Fractal curves from prime trigonometric series
FFRACTAL CURVES FROM PRIME TRIGONOMETRIC SERIES
DIMITRIS VARTZIOTIS AND DORIS BOHNET
Abstract.
We study the convergence of the parameter family of series V α,β ( t ) = (cid:88) p p − α exp(2 πip β t ) , α, β ∈ R > , t ∈ [0 , p , and subsequently, their differentiability proper-ties. The visible fractal nature of the graphs as a function of α, β is analyzed interms of H¨older continuity, self similarity and fractal dimension, backed withnumerical results. We also discuss the link of this series to random walks andconsequently, explore numerically its random properties. Introduction
The prime numbers are not randomly distributed but, there are random modelsthat capture well important properties of the distribution of prime numbers (e.g.[Cram´er, 1936]). The random behavior of a deterministic mathematical object canbe found elsewhere: there are classical function series that can be approximated byrandom processes. Let us briefly describe these series:consider the two functions f n ( x ) = sin(2 πnx ) and f n +1 ( x ) = sin(2 π ( n + 1) x ) for anarbitrary integer n ∈ N . These behave as strongly dependent random variables if weconsider x to be a random real variable uniformly distributed on some interval. Butif one picks from the sequence of frequencies (2 πnx ) n ≥ a sub sequence (2 πn k ) k ≥ such that the integer sequence grows sufficiently fast, i.e. n k +1 / n k ≥ ρ, ρ >
0, thequantities f n k ( x ) and f n k +1 ( x ) behave like independent random variables (see Fig. 1 asan example and § k move f n k ( x ) up. At time N , we find ourselves at S ( x, N ) = (cid:80) Nk =0 f n k ( x ). This sum is displayed for N = 1000 in Fig. 2 on the left.Our example is known as a lacunary Fourier series , that is, its frequencies fulfill thegrowth condition given above. Its random properties are a classical field of research.In the literature, the sequence of prime numbers (2 πp k ) k ≥ is often cited as a coun-terexample for a sequence of frequencies which does not give rise to a lacunaryFourier series: it does neither fulfill the growth condition nor alternative conditionson arithmetic patterns which exist in the literature. However, experiments in this Figure 1.
Graph of sin(2 πx ) and sin(2 πx ) on the left and ofsin(5 πx ) and sin(6 πx ) on the right. a r X i v : . [ m a t h . D S ] A ug DIMITRIS VARTZIOTIS AND DORIS BOHNET
Figure 2.
Graph of (cid:80) n =1 sin(2 n πx ) on the left, of (cid:80) n =1 sin( nπx ) in the middle and of (cid:80) p ≤ sin( pπx ) onthe right.article suggest that (cid:80) k sin( πp k x ) share a lot of the random properties of lacunaryseries (see Fig. 2 for a first impression or [Vartziotis and Wipper, 2016]): for in-stance, the central limit theorem seems to hold. Unfortunately, this looks difficultto prove.On the other hand, we can look at other manifestations of randomness in la-cunary series (e.g. in the example in Fig. 2) and try to see if they are alsopresent in our prime series V α,β : by introducing appropriate coefficients a k , thewalk (cid:80) k a k sin( πn k x ) can be approximated by a Wiener process which is an al-most everywhere continuous random walk with independent normally distributedincrements (see [Philipp and Stout, 1975] and § V α,β . Again,we were not able to prove this.But we can show that our series V α,β has in fact for specific α, β properties incommon with a Wiener process, e.g. its regularity and fractality (see § (cid:80) k a k sin(2 k πx ) is in fact famous for these reasons: Itbelongs to the family of Weierstrass functions F a,b ( x ) = (cid:80) ∞ n =0 a n sin( b n t ) whichhave been extensively studied for its differentiability properties. Under certain con-ditions on a, b this function is nowhere differentiable, but H¨older continuous.Another historical example which is non-differentiable, but multifractal, is the Rie-mann function R ( x ) = (cid:80) inf n =1 n − sin( n x ). Note, that it is not a lacunary seriesas ( n + 1) / n →
1. With our prime series, we place ourselves in between these twohistorical examples with respect to the growth of its frequencies.While prime sums are extensively studied in the context of the famous primeconjectures (e.g. for Vinogradov’s theorem and the like), we have not found atreatment of trigonometric series over prime numbers. The reason for this is mostprobably that these series have not the necessary form to help to progress in theproofs of the prime conjectures where prime exponential sums play a dominantrole. As mentioned above, these series have not been studied in the context of
RACTAL CURVES FROM PRIMES 3 lacunary series as prime numbers do neither grow fast enough nor do they haveknown arithmetic properties which are necessary for a straightforward analysis.By using results of prime number theory, we are nevertheless able to show conditionson the differentiability and self similarity of our prime series. Experimentally, weexplore also its box dimension in dependence on α, β . Remark 0.1.
