[Paper Review] Mertens' theorem and prime number theorem for Selberg class

Yoshikatsu Yashiro|arXiv (Cornell University)|Nov 4, 2013

Analytic Number Theory Research8 references4 citations

TL;DR

This paper generalizes Mertens' third theorem and the prime number theorem to the Selberg class of L-functions. By leveraging Perron's formula, complex analysis, and zero-free region estimates, it establishes an asymptotic for the Euler product over primes in the Selberg class, showing the product grows as $ c_{-m} e^{\gamma m} (\log x)^m $, with a refined error term $ O(e^{-C_F\sqrt{\log x}}) $, and proves a corresponding prime number theorem with a similar error term for the summatory function of coefficients.

ABSTRACT

In 1874, Mertens proved the approximate formula for partial Euler product for Riemann zeta function at $s=1$, which is called Mertens' theorem. In this paper, we generalize Mertens' theorem for Selberg class and show the prime number theorem for Selberg class.

Motivation & Objective

• To extend Mertens' third theorem, originally for the Riemann zeta function, to the broader Selberg class of L-functions.
• To establish a prime number theorem for functions in the extended Selberg class $\mathcal{S}^\#$, generalizing the classical result for $\zeta(s)$.
• To derive precise asymptotic formulas for Euler products and summatory functions of coefficients in the Selberg class, with explicit error terms.
• To demonstrate that the error term in the prime number theorem improves when the zero-free region condition (II) is assumed, rather than just the prime number theorem.

Proposed method

• Applies Perron's formula to relate the logarithmic derivative of $F(s)$ to sums over primes and prime powers.
• Uses complex analysis and contour integration to evaluate the integral representation of the partial Euler product.
• Employs the functional equation and analytic continuation properties of $F(s)$ in $\mathcal{S}^\#$ to control the behavior near $s=1$.
• Imposes the zero-free region condition (II) to refine the error term from $O(1/\log x)$ to $O(e^{-C_F\sqrt{\log x}})$.
• Applies partial summation techniques to transform sums over $b_F(p)/p$ into sums over $b_F(p)\log p / p$, enabling the derivation of the prime number theorem.
• Uses the Ramanujan conjecture and the logarithmic derivative expansion $\log F(s) = \sum b_F(n)n^{-s}$ to control the growth of coefficients.

Experimental results

Research questions

• RQ1 Can Mertens' third theorem, which gives the asymptotic for the product $\prod_{p\leq x} (1 - 1/p)^{-1}$, be extended to L-functions in the Selberg class?
• RQ2 Does the prime number theorem for $\zeta(s)$ generalize to functions in the extended Selberg class $\mathcal{S}^\#$, with a similar error term?
• RQ3 What is the precise asymptotic form of the Euler product $\prod_{p\leq x} \prod_{j=1}^k (1 - \alpha_j(p)/p)^{-1}$ for $F \in \mathcal{S}^\#$, and how does the error term depend on the zero-free region?
• RQ4 How does the assumption of a zero-free region (II) improve the error term in the prime number theorem compared to just assuming the PNT?
• RQ5 Can the generalized Mertens' constant $M$ be explicitly defined in terms of the coefficients $b_F(p^r)$ and the residue $c_{-m}$?

Key findings

• The generalized Mertens' theorem for $F \in \mathcal{S}^\#$ states that $\prod_{p\leq x} \prod_{j=1}^k (1 - \alpha_j(p)/p)^{-1} = c_{-m} e^{\gamma m} (\log x)^m (1 + O(e^{-C_F\sqrt{\log x}}))$, where $c_{-m} = \lim_{s\to 1} (s-1)^m F(s)$.
• The error term in the Mertens-type product is improved to $O(e^{-C_F\sqrt{\log x}})$ under the zero-free region condition (II), which is stronger than the $O(1/\log x)$ error from just assuming the prime number theorem.
• The prime number theorem for the Selberg class is established as $\sum_{n\leq x} b_F(n) \log n = m x + O(x e^{-C_F^{\prime\prime}\sqrt{\log x}})$, with $C_F^{\prime\prime} < C_F$.
• The generalized Mertens' constant $M$ is explicitly defined as $\log c_{-m} + m\gamma - \sum_p \sum_{r=2}^\infty \frac{b_F(p^r)}{p^r}$, which controls the constant term in the logarithmic sum over primes.
• The proof shows that the error term in the prime number theorem is directly tied to the zero-free region: the larger the region, the smaller the error term.
• The results recover and refine known theorems for Dedekind zeta functions $\zeta_K(s)$, where $\alpha_K = \text{Res}_{s=1} \zeta_K(s)$, and the error term is $O(e^{-C_K\sqrt{\log x}})$.