Zeros Near s=1 for L-functions in the Selberg Class and Their Connection to the Constant Term of L'/L

Q: How do the findings of this paper relate to other known results about the distribution of zeros of L-functions, such as the Riemann Hypothesis or the Generalized Riemann Hypothesis?

This paper explores the connection between the zeros of an $L$-function near the line $\Re(s) = 1$ and the constant term, denoted $\ell_0(L)$, in the Laurent expansion of $L'/L$ at $s=1$. This is particularly interesting in light of known results and conjectures about the distribution of zeros of $L$-functions: Riemann Hypothesis (RH) and Generalized Riemann Hypothesis (GRH): The RH and GRH are some of the most profound conjectures in number theory. They assert that all non-trivial zeros of the Riemann zeta function and, more generally, of Dirichlet $L$-functions (and certain other $L$-functions in the case of GRH) lie on the critical line $\Re(s) = 1/2$. While this paper doesn't directly prove or disprove RH or GRH, it provides a new perspective on the behavior of zeros near $s=1$. The existence of a Siegel zero (a real zero very close to $s=1$) would have significant implications for the distribution of prime numbers, and this paper shows how the behavior of $\ell_0(L)$ can preclude their existence within certain families. Zero-Free Regions: Classically, zero-free regions for $L$-functions are established to the right of the critical line $\Re(s) = 1/2$. These regions are crucial for proving results about the distribution of prime numbers. This paper, instead of focusing on classical zero-free regions, investigates the relationship between the constant term $\ell_0(L)$ and the existence of zeros in a specific region near $s=1$. This region, defined in Theorem 1.1, has a shape that narrows as $\Re(s)$ approaches 1, differing from the typical rectangular zero-free regions. Connection to Analytic Number Theory: The results of this paper highlight a deep connection between the analytic properties of $L$-functions (specifically, the constant term of $L'/L$) and the distribution of their zeros. This connection is a recurring theme in analytic number theory, where analytic tools are used to study arithmetic objects.

Q: Could there be families of L-functions outside the considered class where the relationship between zeros near s=1 and the constant term of L'/L behaves differently?

It is certainly possible that the relationship between zeros near $s=1$ and the constant term of $L'/L$ could behave differently for families of $L$-functions outside the Selberg class with polynomial Euler product considered in the paper. Here's why: Specificity of the Selberg Class: The Selberg class is already a rather restricted class of $L$-functions. It imposes stringent conditions like the analytic continuation, functional equation, and Ramanujan Hypothesis, which are not satisfied by all Dirichlet series with arithmetic significance. Role of the Euler Product: The polynomial Euler product (axiom (S5')) plays a crucial role in the proofs of the paper. It allows for a more precise analysis of the logarithmic derivative $L'/L$. $L$-functions without Euler products, or with more complicated Euler products, might exhibit different behavior. Potential Counterexamples: While concrete counterexamples are not immediately obvious, it's conceivable that certain families of $L$-functions associated with more exotic objects (e.g., automorphic forms on higher-rank groups, or $L$-functions in other settings like function fields) might not adhere to the same relationship between $\ell_0(L)$ and zeros near $s=1$. Investigating such families would be an interesting research direction. It could reveal whether the results of this paper are specific to the Selberg class with polynomial Euler product or represent a more universal phenomenon in the theory of $L$-functions.

Core Concepts

For certain families of L-functions in the Selberg class, the existence of zeros near s=1 is closely related to the magnitude of the real part of the constant term in the Laurent expansion of L'/L at s=1.

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arxiv.org

T´afula, C. (2024). Zeros near s = 1 and the constant term of L′/L for L-functions in the Selberg class. arXiv:2001.02405v3.

This paper investigates the relationship between the location of zeros near s=1 and the magnitude of the constant term in the Laurent expansion of L'/L at s=1 for specific families of L-functions within the Selberg class. The author aims to generalize the established connection between Siegel zeros and Euler-Kronecker constants observed in L-functions of global fields to a broader class of L-functions.

Key Insights Distilled From

Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class

by Chri... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2001.02405.pdf

Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class

Deeper Inquiries

How do the findings of this paper relate to other known results about the distribution of zeros of L-functions, such as the Riemann Hypothesis or the Generalized Riemann Hypothesis?

