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Computing Mertens product in quadruple precision

We can define the Mertens product, Mertens(x), named after Franz Mertens (1840-1927) as:

Mertens(x) := Product_{primes ‘p’ <= x} ( 1 – 1/p) .

A result known as Mertens’ Third Theorem is that , with

C = e^(-gamma), gamma being the Euler-Mascheroni constant, and the number ‘e’ being the base for natural logarithms,

lim_{ x -> oo} Mertens(x)/g(x)  = 1  ,

where

g(x) := C/ln(x) .

For example, Mertens(10^6) is about 0.04063821 and g(10^6) is about 0.04063979.

Srinivasa Ramanujan did his thesis work on highly composite numbers, nominally under the Cambridge mathematician Hardy. This resulted in a manuscript too long to appear as a single paper in 1915 England. So the first part of ” Ramanujan’s Thesis ” was published in 1915, and the second part only in 1997 in a Mathematics Journal.

The 1997 article (posthumous) was annotated by Jean-Louis Nicolas and Guy Robin, two French mathematicians who had both published a areas close Ramanujan’s highly composite numbers (among other things).

The result is the paper:

Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal volume 1 , number 2, 1997.

For the sections of this second paper dealing with highly composite numbers and colossally abundant numbers, Ramanujan assumers the Riemann Hypothesis. By assuming the Riemann Hypothesis, some delicate estimates can be proven [ by Ramanujan, ].

One of these is formula (362) in section 68, on page 139 of the paper referred to above. Jean-Louis Nicolas answered some of my questions by email on Section 68. Ramanujan writes in a very terse style, it seems.

By doing some routine although very lengthy computations (calculus with special functions, rarely used), I may have improved marginally on Ramanujan’s formula (362).

I wrote about this in the newsgroup sci.math in posts, beginning with:

Approximations to the product in Mertens Third Theorem, and Ramanujan’s “Highly Composite Numbers”, Part 2 (1997)

It’s archived at Google groups,

https://groups.google.com/forum/#!topic/sci.math/YArrppXRlD0

The last post there was dated June 7.

I expect to do numerical computations of Mertens(x) for x up to approximately 10^14, and compare with the approximation of June 7’th that uses the zeta zeros.

The accuracy (or error term) is conditional on the Riemann Hypothesis being true. The Riemann Hypothesis has been verified for the first 10^13 non-trivial zeros.

 

 

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Written by meditationatae

July 1, 2016 at 5:48 pm

Posted in History

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