a product of shifted zeta-values, matching the expected main-term magnitude and aligning with conjectural predictions. The methods extend prior upper-bound frameworks to produce matching lower bounds for the shifted moments of Dirichlet L-functions.

Abstract

We establish sharp lower bounds for shifted (with two shifts) moments of Dirichlet -function of fixed modulus under the generalized Riemann hypothesis.

Paper Structure

This paper contains 15 sections, 20 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Sums over primes
  4. Upper Bounds for $\log |L(\tfrac{1}{2}+it,\chi)|$
  5. Approximate functional equation
  6. Estimations on Dirichlet polynomials
  7. Proof of Theorem \ref{['thm:main']}
  8. Proof of Proposition \ref{['prop:JBound']}
  9. Proof of Proposition \ref{['prop:I0Bound']}
  10. Initial Treatments
  11. Proof of Proposition \ref{['prop:J1Lower']}
  12. Proof of Proposition \ref{['prop:J2J1comp']}
  13. Proof of Proposition \ref{['prop:RHtwisted']}
  14. Conclusion
  15. Proof of Proposition \ref{['prop:IkBound']}

Key Result

Theorem 1.1

With the notation as above and the truth of GRH, let $k=2$ and $|t_j| \leq q^A$ for $j = 1, \ldots , k$ for a fixed positive real number $A$. Assume also that $\varphi^*(q) \geq \eta \varphi(q)$ for a fixed constant $\eta>0$. we have

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.8
  • Proposition 2.9
  • Lemma 2.10
  • ...and 16 more