For most of our questions, we can restrict ourselves – without lossof generality – to the real part (cid:80) p p − α cos(2 πp β t ) of the series which we denote by V α,β ( t ) , too. Acknowledgement
We would like to thank the referee for his remarks and Florian Pausinger for hishelpful hints and questions.2.
Convergence and differentiability
There are basically two factors which influence the smoothness and convergenceof a function series (cid:80) k a k exp(2 πin k t ) as ours:(1) The faster the coefficients a k decrease for k → ∞ , the smaller is the influ-ence of the higher frequencies. This implies that the series converges betterand the resulting function is smoother.(2) The faster the frequencies n k increase or equivalently, the greater the gaps,the smaller gets the period of the oscillation so that one obtains more peaksand sinks in one interval which increases the fractal character.2.1. Historical remarks.
The nature of these influences, easily deduced, are alsobacked by the long history of studies on the following two families of functions (andderived families):Let F a,b ( t ) = ∞ (cid:88) n =0 a n cos( b n t )be the family of Weierstrass functions which have been extensively studied. Oneknows the following: Theorem 1 ([Hardy, 1916],[Jaffard, 2010]) . (1) If < ab < , a < , b > ,then F a,b is differentiable. (2) If < a < < ab , then F a,b is nowhere differentiable. Further, the H¨olderexponent is a constant function s = − log a log b , i.e. for all t, t it holds | F a,b ( t ) − F a,b ( t ) | ≤ C | t − t | s . On the other hand, one has the family of Riemann’s function (whose authorshipby Riemann is apparently only confirmed by Weierstrass) defined by R α ( x ) = ∞ (cid:88) n =1 n − α cos( n x )which has the following proven properties: Theorem 2 ([Hardy, 1916], [Hardy and Littlewood, 1912],[Gerver, 1970a, Gerver, 1970b],[Chamizo and C´ordoba, 1996]) . (1) If < α ≤ , then the series is not a Fourier series of a L -function. If < α < , then R α converges at x if and only if x = aq , where a, q arecoprime and divides q − . (2) If < α < , then the series converges in p -norm to a L p -function for p < − α . (3) If α = 1 , then the series has bounded mean oscillation. DIMITRIS VARTZIOTIS AND DORIS BOHNET (4) If α < , then R α is not differentiable at any irrational value of x , and itsHausdorff dimension for ≤ α ≤ is equal to dim H ( R α ) = 94 − α . If α = 2 , then R is differentiable at x if and only if x = aq where a, q arecoprime and divides q − . (5) If α = 2 , the Hoelder exponent is discontinuous everywhere. In fact, R isa function with unbounded variation and multifractal. In the following, we aim to give a similar description for our function series. Letus start with some preliminary definitions which are necessary for what follows:2.2.
Preliminary definitions.
