Prime Reciprocal Digit Frequencies and the Euler Zeta Function

Transcription

1 Prime Recirocal Digit Frequencies and the Euler Zeta Function Subhash Kak. The digit frequencies for rimes are not all equal. The least significant digit for rimes greater than 5 can only be, 3, 7, or 9. For binary reresentation of rimes, the frequency of is higher than that of Frequency of Series Number of bits Figure : Frequency of in a binary, constant bit reresentation of rimes The number of 0s and s for all rimes of with resect to different binary lengths from to 7 is given in Table. Thus for all rimes of binary length 3, we have the rimes, 3, 5, and 7 which in the binary form are 00, 0, 0, and, with four 0s and 8 s. Likewise, for all rimes of bit length 4, we count the rimes, 3, 5, 7,, and 3 corresonding to the sequences 000, 00, 00, 0, 0, 0, which gives us ten 0s and 4 s. If one were to take a variable binary reresentation of rimes, the frequency of 0s will naturally be less. Thus for 7-bit reresentation, the count of 0s in a variable bit reresentation is and that of s is (comare the constant bit values in Table ).

2 Table : Frequency of 0 and in a constant bit reresentation of rimes Bits Count 0 Count Total The rincial question of interest to us is whether the excess of s is counterbalanced by the higher frequency of 0 comared to in the reresentation of rime recirocals?. There is indeed evidence [],[] that for rime recirocals in binary exansions the frequency of 0 is slightly larger than that of. For examle, the number of cases where 0s exceed s comared to where s exceed 0s is in the roortion 7: in the rimes in the range 50,000 to 60,000. Likewise, in the range 800,0 to 999,983, the number of cases where 0s exceed s in the rime recirocals is 3609, whereas the number of cases where s exceed 0s is only 64 (the number of cases where 0s and s were equal was 097, which are the maximum-length cases).

3 3. It is ossible that the frequency of 0 being slightly larger than that of does not hold u as the number of rimes tested increases much beyond 999,983, which was the limit of the most recent exeriment. But it is more likely that the difference will continue to hold u, esecially if we see this excess as counterbalancing the excess of s in the binary reresentation of rimes. 4. One would like to know how the difference between 0s and s changes as the range increases. With this in mind, it may be well worth studying the ossible relationshi of the recirocal frequencies with the Euler and the Riemann zeta functions. 5. The Euler zeta function ζ(s) is given by: ζ(s) n /n s s + s s s ζ() This is the sum of all recirocals, which is the harmonic series whose sum is infinity (the roof of which is elementary). 6. By the Euler identity ζ(s) rime s Therefore, ζ () rime rime s Or, 3

4 ζ () rime Since is merely a cyclic dislacement [3],[4] of the digits of the recirocal exansion of /, it is clear that there is some relationshi between the Euler zeta function and the rime recirocals. But it is not a direct relationshi since it involves the multilication of rime recirocals. Perhas this multilication imlies a frequency roerty related to all rime recirocals. ζ () is also written as a Dirichlet series n μ( n), where μ(n) is the Möbius function which n is + if n is square-free with even number of distinct factors, - if it is square free with odd number of distinct factors, and 0 if it is not square free. Therefore, ζ () s -/ -/3 -/5 +/5 /7 + /0 / -/3 + /4 + /5 - Considering binary rime recirocal sequences, the difference between addition and subtraction may not matter in the comutation of grou frequencies. 7. Similarly, ζ () rime This suggests that it would also be useful to study the exansions of recirocals of rime owers. Since ζ() is π /6, could this imly a corresonding interesting roerty for such exansions? 8. By taking logs of both sides in the inverse exansion of the Euler zeta function, ζ () rime ( ) 4

5 Likewise, we have ζ () rime ( 9. It is ossible that further insights into rime recirocals may be obtained from s comlex, or, in other words, from the Riemann zeta function. 0. It is convenient to generate binary exansions of the rime recirocal / by [4],[5]: a(i) i mod mod ) It is worthwhile to determine if this henomenon holds for non-binary cases. To generate decimal exansions of /, one may use the following formula: A reliminary exeriment indicates that in base 0 also the frequency of 0 is larger than that of the other digits.. Another interesting question to ask is what is the largest grou of consecutive rime recirocals that are non-maximum length in base? For examle, the 6 consecutive rime recirocals 97079, 97097, , 97033, 97035, and , , 98967, , are all non-maximum length in base.. For some alications of rime recirocal sequences, see [6]-[0]. References. S. Kak, A structural redundancy in d-sequences. IEEE Transactions on Comuters, vol. C-3, , S.K.R. Gangasani, htt://arxiv.org/abs/ S. Kak and A. Chatterjee, On decimal sequences. IEEE Transactions on Information Theory, vol. IT-7, , S. Kak, Encrytion and error-correction coding using D sequences. IEEE Transactions on Comuters, vol. C-34, , S. Kak, New results on d-sequences. Electronics Letters, vol. 3,. 67, N. Mandhani and S. Kak, Watermarking using decimal sequences. Crytologia, vol. 9, , 005; arxiv:cs/060003v 7. A. Parakh, A d-sequence based recursive random number generator. Proceedings of International Symosium on System and Information Security -- Sao Jose dos Camos: CTA/ITA/IEC, 006; arxiv:cs/060309v 5

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