On Harmony in Number Theory – A Journey from Primes to Means and Logs

 

Mathematik | Informatik

 

Sinan Deveci, 2003 | Männedorf, ZH

 

We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the Hölder mean based on the functional equation for the Riemann zeta function. We find a harmonic mean analogue of Chebyshev’s inequality for the prime counting function involving the Euler-Mascheroni constant. Furthermore, we define a function taking the Hölder mean of all positive integers up to a given number and investigate its asymptotic behavior, finding two different patterns which are separated by the harmonic mean. Additionally, we discuss the behavior of said function at zero and discover a formula involving the Riemann zeta function, whose continuity we prove with Riemann’s functional equation. Inspired by the alternating harmonic series, we find a double series formula for the natural logarithm, resulting in identities involving the Riemann zeta function, binomial coefficients, and logarithms.

Introduction

We consider the following list of questions. (I) What happens if we replace log with the harmonic mean in Chebyshev’s inequality for the prime counting function? (II) We define a function taking the Hölder mean of all positive integers less than a given number. How does it behave? What patterns can we find? (III) Looking at the alternating series formula for log 2, we wonder whether the number 2 has any special properties that give rise to this identity or, if not, whether we can generalize it.

Methods

I used the document preparation system LaTeX and the computational knowledge engine WolframAlpha. Whenever I discovered a number sequence, I checked the OEIS (On-Line Encyclopedia of Integer Sequences) for a match and when an unfamiliar topic appeared, I looked it up at MathWorld. I have done lots of trial & error style numerical experimentation, taking limits, seeking to generalize, and extending beyond original definitions.

Results

We generalize the equivalence between the alternating harmonic series and the natural logarithm of 2 and find a double series formula for the natural logarithm of all positive integers. We then proceed by raising the individual terms to positive integer powers in two different ways and find formulae involving once again the Riemann zeta function and powers of pi as well as central entries of Pascal’s triangle and thus binomial coefficients. Modifying Chebyshev’s inequality, we find a bound for the prime counting function that is dependent on the harmonic mean and hence computable only with elementary operations. We give an introduction to the Hölder mean, define a function that takes the m-mean of all positive integers less than or equal to a given number, and investigate the dominant term of its asymptotic behavior. Approaching the value -1 for m, we find an interesting sequence. We also find a relation to the Riemann zeta function and thus the Bernoulli numbers. We find a proof at the discontinuity m=0 involving the functional equation for the Riemann zeta function. Then we investigate the behavior around zero and find similar values. Analogous to how adding factorials in the denominators of the geometric series gives the Taylor series expansion of the exponential functions, we add factorials in the denominators of arithmetic and harmonic series and thereby rediscover the Bell numbers.

Discussion

Unfortunately, our modification made the approximation for the prime counting function less strict. We have found many interesting patterns studying the Hölder mean function. Indeed, the number 2 is not special; we have generalized the alternating series formula for the logarithm. Our tools have been very helpful, and I think our methods were successful. However, the phase transition of the mean function has not been investigated sufficiently and not all of our hypotheses could be proved yet.

Conclusions

It is truly remarkable how such simple questions as expanding the notion of the logarithm to powers akin to the Riemann zeta function or investigating a process as simple as averaging different quantities can produce an array of numbers so rich that it encompasses many major fields in recreational number theory, such as Pascal’s triangle, powers of pi, Bernoulli numbers, and even a proof involving the functional equation of zeta(s). Potential next steps would be to extend the log series to rational numbers and investigate why the zeta function appears in the means. We realize that we have barely scratched the surface. But that means there is still something left to do!

 

 

Würdigung durch den Experten

Dr. Richard Bödi

This is a work on elementary number theory that combines various topics such as the prime number theorem, harmonic numbers, arithmetic, geometric, Pythagorean and Hölder means in novel and unexpected ways. Many of the results cannot be found elsewhere. It nicely demonstrates the author’s talent, his mathematical skills and the way of asking the right questions. Moreover, it showcases a modern approach on how to use state-of-the-art tools like Wolfram Alpha and the On-Line Encyclopedia of Integer Sequences (OEIS) for gaining interesting insights into mathematical relationships.

Prädikat:

hervorragend

Sonderpreis Metrohm – Stockholm International Youth Science Seminar (SIYSS)

 

 

 

Rämibühl-Realgymnasium, Zürich
Lehrer: Lucas Enz