We call a function f : R → C locally H¨older con-tinuous at x ∈ R , if there exist s ∈ (0 ,
1] and
C, (cid:15) > | f ( x ) − f ( y ) | ≤ C | x − y | s , for all x, y ∈ B (cid:15) ( x ) . We call the supremum of s for which these inequality holds at x the local H¨olderexponent .Let φ : R → C be a smooth function with compact support supp( φ ) ⊂ C . We writeˆ φ ( u ) = (cid:90) R φ ( t ) exp( − iut ) dt for the Fourier transform of φ . Further, let φ : R → R be given such that thesupport of its Fourier transform is contained in [ − , Gabor wavelettransform of a function f : R → C is defined by G ( a, b, λ ) = 1 a (cid:90) R f ( t ) exp( − iλt ) φ (cid:18) t − ba (cid:19) dt. With these notation, we have the following estimation which is a special case ofProposition 5 in [Jaffard, 2010]:
Proposition 2.1 (Jaffard) . Let f : R → C be a bounded function. Let G ( a, b, λ ) bethe Gabor wavelet transform of f . If f is locally H¨older continuous at x ∈ R withH¨older coefficient s , then there exists C > such that for all a ∈ (0 , and for all b ∈ B ( x ) and for all λ ≥ a − we have | G ( a, b, λ ) | ≤ Ca s (cid:18) | x − b | a (cid:19) s . Differentiability of V α,β . In the spirit of the results in § V β ( n, t ) = (cid:88) p ≤ n f ( p ) cos(2 πp β t ) , β > , where f is any function of prime numbers. We can state the trivial fact that Proposition 2.2.
For any β ≥ , if (cid:82) ∞ | f ( x ) | ln( x ) dx < ∞ , then the partial sums V β ( n, t ) converges uniformly and absolutely to a continuous function denoted by V β .Proof. We have (cid:12)(cid:12) f ( p ) cos(2 πp β t ) (cid:12)(cid:12) ≤ | f ( p ) | for all p . By the Weierstrass M -test thepartial sums V β ( n, t ) converges uniformly and absolutely if (cid:80) p | f ( p ) | < ∞ . Usingthe Riemann-Stieltjes Integral and the Prime number theorem we get (cid:88) p f ( p ) = (cid:90) ∞ f ( x ) dπ ( x ) = (cid:90) ∞ f ( x )ln( x ) dx, RACTAL CURVES FROM PRIMES 5 where π ( x ) denotes the number of primes ≤ x finishing the proof. (cid:3) We take now V α,β ( n, t ) = (cid:88) p ≤ n p − α cos(2 πp β t )and denote with V α,β ( t ) its limit whenever it exists. Then one can show the follow-ing statement: Theorem 3.
Let α ∈ R and α > . (1) Then the series V α,β ( n, t ) converges uniformly and absolutely to a continu-ous function V α,β ( t ) . (2) For m ≥ , if further α − mβ > , then the function V α,β ( t ) is C m , i.e. m times continuously differentiable.Proof. For the first result we use the properties of the prime zeta function P ( α ) = (cid:80) p p − α : it converges absolutely for α > , α ∈ R , and diverges for α = 1 (seee.g. [Landau and Walfisz, 1920],[Fr¨oberg, 1968]). The coefficients p − α are an upperbound for the terms p − α cos (cid:0) πp β t (cid:1) . Consequently, the Weierstrass M -test impliesthat for α > t ∈ [0 , V α,β ( n, t ) converges uniformly and absolutely to V α,β ( t ). As any partial sum V α,β ( n, t ) is continuous, the limit is a continuousfunction, too.Secondly, for any n and t we can differentiate the partial sums V (cid:48) α,β ( n, t ) = − π (cid:88) p ≤ n p − α + β sin(2 πp β t ) . This sequence of derivatives converges uniformly with the same argument as abovefor α − β >
1, so that one concludes that V α,β ( t ) is continuously differentiableitself with derivative V (cid:48) α,β ( t ) = − π (cid:80) p p − α + β sin(2 πp β t ). By induction over m ,one proves the m -time differentiability of the function. (cid:3) Remark 3.1.
