uniform distribution theory

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1 Uniform Distribution Theory 0 205, no., uniform distribution theory THE TAIL DISTRIBUTION OF THE SUM OF DIGITS OF PRIME NUMBERS Eric Naslund Dedicated to Professor Harald Niederreiter on the occasion of his 70th birthday ABSTRACT. Lets qndenote the baseq sumofdigitsfunction, whichforn x, is centered around 2 log q x. In this paper we provide bounds on the tails of the distribution of s qn, and prove that given α in the range α < , and 2 any ǫ > 0, there exists a constant c depending on ǫ such that { p x, p prime : s qp αq log q x } 2 25 x2 α e c logqlogx /2+ǫ for sufficiently large x. In particular, this shows that there are infinitely many primes with more than twice as many ones than zeros in their binary expansion. Communicated by Michael Drmota. Introduction A prime number of the form 2 n is called a Mersenne prime and it will only have ones in its binary expansion. The first few such primes are 3, 7, 3, and 27, and currently the largest known prime is of this form with over 2.9 million digits. It is a long standing conjecture that there are infinitely many Mersenne primes, and this currently seems entirely out of reach of modern analytic methods. However, we may weaken the condition and ask about primes with a large number of s in their base 2 expansion. With this in mind, we ask: ÈÖÓ Ð Ñ º Are there infinitely many primes with more than twice as many ones than zeros in their binary expansion? 200 Mathematics Subject Classification: N05,N36. Keywords: Mersenne primes, Sum of Digits, Digits of primes, Prime numbers. I would like to thank Didier Piau for his help, and Gil Kalai for his motivating Math Overflow question. I am also grateful to Greg Martin for his comments and suggestions. 63

2 ERIC NASLUND The set of integers with more than twice as many ones than zero s is very small as most integers have approximately half of their digits equal to. If we let s q n denote the sum of the digits of n written in base q, then we are asking if there exists infinitely many primes p satisfying s 2 p 2 3 log 2p. Moving to a slightly more general setting, we will look at the sum of digits base q rather than just the binary case. On average s q n is 2 multiplied by the number of digits, so we have the asymptotic s q n q log 2 q x. n x However, things become more complicated when we restrict ourselves to the prime numbers. In 946 Copeland and Erdos [2] proved that s q p q log πx 2 q x p x where πx = p x is the prime counting function, and a more precise error term was subsequently given by Shiokawa [4]. In 2009, Drmota, Mauduit and Rivat [3] gave exact asymptotics for the set { p x, p prime sq p = αq log q x } where α lies in the range α 2 Kloglogx 2 ǫ, logx 2 +Kloglogx 2 ǫ, logx and is chosen so that αq log q x is an integer which avoids certain congruence conditions. However, these results only apply to the center of the distribution, and they don t allow us to make any conclusions about problem. In [3] they ask about finding non-trivial bounds for the sum p x 2sqp, as this would yields results regarding the tail distribution of the sum of digits of primes. That is, they ask about lower bounds for the size of { p x, p prime : sq n αq log q x } where α > 2 does not depend on x. These are exactly the type of bounds we are looking for in order to answer our question, as problem is the case when α = 2 3 and q = 2. In this note, we provide such lower bounds, and prove the following theorem: Ì ÓÖ Ñ º Given < β 2 and 2 α < , there exists a constant c depending on ǫ such that for sufficiently large x we have { p x, p prime : s q n αq log q x } 2 25 x2 α e c logqlogx /2+ǫ 64

