Prime Number Theorem Equivalences and Non‐equivalences

Diamond, Harold G.; Zhang, Wenbin

doi:10.1112/s0025579317000201

Cited by 4 publications

(1 citation statement)

References 2 publications

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“…Understandably, as in the classical case, this relation requires only an analysis of the ζ function of Beurling in, and on the boundary of the convergence halfplane. Second, conditions for the prime number theorem to hold, were sought see, for example, [3, 5, 6, 10, 18, 49, 50]. Again, this relies on the boundary behavior of ζ on the one‐line

\sigma =1

.…”

Section: Introductionmentioning

confidence: 99%

Density estimates for the zeros of the Beurling ζ function in the critical strip

Révész

2022

Mathematika

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In this paper, we prove three results on the density, respectively, local density and clustering of zeros of the Beurling zeta function 𝜁(𝑠) close to the one-line 𝜎 ∶= ℜ𝑠 = 1. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. As a complement to known results for the Selberg class, first we prove a Carlson type zero density estimate. Note that density results for the Selberg class rely on use of the functional equation of 𝜁, not available in the Beurling context. Our result sharpens results of Kahane, who proved an 𝑂(𝑇) estimate for zeros lying precisely just on a vertical line ℜ𝑠 = 𝑎 in the critical strip. Next we deduce a variant of a well-known theorem of Turán, extending its range of validity even for rectangles of height only ℎ = 2. Finally, we extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result -which, on the other hand, is a strong sharpening of the average result from the classic book of Montgomery -was worked out by Diamond, Montgomery and Vorhauer. On our way, we show that some obscure technicalities of the Ramachandra paper can be avoided.

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\sigma =1

.…”

Section: Introductionmentioning

confidence: 99%

Density estimates for the zeros of the Beurling ζ function in the critical strip

Révész

2022

Mathematika

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On the estimate $$M(x)=o(x)$$ for Beurling generalized numbers

Vindas

2024

Anal Math

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Oscillation of the Remainder Term in the Prime Number Theorem of Beurling, “Caused by a Given ζ-Zero”

Révész

2022

International Mathematics Research Notices

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Continuing previous studies of the Beurling zeta function, here, we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $\rho _0$ of the Beurling zeta function $\zeta _{{\mathcal {P}}}$ for a given number system generated by the primes ${\mathcal {P}}$, the corresponding error term $\Delta (x):=\psi _{{\mathcal {P}}}(x)-x$, where $\psi _{{\mathcal {P}}}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $\frac {\pi /2-\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. The somewhat mysterious appearance of the constant $\pi /2$ is explained in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes ${\mathcal {P}}$, such that $|\Delta (x)| \le \frac {\pi /2+\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. In this 2nd part, a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery, and Vorhauer.

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Prime Number Theorem Equivalences and Non‐equivalences

Cited by 4 publications

References 2 publications

Density estimates for the zeros of the Beurling ζ function in the critical strip

Density estimates for the zeros of the Beurling ζ function in the critical strip

On the estimate $$M(x)=o(x)$$ for Beurling generalized numbers

Oscillation of the Remainder Term in the Prime Number Theorem of Beurling, “Caused by a Given ζ-Zero”

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