The result is in accordance with the intuitive smoothness of theseries: for fixed α > , the series gets the smoother, the smaller the frequency p β , β → , or equivalently, the larger the period. Therefore, the peaks and sinks ofthe oscillation are more and more separated so that the series gets smoother (seeFig. 3-5). Theorem 4. If < α ≤ β + 1 , then the function is H¨older continuous with H¨oldercoefficient s ≤ αβ .Proof. First of all, let f : R → C be an integrable function and N >
0, then byusing the Riemann-Stieltjes integral (see e.g. [Rosser and Schoenfeld, 1962]) andthe Prime number theorem as above one knows (cid:88) p ≤ N p − α = (cid:90) N x − α dπ ( x ) ∼ (cid:90) N x α ln( x ) dx. From this formula and li( x ) denoting the logarithmic integral function, one deduces(substituting dx by d (cid:0) x − α (cid:1) ) for α < (cid:88) p ≤ N p − α = (1 − α ) − li (cid:0) N − α (cid:1) + O (cid:16) N − α exp( − c (cid:112) ln( N ) (cid:17) . Approximating the logarithmic integral this implies(1) (cid:88) p ≤ N p − α ∼ N − α (1 − α ) ln( N ) . DIMITRIS VARTZIOTIS AND DORIS BOHNET
Figure 3.
Graph of V . , (10 , t ) at 5 ∗ discrete points in eachdirection (interpolated). Figure 4.
Graph of V . , . (10 , t ) at 5 ∗ discrete points ineach direction (interpolated). RACTAL CURVES FROM PRIMES 7
Figure 5.
Graph of V . , (10 , t ) at 5 ∗ discrete points in eachdirection (interpolated).If α >
1, we have to use the explicit formula for the prime zeta function to get anestimate for the speed of convergence (see e.g. [Cohen, 2000] for a derivation of theformula). We then have by partial summation (cid:88) p p − α = (cid:88) p ≤ N p − α + ∞ (cid:88) n =1 µ ( n ) n ln ( ζ ( N, αn )) , with ζ ( N, α ) = ζ ( α )Π p ≤ N (cid:0) − p − α (cid:1) , where ζ ( α ) = (cid:80) ∞ n =1 n − α denotes the Riemann zeta function and µ the Moebiusfunction . So we get for the tail of the prime zeta function (cid:88) p>N p − α = ∞ (cid:88) n =1 µ ( n ) n ln ( ζ ( N, αn )) with(2) ln ( ζ ( N, α )) = O (cid:0) N − α (cid:1) and ∞ (cid:88) n =1 µ ( n ) n = 0 . Combining Eq. (1)-(2) on the asymptotic of the prime zeta function, we can estimatenow the regularity of our function V α,β ( t ).For any t, t ∈ [0 , N = | t − t | − α . Then we have with the mean value
DIMITRIS VARTZIOTIS AND DORIS BOHNET theorem and using the absolute convergence of the series | V α,β ( t ) − V α,β ( t ) | ≤ (cid:88) p ≤ N p − α (cid:12)(cid:12) cos(2 πp β t ) − cos(2 πp β t ) (cid:12)(cid:12) + 2 (cid:88) p>N p − α ≤ (cid:88) p ≤ N p − α + β | t − t | + 2 (cid:88) p>N p − α ≤ N − α + β +1 ( β − α + 1) ln( N ) | t − t | + 2 CN − α ≤ C | t − t | − β +1 α . The exponent 1 − β < − β +1 α ≤ V α,β ( t ) is picked up. Let θ m = min (cid:110) p βm − p βm − , p βm +1 − p βm (cid:111) and∆ m = p m − p m − .We choose a function φ whose Fourier transform ˆ φ has compact support supp( ˆ φ ) ⊂ [0 ,
