# Star refinement

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.

In number theory, **Mertens' theorems** are three 1874 results related to the density of prime numbers proved by Franz Mertens.^{[1]} "Mertens' theorem" may also refer to his theorem in analysis.

## In number theory

In the following, let mean all primes not exceeding *n*.

**Mertens' 1st theorem**:

does not exceed 2 in absolute value for every .

**Mertens' 2nd theorem**:

where *M* is the Meissel–Mertens constant. More precisely, Mertens^{[1]} proves that the expression under the limit does not in absolute value exceed

**Mertens' 3rd theorem**:

where γ is the Euler–Mascheroni constant.

### Changes in sign

In a paper ^{[2]} on the growth rate of the sum-of-divisors function published in 1983, Guy Robin proved that in Mertens' 2nd theorem the difference

changes sign infinitely often, and that in Mertens' 3rd theorem the difference

changes sign infinitely often. Robin's results are analogous to Littlewood's famous theorem that the difference π(*x*) − li(*x*) changes sign infinitely often. No analog of the Skewes number (an upper bound on the first natural number *x* for which π(*x*) > li(*x*)) is known in the case of Mertens' 2nd and 3rd theorems.

### Mertens' second theorem and the prime number theorem

Regarding this asymptotic formula Mertens refers in his paper to "two curious formula of Legendre",^{[1]} the first one being Mertens' second theorem's prototype (and the second one being Mertens' third theorem's prototype: see the very first lines of the paper). He recalls that it is contained in Legendre's third edition of his "Théorie des nombres" (1830; it is in fact already mentioned in the second edition, 1808), and also that a more elaborate version was proved by Chebyshev in 1851.^{[3]} Note that, already in 1737, Euler knew the asymptotic behaviour of this sum.^{[4]}

Mertens diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs is acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic logarithm of infinity, and the logarithm of the logarithm of infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which will be proved in 1896 and then be better known as the prime number theorem.

Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, on the other hand, will heavily exploit the behaviour of the Riemann zeta function in the complex plane. It is in that respect a remarkable proof. Indeed, with modern notation it yields

whereas the prime number theorem (in its simplest form, without error estimate), can be shown to be equivalent to^{[5]}

## In summability theory

In summability theory, **Mertens' theorem** states that if a real or complex infinite series

converges to *A* and another

converges absolutely to *B* then their Cauchy product converges to *AB*.

## References

- ↑
^{1.0}^{1.1}^{1.2}F. Mertens. J. reine angew. Math. 78 (1874), 46-62 Ein Beitrag zur analytischen Zahlentheorie - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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- ↑ Leonhard Euler. Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae 9 (1737), 160-188.
- ↑ Although this equivalence is not explicitly mentioned there,it can for instance be easily derived from the material in chapter I.3 of: G. Tenenbaum. Introduction to analytic and probabilistic number theory. Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge,1995.

## Further reading

- Yaglom and Yaglom
*Challenging mathematical problems with elementary solutions*Vol 2, problems 171, 173, 174

## External links

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