3 THE TAIL DISTRIBUTION OF THE SUM OF DIGITS OF PRIME NUMBERS and { p x, p prime : sq n βq logqx } 2 25 x2β e c logqlogx /2+ǫ. Toapproachthisproblemwedonotexaminethesum p x 2sqp,andinstead exploit the fact that the multinomial distribution is sharply peaked, using results regarding primes in small intervals to attain the desired lower bound. From theorem, problem follows as a corollary. In fact, we have that for any α < there are infinitely many primes where the proportion of s in their binary expansion greater than α. 2. The Tail Distribution We start by providing bounds on the size of the tails of the multinomial distribution. Ä ÑÑ º Chernoff bound Given 0 < a < 2 < b <, we have that { n < q k : bq k s q n } q k exp 2k b 2, 9 2 and { n < q k : s q n aq k } q k exp 2k 9 a Proof. Each integer in the interval [ 0,q k ] can be written so that it has exactly k digits base q, by adding zeros in front where neccesary. The distribution of each of the k digit s is an independent random variable which corresponds to the roll of a q sided dice with sides 0,,...,. Normalizing, let ξ be a random variable where P ξ i = 2 q j = q for 0 j q, and for each i let ξ i = ξ. Our goal is then to examine P γ ξ +ξ 2 + +ξ k. k For any nonnegative t, P γ ξ +ξ 2 + +ξ k k E e tξ+ +ξ k e tkγ = e tγ E e tξ k 65

4 where ERIC NASLUND = e kit,γ It,γ = tγ loge e tξ. Evaluating the expectation, we find that E e tξ = j=0 2j et = e t q q This gives rise to the series expansion log sinh t+ t q sinh t j=0 where the error term holds uniformely for inequality e 2t j sinh = q sinh = q2 6q 2t2 q4 80 4t4 +O loge e tξ q2 t 2 6q 2, t+ t. t q 6 q 6t6, qt <. This allows us to prove the for q,t satisfying qt <. To maximize It,γ, we choose t = γ 3 q+, and obtain the upper bound P γ ξ +ξ 2 + +ξ k k exp k 6 q γ 2, q + which provesequation upon taking γ = 2b,and noting that q+ 3 since q 2. The proof of equation 2 is identical, as the distribution is symmetric. Next, we will need the best existing results on prime gaps. In 200, Baker, Harman and Pintz proved that for x x 0, x θ π x+x θ πx 9 00logx for any θ []. Armed with equation 3 and lemma, we are now ready to prove theorem. Proof. Let α = α + rx where rx is chosen so that α < Let k = [ log q x ], l = 2 α k so that q k x and q l x Consider the interval [ q k q l, q k ], which is an interval whose first k l digits base q are 66 3

5 THE TAIL DISTRIBUTION OF THE SUM OF DIGITS OF PRIME NUMBERS equal to q. By Baker, Harman and Pintz, if x is sufficiently large, there will be 9 q l 00logq k 9 q l 00logx primes in this interval, where the constant is explicit. By equation 2, there are at most q l exp 2lδ2 9 integers between 0 and q l which have digit sum less than q l 2 δ loglogx. Letting δ = 9 l, it follows that there are at most q l /log 2 x integers in the interval [ q k q l, q k ] whose digit sum is less than logl q k l+q l 2. l For x e 00, logx 00, which implies that for sufficiently large x there are more than 2 q l 25 logx primes in the interval [ q k q l, q k ] with digit sum larger than α q k q lloglogx. Expanding α = α+rx, and taking rx = than αq log q x, loglogx log q x yields a digit sum greater which proves the result since q l logx x2 α x 2rx x 2 α exp 4 logq logx loglogx. logx The proof of the lower bound for the size of the corresponding set of primes with s q p βq log q x for < β 2 is identical. REFERENCES [] R. C. Baker, G. Harman, and J. Pintz. The difference between consecutive primes. II. Proc. London Math. Soc. 3, , [2] Arthur H. Copeland and Paul Erdös. Note on normal numbers. Bull. Amer. Math. Soc., , [3] Michael Drmota, Christian Mauduit, and Joël Rivat. Primes with an average sum of digits. Compos. Math., , [4] Iekata Shiokawa. On the sum of digits of prime numbers. Proc. Japan Acad., , ,. 67

6 ERIC NASLUND Received May 8, 203 Accepted August 20, 203 Eric Naslund Fine Hall, Washington Road Princeton NJ USA 68