Benford's Law for Coefficients of Newforms
BBENFORD’S LAW FOR COEFFICIENTS OF NEWFORMS
MARIE JAMESON, JESSE THORNER, AND LYNNELLE YE
Abstract.
Let f ( z ) = (cid:80) ∞ n =1 λ f ( n ) e πinz ∈ S new k (Γ ( N )) be a newform of even weight k ≥ ( N ) without complex multiplication. Let P denote the set of all primes. Weprove that the sequence { λ f ( p ) } p ∈ P does not satisfy Benford’s Law in any integer base b ≥ b ≥ S in base b , the set A λ f ( b, S ) := { p prime : the first digits of λ f ( p ) in base b are given by S } has logarithmic density equal to log b (1 + S − ). Thus { λ f ( p ) } p ∈ P follows Benford’s Law withrespect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture. Introduction and Statement of Results
In 1881, astronomer Simon Newcomb [13] made the observation that certain pages of log-arithm tables were much more worn than others. Users of the tables referenced logarithmswhose leading digit is 1 more frequently than other logarithms, contrary to the naive expec-tation that all of the logarithms would be referenced uniformly. In 1938, Benford [3] madea similar observation for a variety of sequences.This bias, now known as Benford’s Law, is given as follows. Let N denote the set ofpositive integers, and let I ⊂ N be an infinite subset. Fix an integer base b ≥ S in base b . For a given function g : N → R , define(1.1) A g ( b, S ) := { i ∈ N : the first digits of g ( i ) in base b are given by S } . We define the arithmetic density of A g ( b, S ) within I by(1.2) δ I ( A g ( b, S )) = lim x →∞ { i ≤ x : i ∈ I ∩ A g ( b, S ) } { i ≤ x : i ∈ I} . We say that the sequence { g ( i ) } i ∈I satisfies Benford’s Law, or that { g ( i ) } i ∈I is Benford, iffor any integer base b ≥ S in base b ,(1.3) δ I ( A g ( b, S )) = log b (1 + S − ) . It is easy to show [5] that { g ( i ) } i ∈I is Benford if and only if the set { log b ( g ( i )) : i ∈ I} is equidistributed modulo 1 for each base b (setting log b ( g ( i )) = 0 if g ( i ) = 0). For somegeneral surveys on Benford’s Law, we refer the reader to [7, 8, 11, 15].Stirling’s approximation of Γ( s ) in conjunction with standard equidistribution resultsquickly yields that { n ! } n ∈ N is a Benford sequence. However, if g ( n ) = n a for any fixed a ∈ R , then { g ( n ) } n ∈ N is not a Benford sequence, forlim sup n ∈ N | n log( | g ( n + 1) /g ( n ) | ) | < ∞ , contradicting equidistribution of log b ( g ( n )) modulo 1. Taking a = 1, one sees that thepositive integers are not Benford. a r X i v : . [ m a t h . N T ] N ov uch real world data, including lengths of rivers, populations of nations, and heights ofskyscrapers exhibit behavior which is suggestive of Benford’s Law (when restricted to base10). There are also several settings in which Benford’s law arises that are of arithmeticinterest. In [4], several dynamical systems such as linearly-dominated systems and non-autonomous dynamical systems are shown to be Benford. Benford’s Law is proven fordistributions of values of L -functions [9] and the 3 x + 1 problem [9, 10]. In [1], the image ofthe partition function is shown to be Benford, as well as the coefficients of an infinite classof modular forms with poles.While the positive integers are not Benford, we can say even more. Specifically, if we take g ( n ) = n , the limits defining δ N ( A g ( b, δ N ( A g (2 , b ≥ lim inf x →∞ { i ≤ x : i ∈ N ∩ A g (10 , } { i ≤ x : i ∈ N } = 19 , lim sup x →∞ { i ≤ x : i ∈ N ∩ A g (10 , } { i ≤ x : i ∈ N } = 59 . However, if we change our notion of density, the first digits of the integers still satisfy thedistribution in Benford’s law. We define the logarithmic density of A g ( b, S ) in I by (1.4) (cid:101) δ I ( A g ( b, S )) = lim x →∞ (cid:88) i ≤ xi ∈I∩ A g ( b,S ) i − (cid:88) i ≤ xi ∈I i − . With this modified notion of density, we have that (cid:101) δ N ( A g ( b, S )) exists and equals log b (1+ S − )for any base b ≥ S in base b [6]. In light of this fact, we say that a sequence { g ( i ) } i ∈I is logarithmically Benford if(1.5) (cid:101) δ I ( A g ( b, S )) = log b (1 + S − )for any base b and any string S in base b . We note that if a set has an arithmetic density,then it also has a logarithmic density, and the two densities are equal. Thus all Benfordsequences are logarithmically Benford. Remark.
