Counting zeros of the Riemann zeta function

Hasanalizade, Elchin; Shen, Quanli; Wong, Peng‐Jie

doi:10.1016/j.jnt.2021.06.032

Cited by 12 publications

(4 citation statements)

References 15 publications

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“…This result is the same as [For02, Lem 4.3], except we use a sharper estimate of S(t), due to Trudgian [Tru14b], and extend the range of permissible values of η up to 2/7. We note that for large t, the estimate of S(t) due to [HSW21] is sharper, however our results are most sensitive to sharpness of the estimate for "small" t. Using Theorem 1.1, it is possible to further refine the arguments of [Tru14b] and [HSW21] to improve bounds on S(t), and thus improve Lemma 4.5. We leave such considerations for possible future work (see remarks in §5).…”

Section: 7)mentioning

confidence: 86%

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Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region

Yang¹

2023

Preprint

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We derive explicit upper bounds for the Riemann zeta-function ζ(σ + it) on the lines σ = 1 − k/(2 k − 2) for integer k ≥ 4. This is used to show that the zeta-function has no zeroes in the regionThis is the largest known zero-free region for exp(209) ≤ t ≤ exp(5 • 10 5 ).Our results rely on an explicit version of the van der Corput A n B process for bounding exponential sums.ζ(σ + it) ≪ t 1/(2 k −2) log t.(1.1) This is sharper than what is achievable via convexity arguments for all k ≥ 4.

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Section: 7)mentioning

confidence: 86%

“…Additionally, Theorem 1.1 can be used to improve explicit bounds on S(t), the argument of the zeta-function along the critical line (see e.g. [Tru14b;HSW21]). In this work, we use Theorem 1.1 to prove an explicit version of Littlewood's [Lit22] zero-free region of the form 1 − σ ≪ log log t/ log t.…”

Section: Introductionmentioning

confidence: 99%

Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region

Yang¹

2023

Preprint

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“…By the classical work of Rosser [28,Theorem 19] in Lemma 2.1 we may take a 1 = 0.137, a 2 = 0.443, and a 3 = 1.588. We note that these constants can be improved, see for instance the work of Hasanalizade, Shen, and Wong in [12] who obtained:…”

Section: 1mentioning

confidence: 95%

Density results for the zeros of zeta applied to the error term in the prime number theorem

Fiori¹,

Kadiri²,

Swidinsky³

2022

Preprint

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We improve the unconditional explicit bounds for the error term in the prime counting function ψ(x). In particular, we prove that, for all x > 2, we haveand that, for all log x ≥ 3 000, |ψ(x) − x| < 4.47 • 10 −15 x. This compares to results of Platt & Trudgian (2021) who obtained 4.51 • 10 −13 x. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning π(x) will appear in a follow up work.

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“…The geometry and patterns of series of positive primes are given in (Hasanalizade E., Shen O. & Wong P.J., 2022;.…”

Section: Finite-dimensional Symmetric Series Wpnmentioning

confidence: 99%

Prime Numbers Among Fractional, Integer and Natural Numbers

Мазуркин

2023

IST

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Series of primes are considered in three levels of sets of numbers (N)(Z)(Q), where – series of prime numbers in the set of natural numbers (PN), – a series of integer primes in the set of integers (WPN), – a series of rational primes in the set of rational (fractional) numbers (FPN). In series and the abscissa is taken as a complete series of integers , and for a series of prime numbers the positive right side of the x-axis is taken as a series of natural numbers . It is proved that the FPN set contains four types in the quadrants of the Cartesian coordinate system, the WPN set includes two mirror-opposite lines (the left one in quadrants II and IV and the right one in quadrants I and III). Subtraction from the next term WPN and PN of the previous component of the series gives new series of increments. Converting from decimal to binary gives for the series WPN and PN, as well as their increments, the rational root 1/2 of the Riemann zeta function on the second vertical of the binary expansion. Such a transformation makes it possible to obtain the block structure WPN and PN of partitioning the original series into blocks, and then the groups starting from the number 2. Using the example of block No. 8168 for the series PN, we proved the decomposition of groups into separate quanta, starting from the number 2, by asymmetric wavelets in the form solitary waves (solitons) with variable amplitude and oscillation period with a maximum relative error of 0.1%.

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Counting zeros of the Riemann zeta function

Cited by 12 publications

References 15 publications

Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region

Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region

Density results for the zeros of zeta applied to the error term in the prime number theorem

Prime Numbers Among Fractional, Integer and Natural Numbers

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