Extremal Properties of L-Functions and Zeta Functions

The extremal analysis of L-functions and zeta functions illuminates the boundary behaviour of these complex-valued analytic objects when their arguments approach regions of critical significance. Central to analytic number theory, the Riemann zeta function and its generalisations—Dirichlet L-functions, Dedekind zeta functions and automorphic L-functions—encode arithmetic data through their zero distributions and magnitude on the critical strip. Extremal investigations focus on lower and upper bounds for values or derivatives on the critical line, with implications for prime distribution, the Lindelöf hypothesis and zero-spacing statistics. Recent work advances our understanding of how these functions attain unusually large or small magnitudes, often revealing deep connections to probabilistic models, random matrix theory and spectral theory. Concrete examples include the quantification of maximal growth rates of ζ(½+it) or its derivatives, and dichotomies which link extremal behaviour across distinct L-functions. These studies not only refine classical conjectures but also inform computational strategies for locating zeros and estimating error terms in prime-counting functions.

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Extremal Properties of L-Functions and Zeta Functions publication trend

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Technical terms

Riemann zeta function: A complex function ζ(s)=∑ₙ₌₁ⁿ⁻ˢ extended by analytic continuation, central to the distribution of prime numbers.

L-function: A broad class of complex analytic functions generalising ζ(s), associated with arithmetic objects such as Dirichlet characters or modular forms.

Critical line: The vertical line in the complex plane ℜ(s)=½, whose zero distribution is linked to deep conjectures in number theory.

Log-derivative: The function ζ′(s)/ζ(s), whose magnitude controls zero distributions and error estimates in prime-counting.

Dickman function: A continuous function arising in probabilistic models of integer factorisation, used in asymptotic estimates of extremal values.

Bondarenko–Seip bound: A known lower bound for large values of Dirichlet L-functions, named after the mathematicians who established it.

References

1. Extreme values of derivatives of zeta and L‐functions. Bulletin of the London Mathematical Society (2023).
2. A dichotomy for extreme values of zeta and Dirichlet L$L$‐functions. Bulletin of the London Mathematical Society (2023).
3. Bounding the log-derivative of the zeta-function. Mathematische Zeitschrift (2021).