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arXiv:2409.04261 (Published 2024-09-06)
Explicit bounds for prime K-tuplets
Categories: math.NTLet $K\geq 2$ be a natural number and $a_i,b_i\in\mathbb{Z}$ for $i=1,\ldots,K-1$. We use the large sieve to derive explicit upper bounds for the number of prime $k$-tuplets, i.e., for the number of primes $p\leq x$ for which all $a_ip+b_i$ are also prime.
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arXiv:2301.03165 (Published 2023-01-09)
Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region
Comments: 51 pages. Comments are welcomeCategories: math.NTWe derive explicit upper bounds for the Riemann zeta-function $\zeta(\sigma + it)$ on the lines $\sigma = 1 - k/(2^k - 2)$ for integer $k \ge 4$. This is used to show that the zeta-function has no zeroes in the region $$\sigma > 1 - \frac{\log\log|t|}{21.432\log|t|},\qquad |t| \ge 3.$$ This is the largest known zero-free region for $\exp(209) \le t \le \exp(5\cdot 10^{5})$. Our results rely on an explicit version of the van der Corput $A^nB$ process for bounding exponential sums.
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arXiv:1802.00085 (Published 2018-01-31)
Explicit bounds for primes in arithmetic progressions
Comments: 66 pages. Results of computations, and the code used for those computations, can be found at: http://www.nt.math.ubc.ca/BeMaObRe/Categories: math.NTWe derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show that $$ \bigg| \theta (x; q, a) - \frac{x}{\phi (q)} \bigg| < \frac1{3600} \frac q{\phi(q)} \frac{x}{\log x}, $$ for all $x \geq 7.94 \cdot 10^9$ (with sharper constants obtained for individual such moduli $q$). We establish inequalities of the same shape for the other standard prime-counting functions $\pi(x;q,a)$ and $\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\pmod q$ when $q\le4500$. For moduli $q>10^5$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,\chi)$ for quadratic characters $\chi$, and an improved explicit upper bound for exceptional zeros.