1] and ˆ φ (0) = 1. We then look at the Gabor-wavelet transform G m ( θ − m , t , p βm ) = θ m (cid:88) k p − αk (cid:90) R exp (cid:16) i (cid:16) p βk − p βm (cid:17) t (cid:17) φ ( θ m ( t − t )) dt = (cid:88) k p − αk exp (cid:16) i (cid:16) p βk − p βm (cid:17) t (cid:17) (cid:90) R exp i (cid:16) p βk − p βm (cid:17) θ m φ ( u ) du, with u = θ m ( t − t ). Substituting ˆ φ ( y ) = (cid:82) R exp( iyu ) φ ( u ) du for y = ( p βk − p βm ) uθ m in theequation we get G m ( θ − m , t , p βm ) = (cid:88) k p − αk exp (cid:16) i (cid:16) p βk − p βm (cid:17) t (cid:17) ˆ φ (cid:16) p βk − p βm (cid:17) uθ m . As the support of ˆ φ is a subset of the unit interval, it does vanish for any k (cid:54) = m ,so the expression it reduced to(3) G m ( θ − m , t , p βm ) = p − αm Recall that we have just proved that V α,β is locally H¨older continuous at t ∈ R .Further, for all m it is p βm ≥ θ m and θ − m ∈ (0 , C > s ∈ (0 , G m ( θ − m , t , p βm ) = p − αm ≤ Cθ − sm . The gap θ m is bounded by p βm from above so that the H¨older coefficient s is boundedby αβ from above finishing the proof. (cid:3) Remark 4.1.
Let α > be fixed. The bigger the gaps of the frequency, β → ∞ ,the stronger the irregularity of V α,β ( t ) . Self similarity and fractal dimension.
The graph of the function V α,β seems to be self similar for certain α, β . There seems to be an approximate scalarinvariance a points q − , where q is prime. Let us make more precise this intuition:look for example on the partial sums V , ( n, t ) = (cid:80) p ≤ n p − exp(2 πipt ) in Fig. 6:Denote by p k the k th prime number. We restrict ourselves again to the real partof V , ( n, t ). The point is a global minimum as V (cid:48) , (cid:0) n, (cid:1) = 0 and V , (cid:0) n, (cid:1) = RACTAL CURVES FROM PRIMES 9
Figure 6.
Graph of V , (10 , t ) at 5 ∗ discrete points. − (cid:80) nk =1 p − k as the primes greater than 2 are odd. Now, consider the point : wehave V , (cid:0) n, (cid:1) = − (cid:80) nk =1 ,k (cid:54) =2 p − k . More generally, one has V , (cid:18) n, q (cid:19) = 1 q + q − (cid:88) l =1 cos (cid:18) πlq (cid:19) n (cid:88) p k = l mod q p − k , q prime= q − (cid:88) l =0 cos (cid:18) πlq (cid:19) R l,q That is, we can decompose the partial sum into residue classes of the prime numbersand the roots of unity of cosine. One knows that the number of primes p ≤ n thatare congruent to l mod q are approximately the same for all l , that is, n Φ( q ) log( n ) where Φ( q ) denotes the Euler totient function and is equal to q − q prime.So for any q , q prime, one can use this distribution and the Riemann-Stieltjesintegral to show that the difference between the sums (cid:80) p k = l mod q, p k ≤ n p − k for each l = 1 , . . . , q − n → ∞ , that is: R l,q = (cid:88) p k = l mod q, p k ≤ n p − k ∼ q − (cid:90) n x ln( x ) dx = 1 q − n ) + C ) . Figure 7.
