A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma &lt; 1$

Yuk-Kam Lau

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We consider the Legendre quadratic forms $ϕ_{a b} (x, y, z) = a x^{2} + b y^{2} - z^{2}$ and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers $1 \leq a \leq A, 1 \leq b \leq B$ , for which the form $ϕ_{a b}$ has a non-trivial rational zero. Under certain mild conditions on the integers $a, b$ , we are able to find the asymptotic formula for the number of such forms.

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Let $𝔖_{n}$ denote the symmetric group with $n$ letters, and $g (n)$ the maximal order of an element of $𝔖_{n}$ . If the standard factorization of $M$ into primes is $M = q_{1}^{α_{1}} q_{2}^{α_{2}} ... q_{k}^{α_{k}}$ , we define $ℓ (M)$ to be $q_{1}^{α_{1}} + q_{2}^{α_{2}} + ... + q_{k}^{α_{k}}$ ; one century ago, E. Landau proved that $g (n) = {max}_{ℓ (M) \leq n} M$ and that, when $n$ goes to infinity, $log g (n) \sim \sqrt{n log (n)}$ . There exists a basic algorithm to compute $g (n)$ for $1 \leq n \leq N$ ; its running time is $𝒪 (N^{3 / 2} / \sqrt{log N})$ and the needed memory is $𝒪 (N)$ ; it allows computing $g (n)$ up to, say, one million. We describe an algorithm to calculate $g (n)$ for $n$ up to $10^{15}$ . The main idea is to use the...

Displaying similar documents to “A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip 3 / 4 ≤ σ &lt; 1 ”

Displaying similar documents to “A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3 / 4 \leq σ < 1$ ”