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A typical book can be printed with 10 6 zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 10 94 such books to print all the zeros of a googolplex (that is, printing a googol zeros). [4] If each book had a mass of 100 grams, all of them would have a total mass of 10 93 kilograms.
In 1914, Godfrey Harold Hardy proved [1] that the Riemann zeta function (+) has infinitely many real zeros. Let () be the total number of real zeros, () be the total number of zeros of odd order of the function (+), lying on the interval (,].
According to van den Essen, [2] the problem was first conjectured by Keller in 1939 for the limited case of two variables and integer coefficients. The obvious analogue of the Jacobian conjecture fails if k has characteristic p > 0 even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least 2.
A googol is the large number 10 100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeros: 10, 000, 000 ...
Jensen's formula can be used to estimate the number of zeros of an analytic function in a circle. Namely, if is a function analytic in a disk of radius centered at and if | | is bounded by on the boundary of that disk, then the number of zeros of in a circle of radius < centered at the same point does not exceed
In 1992 Karatsuba proved [5] that an analog of the Selberg conjecture holds for "almost all" intervals (T, T + H], H = T ε, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits one to investigate zeroes of the Riemann zeta function on "supershort" intervals of the critical line, that is, on the intervals (T, T ...
Based on a new algorithm developed by Odlyzko and Arnold Schönhage that allowed them to compute a value of ζ(1/2 + it) in an average time of t ε steps, Odlyzko computed millions of zeros at heights around 10 20 and gave some evidence for the GUE conjecture. [3] [4] The figure contains the first 10 5 non-trivial zeros of the Riemann zeta ...
Because of the order of zeros and poles being defined as a non-negative number n and the symmetry between them, it is often useful to consider a pole of order n as a zero of order –n and a zero of order n as a pole of order –n. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.