The graph of the real part of − V , (10 , t ) (in black)and V , (10 , t/
3) + (in grey).The factors cos (cid:16) πlq (cid:17) are exactly the prime roots of unity and the sum (cid:80) q − l =0 cos (cid:16) πlq (cid:17) =0. Consequently, one computes V , (cid:18) n, q (cid:19) ∼ q − q − n ) + C ) . As we have V , ( n,
1) = (cid:80) p ≤ n p − ∼ ln ln( n ) + M , one could argue that V , ( n, tq ) ≈ − q V , ( n, t ) + 1 q , q ≥ , prime , see Fig. 7. But keep in mind that these are only asymptotic equivalences while ourpartial sum V , ( n, t ) do not converge for n → ∞ , so the self similarity of the graphis certainly not strict.2.4.1. Fractal dimension of V α,β . Further, we compute numerically the box dimen-sion of the graph of V α,β defined in the following way: let A := [ a, b ] × [ c, d ] bethe rectangle such that the graph V α,β ( n, t ) ⊂ A is contained. We compute thenfor i, j = 0 , . . . N − V α,β ( n, t ) ∩ [ a + i ( b − a ) /N, a + ( i + 1)( b − a ) /N ] × [ c + j ( d − c ) /N, c + ( j + 1)( d − c ) /N ]. We denote the number of non-emptyintersections by M ( N ). The box dimension is then given bydim B ( V α,β ( n, t )) = lim N →∞ ln( M ( N ))ln( N ) . In accordance to our results on regularity of V α,β we obtain the following Fig. 8 forthe (numerically computed) box dimension dim B over the fraction αβ . For α > β → ∞ , that is, αβ →
0, we expect that the fractal dimension converges
RACTAL CURVES FROM PRIMES 11
Figure 8.
Box dimension of V α,β in dependence on the fractionof the powers αβ with α ∈ [1 , .
5] and β ∈ [0 . , V α,β is not convergent for α = 1.to 2. On the other hand, for β →
0, the fractal dimension should converge to 1 asthe graph gets continuously differentiable if α − β > β .3. Random properties for V α,β The quite similar behavior of lacunary and random Fourier series let us thinkthat it might be possible to capture the random character of the series V α,β whichis the subject of this section. Let us briefly review what it is known in the contextof lacunary sequences and random variables:3.1. Lacunary sequences behaving as independent random variables: shortoverview.
The terms (sin(2 πkx ) k and (cos(2 πkx ) k behave like random variables,but strongly dependently. But if one restricts the sequence of frequencies (2 πk ) k ≥ to (2 πn k ) k ≥ where the sequence ( n k ) k ≥ has sufficiently fast growing gaps, i.e.(4) n k +1 n k ≥ ρ, ρ > , then the sequences (sin(2 πn k x ) behave like independent random variables. Forexample, one has 1 √ N N (cid:88) k =1 sin(2 πn k x ) → N (0 , , where N (0 ,
1) is the normal distribution. This was the main observation whichhas led to study the connections between lacunary and random Fourier series,most importantly the question which are the optimal growth conditions on thesequence ( n k ) k such that the sequence ( f ( n k )) k ≥ for general periodic measur-able functions f with vanishing integral exhibits random properties (see historical overview [Kahane, 1997]). By introducing weights a k which obey certain growthconditions themselves, one can recover several limit theorems in complete anal-ogy to random variables. In particular, the Central Limit Theorem (CLT) andthe Law of Iterated Logarithm (LIL) are true (see results by Salem-Zygmund in[Salem and Zygmund, 1947], [Salem and Zygmund, 1948], Erd¨os-G´al in [Erd¨os and G´al, 1955]and Weiss in [Weiss, 1959]). Further, it can be shown that the process can be ap-proximated by a standard Brownian motion: Theorem 5 (Philipp-Stout [Philipp and Stout, 1975]) . Assume the Hadamard gapcondition. Assume further that A N := (cid:113) (cid:80) Nk =1 a k → ∞ and there exists δ > such that lim N →∞ a N A − δN = 0 . Then without changing the distribution of the process S ( t, x ) = (cid:88) k ≤ t a k cos(2 πn k x ) , t ≥ , it can be redefined on a suitable probability space together with a Wiener process (cid:8) W ( t ) (cid:12)(cid:12) t ≥ (cid:9) such that