منابع مشابه
Simple Zeros of the Riemann Zeta-function
Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.
متن کاملA NOTE ON S(t) AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION
Let πS(t) denote the argument of the Riemann zeta-function at the point 1/2 + it. Assuming the Riemann hypothesis, we sharpen the constant in the best currently known bounds for S(t) and for the change of S(t) in intervals. We then deduce estimates for the largest multiplicity of a zero of the zeta-function, and for the largest gap between the zeros.
متن کاملZeros of the Riemann Zeta-Function on the Critical Line
It was shown by Selberg [3] that the Riemann Zeta-function has at least cT log T zeros on the critical line up to height T, for some positive absolute constant c. Indeed Selberg’s method counts only zeros of odd order, and counts each such zero once only, regardless of its multiplicity. With this in mind we shall write γ̂i for the distinct ordinates of zeros of ζ(s) on the critical line of odd m...
متن کاملOn simple zeros of the Riemann zeta-function
We investigate the distribution of simple zeros of the Riemann zeta-function. Let H ≤ T and L = log T . We calculate in a new way (following old ideas of Atkinson and new ideas of Jutila and Motohashi) the mean square of the product of F (s) = ζ(s) + 1 Lζ ′(s) and a certain Dirichlet polynomial A(s) = ∑ n≤M a(n) ns of length M = T θ with θ < 38 near the critical line: if R is a positive constan...
متن کاملOn the Zeros of the Riemann Zeta Function
We describe extensive computations which show that Riemann's zeta function f(s) has exactly 200,000,001 zeros of the form a + it in the region 0 < t < 81,702,130.19; all these zeros are simple and he on the line a = j. (This extends a similar result for the first 81,000,001 zeros, established by Brent in Math. Comp., v. 33, 1979, pp. 1361-1372.) Counts of the numbers of Gram blocks of various t...
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ژورنال
عنوان ژورنال: Proceedings of the Japan Academy, Series A, Mathematical Sciences
سال: 1936
ISSN: 0386-2194
DOI: 10.3792/pia/1195580057