This paper explores the connection between the zeros of an $L$-function near the line $\Re(s) = 1$ and the constant term, denoted $\ell_0(L)$, in the Laurent expansion of $L'/L$ at $s=1$. This is particularly interesting in light of known results and conjectures about the distribution of zeros of $L$-functions:

Riemann Hypothesis (RH) and Generalized Riemann Hypothesis (GRH):  The RH and GRH are some of the most profound conjectures in number theory. They assert that all non-trivial zeros of the Riemann zeta function and, more generally, of Dirichlet $L$-functions (and certain other $L$-functions in the case of GRH) lie on the critical line $\Re(s) = 1/2$.  While this paper doesn't directly prove or disprove RH or GRH, it provides a new perspective on the behavior of zeros near $s=1$.  The existence of a Siegel zero (a real zero very close to $s=1$) would have significant implications for the distribution of prime numbers, and this paper shows how the behavior of $\ell_0(L)$ can preclude their existence within certain families.

Zero-Free Regions:  Classically, zero-free regions for $L$-functions are established to the right of the critical line $\Re(s) = 1/2$. These regions are crucial for proving results about the distribution of prime numbers. This paper, instead of focusing on classical zero-free regions, investigates the relationship between the constant term $\ell_0(L)$ and the existence of zeros in a specific region near $s=1$. This region, defined in Theorem 1.1, has a shape that narrows as $\Re(s)$ approaches 1,  differing from the typical rectangular zero-free regions.

Connection to Analytic Number Theory: The results of this paper highlight a deep connection between the analytic properties of $L$-functions (specifically, the constant term of $L'/L$) and the distribution of their zeros. This connection is a recurring theme in analytic number theory, where analytic tools are used to study arithmetic objects.

Could there be families of L-functions outside the considered class where the relationship between zeros near s=1 and the constant term of L'/L behaves differently?

It is certainly possible that the relationship between zeros near $s=1$ and the constant term of $L'/L$ could behave differently for families of $L$-functions outside the Selberg class with polynomial Euler product considered in the paper. Here's why:

Specificity of the Selberg Class: The Selberg class is already a rather restricted class of $L$-functions. It imposes stringent conditions like the analytic continuation, functional equation, and Ramanujan Hypothesis, which are not satisfied by all Dirichlet series with arithmetic significance.

Role of the Euler Product: The polynomial Euler product (axiom (S5')) plays a crucial role in the proofs of the paper. It allows for a more precise analysis of the logarithmic derivative $L'/L$.  $L$-functions without Euler products, or with more complicated Euler products, might exhibit different behavior.

Potential Counterexamples:  While concrete counterexamples are not immediately obvious, it's conceivable that certain families of $L$-functions associated with more exotic objects (e.g., automorphic forms on higher-rank groups, or $L$-functions in other settings like function fields) might not adhere to the same relationship between $\ell_0(L)$ and zeros near $s=1$.
Investigating such families would be an interesting research direction. It could reveal whether the results of this paper are specific to the Selberg class with polynomial Euler product or represent a more universal phenomenon in the theory of $L$-functions.

What are the implications of these findings for understanding the statistical distribution of prime numbers in arithmetic progressions?

While the paper primarily focuses on the analytic properties of $L$-functions, its findings have indirect implications for understanding the distribution of prime numbers in arithmetic progressions, particularly when considered in the context of Dirichlet $L$-functions:

Dirichlet $L$-functions and Primes in Arithmetic Progressions: Dirichlet $L$-functions are directly related to the distribution of prime numbers within arithmetic progressions. Specifically, the non-vanishing of $L(s, \chi)$ for a Dirichlet character $\chi$ at $s=1$ is equivalent to Dirichlet's theorem on arithmetic progressions, which states that there are infinitely many primes in the arithmetic progression  $a, a+m, a+2m, ...$ if $a$ and $m$ are relatively prime.

Siegel Zeros and Irregularities: The possible existence of a Siegel zero for a Dirichlet $L$-function would indicate a significant irregularity in the distribution of primes in the corresponding arithmetic progression.  It would imply that this progression contains fewer primes than expected up to a certain point.

Theorem 1.1 and Prime Distribution: Theorem 1.1, when applied to families of Dirichlet $L$-functions, provides a condition under which Siegel zeros can be ruled out. If the constant term $\ell_0(L)$ is controlled for all $L$-functions in the family, then no Siegel zeros can exist. This implies a certain regularity in the distribution of primes across the arithmetic progressions associated with the characters in that family.

Limitations: It's important to note that the results are not strong enough to prove the non-existence of Siegel zeros in general. They only provide a conditional statement: if $\ell_0(L)$ is well-behaved, then Siegel zeros are absent.  Proving the non-existence of Siegel zeros unconditionally is a much harder problem, closely related to the celebrated "Class Number Problem" in algebraic number theory.
In summary, while not directly addressing the distribution of primes, the paper's findings offer a new perspective on a problem closely related to this fundamental topic in number theory. By connecting the constant term $\ell_0(L)$ to the absence of Siegel zeros, the paper indirectly sheds light on the regularity of prime distribution in arithmetic progressions, at least within certain families of Dirichlet $L$-functions.