S ( t, x ) = W ( A t ) + O (cid:16) A − ρt (cid:17) , almost surely for some ρ > . While the Hadamard growth condition (4) for CLT can be weakened for generalsequences ( n k ) k (see [Erd¨os, 1962]) for coefficients a k = 1 to the optimal growthcondition n k +1 n k ≥ c k √ k with c k → ∞ , one has observed that sequences with muchslower growth can nevertheless satisfy the CLT if they fulfill certain arithmeticconditions, more precisely, bounds on the number of solutions for the diophantineequation. Results in this direction started with Gaposhkin ([Gaposhkin, 1966]) andwere recently sharpened by Berkes, Philipp and Tichy ([Berkes et al., 2008]). Thedifficulties with the prime sequence ( p k ) k ≥ are on both sides: Firstly, while it issure that the prime sequence is not a Hadamard sequence, neither precise lowernor upper bounds for the prime gap p k +1 − p k are known. The best results for alower bound which would be of interest for us do not hold for all k ≥ k →∞ ∆ k log p k = 0 . Secondly, there is nobuilding law for prime numbers known and the infinite recurrence of certain patternslike twin primes are only conjectured but not completely proved. On the other hand,the random character of prime numbers is often invoked without being analyticallyestablished anywhere although the random model by Cram´er ([Cram´er, 1936]) iswidely used and reproduces some results very efficiently (but fail in other aspects,e.g. in forecasting the size of the prime gap). In the question on convergenceof functions f ( n k ) random models were also introduced (see e.g. [Schatte, 1988]).Obviously, this is a broad and intensively studied mathematical subject where wedo not dare to make contributions. Therefore, we stay more closely to our studiedseries:3.2. The central limit theorem.
Because of the reason mentioned above we havenot been able to show the central limit theorem for the random variables sin( πp k x )or cos( πp k x ), the base of our series V α,β . Nevertheless, numerical computationsstrongly suggest that the central limit theorem holds, see Fig. 9: we took 10 uniformly distributed points x of the interval (cid:2) − π , π (cid:3) and computed the sampleaverage N (cid:80) Nk =1 sin( p k x ) for N = 78498, that is, the number of primes ≤ . Wecomputed the histogram for the values of the sample average which experimentallytends to a normal distribution as the size of the sample tends to infinity. RACTAL CURVES FROM PRIMES 13
Figure 9.
Normal distribution of N (cid:80) Nk =1 sin( πp k x ) for x uni-formly distributed in (cid:2) − π , π (cid:3) .4. Concluding remarks
The properties of the series V α,β which we have discussed in this article areintimately related to the distribution of prime numbers, and it was mostly dueto the unanswered questions on prime numbers that the analytical access to ourseries is limited. Therefore, knowledge on the distribution and bounds for thegaps of prime numbers would imply more or less directly properties where we wererestricted to a numerical approach.Although the series might remind of the Riemann zeta function or other number-theoretical functions, we did not construct V α,β in this way and do not see a pos-sibility to deduce it from any of them, besides from the trivial fact, that V α,β (0) isequal to the prime zeta function P ( α ) = (cid:80) p p − α . Figure 10.
Normal distribution of N (cid:80) Nk =1 sin( πp k x ) for x uni-formly distributed in (cid:2) − π , π (cid:3) . References [Berkes et al., 2008] Berkes, I., Philipp, W., and Tichy, R. (2008).
Metric Discrepancy Results forSequences { nkx } and Diophantine Equations , pages 95–105. Springer Vienna, Vienna.[Chamizo and C´ordoba, 1996] Chamizo, F. and C´ordoba, A. (1996). Differentiability and dimen-sion of some fractal fourier series. Advances in Mathematics
Acta Arith. , 2:23–46.[D. Goldston and Yildirim, 2009] D. Goldston, J. P. and Yildirim, C. (2009). Primes in tuples i.
Ann. Math. , 170(2):819–862.[Erd¨os, 1962] Erd¨os, P. (1962). On trigonometric sums with gaps.
Publ. Math. Inst. Hung. Acad.Sci., Ser. A , 7:37–42.[Erd¨os and G´al, 1955] Erd¨os, P. and G´al, I. (1955). On the law of iterated logarithm i + ii.
Nederl.Akad. Wetensch. Proc. Ser. A. , 17(58):65–84.[Fr¨oberg, 1968] Fr¨oberg, C.-E. (1968). On the prime zeta function.
BIT , 8:187–202.[Gaposhkin, 1966] Gaposhkin, V. F. (1966). Lacunary series and independent functions.
UspehiMat. Nauk. 21 , 132(6):3–82.[Gerver, 1970a] Gerver, J. (1970a). The differentiability of the riemann function at certain rationalmultiples of π . Amer. J. Math. , 92:33–55.[Gerver, 1970b] Gerver, J. (1970b). More on the differentiability of the riemann function.
Amer.J. Math. , 93:33–41.[Hardy and Littlewood, 1912] Hardy and Littlewood (1912). Contributions to the arithmetic the-ory of series.
Proceedings of the London Mathematical Society , 11(2):411–478.[Hardy, 1916] Hardy, G. H. (1916). Weierstrass’s non-differentiable function.
Transactions of theAmerican Mathematical Society , 17(3):301–325.[Jaffard, 2010] Jaffard, S. (2010). Pointwise and directional regularity of nonharmonic fourierseries.
Applied and Computational Harmonic Analysis , 22(3):251–266.[Kahane, 1997] Kahane, J.-P. (1997). A century of interplay between taylor series, fourier seriesand brownian motion.
Bull. London Math. Soc. , 29:257–279.[Landau and Walfisz, 1920] Landau, E. and Walfisz, A. (1920). ¨Uber die nichfortsetzbarkeiteiniger durch dirichletsche reihen definierter funktionen.
Rend. Circ. Math. Palermo , 44:82–86.[Philipp and Stout, 1975] Philipp, W. and Stout, W. F. (1975).
Almost Sure Invariance Principlesfor Partial Sums of Weakly Dependent Random Variables . Mem. Am. Math. Soc. AMS.[Rosser and Schoenfeld, 1962] Rosser, J. and Schoenfeld, L. (1962). Approximate formulas forsome functions of prime numbers.
Illinois J. Math. , 6(1):64–94.
RACTAL CURVES FROM PRIMES 15 [Salem and Zygmund, 1947] Salem, R. and Zygmund, A. (1947). On lacunary trigonometric series.
Proc. Nat. Acad. Sci. U.S.A. , 33:333–338.[Salem and Zygmund, 1948] Salem, R. and Zygmund, A. (1948). On lacunary trigonometric series.
Proc. Nat. Acad. Sci. U.S.A. , 34:54–62.[Schatte, 1988] Schatte, P. (1988). On a law of iterated logarithm for sums mod 1 with applicationsto benford’s law.
Prob. theory Rel. Fields , 77:167–178.[Vartziotis and Wipper, 2016] Vartziotis, D. and Wipper, J. (2016). The fractal nature of an ap-proximate prime counting function.
ArXiv e-prints .[Weiss, 1959] Weiss, M. (1959). The law of the iterated logarithm for lacunary trigonometric series.
Trans. Amer. Math. Soc. , 91:444–469.
NIKI Ltd. Digital Engineering, Research Center, 205 Ethnikis Antistasis Street,45500 Katsika, Ioannina, Greece and, TWT GmbH Science & Innovation, Department forMathematical Research & Services, Ernsthaldenstr. 17, 70565 Stuttgart, Germany
E-mail address : [email protected] TWT GmbH Science & Innovation, Department for Mathematical Research & Ser-vices, Ernsthaldenstr. 17, 70565 Stuttgart, Germany
E-mail address ::