Logarithmic density is closely related to Dirichlet density, which is ubiquitous innumber theory. For integer sequences, logarithmic density and Dirichlet density have equiv-alent definitions; this (and much more) is proven in Part 3 of [17]. For a discussion onDirichlet density in the context of the prime number theorem for arithmetic progressions orthe Chebotarev density theorem, see Chapter 7 of [12].Since the n -th prime is asymptotically equal to n log( n ) by the prime number theorem,one sees that the primes are not Benford. However, Whitney [18] proved that for g ( n ) = n ,we have (cid:101) δ P ( A g (10 , S )) = log (1 + S − ) for any string S in base 10. The fact that the primesare logarithmically Benford follows easily from Whitney’s proof. Serre briefly discusses thisproblem for the primes in Chapter 4, Section 5 of [16].In this paper, we consider sequences given by Fourier coefficients of certain modular formswithout complex multiplication (see [14]). Specifically, let(1.6) f ( z ) = ∞ (cid:88) n =1 λ f ( n ) q n ∈ S new k (Γ ( N )) , q = e πiz e a newform (i.e., a holomorphic cuspidal normalized Hecke eigenform) of even weight k and trivial nebentypus on Γ ( N ) that does not have complex multiplication. The Fouriercoefficients λ f ( n ) of such a newform will be real. We consider sets of the form A λ f ( b, S ) = { n ∈ N : the first digits of λ f ( n ) in base b are given by S } . One important example of the newforms under our consideration is the weight 12 newformon Γ (1) given by ∆( z ) = q ∞ (cid:89) n =1 (1 − q n ) = ∞ (cid:88) n =1 τ ( n ) q n , where τ ( n ) is the Ramanujan tau function. Consider the following table. x { p ≤ x : the first digit of τ ( p ) is 1 } /π ( x )10 . . . . . . . . . . . . If { τ ( p ) } p ∈ P were Benford, then we would have δ P ( A τ (10 , (2) ≈ . δ P ( A τ (10 , (2). However, the plot in Figure 1 indicates that this conclusionis very far from the truth. Figure 1.
The proportion of primes p ≤ x for which τ ( p ) has leading digit 1for x ≤ × .It turns out to be the case that the arithmetic density δ P ( A τ (10 , (cid:101) δ P ( A τ (10 , (2). More generally, we prove the twofollowing results. Theorem 1.
Let f ( z ) = (cid:80) ∞ n =1 λ f ( n ) q n ∈ S new k (Γ ( N )) be a newform of even weight k ≥ without complex multiplication. The arithmetic density δ P ( A λ f ( b, does not exist for anyinteger base b ≥ , and the arithmetic density δ P ( A λ f (2 , does not exist. Thus the sequence { λ f ( p ) } p ∈ P is not Benford. emark. For a string S in base b , the method used in the proof of Theorem 1 can be modifiedto show that there are infinitely many primes p such that λ f ( p ) begins with S . Since thisfact is also a direct consequence of Theorem 2, we omit the details. Theorem 2.
Let f ( z ) = (cid:80) ∞ n =1 λ f ( n ) q n ∈ S new k (Γ ( N )) be a newform of even weight k ≥ without complex multiplication. Let b ≥ be a given integer base, and let S be an initial stringof digits in base b . We have (cid:101) δ P ( A λ f ( b, S )) = log b (1 + S − ) . Thus { λ ( p ) } p ∈ P is logarithmicallyBenford. Acknowledgements.
The authors thank Ken Ono and the anonymous referee for theircomments and Ken Ono for suggesting this project. The authors used Maple 18, Mathemat-ica 9, and SAGE for the numerical computations and plots.2.
The Sato-Tate Conjecture
Let f ( z ) = ∞ (cid:88) n =1 λ f ( n ) q n ∈ S new k (Γ ( N ))be a newform of even weight k ≥ ( N ). The Fourier coefficients λ f ( n ) will lie in the ring of integers of a totally real number field. By Deligne’s proof of theWeil conjectures, we have that for every prime p , | λ f ( p ) | ≤ p k − . Thus there exists θ p ∈ [0 , π ] satisfying λ f ( p ) = 2 p k − cos θ p . Around 1960, Sato and Tate studied the sequence { cos θ p } as p varies through the primeswhen f is the newform associated to an elliptic curve E/ Q without complex multiplication.All such newforms have weight k = 2, and if p is prime, then λ f ( p ) = p + 1 − E ( F p ) , | λ f ( p ) | ≤ √ p. where E ( F p ) is the number of F p -rational points on E/ Q . Thus cos θ p is the normalizederror in approximating E ( F p ) with p + 1. Sato and Tate conjectured a distribution forthe sequence { cos θ p } , and this conjecture was later generalized to a much larger class ofnewforms. This conjecture, which we now state, was proven by Barnet-Lamb, Geraghty,Harris, and Taylor [2]. Theorem 3 (The Sato-Tate Conjecture) . Let f ( z ) = (cid:80) ∞ n =1 λ f ( n ) q n ∈ S new k (Γ ( N )) be anewform of even weight k ≥ without complex multiplication. The sequence { cos θ p } isequidistributed in the interval [ − , with respect to the measure dµ ST = 2 π √ − t dt. In other words, if I ⊂ [ − , is a subinterval and we define π f,I ( x ) = { p ≤ x : cos θ p ∈ I } , then as x → ∞ , π f,I ( x ) ∼ µ ST ( I ) π ( x ) . he following immediate corollary of the Sato-Tate Conjecture plays an important role inthe proof of Theorem 2. Corollary 1.
Let f ( z ) = (cid:80) ∞ n =1 λ f ( n ) q n ∈ S new k (Γ ( N )) be a newform of even weight k ≥ without complex multiplication. Let I ⊂ [ − , be an interval. As x → ∞ , we have (cid:88) p ≤ x cos θ p ∈ I p − ∼ µ ST ( I ) (cid:88) p ≤ x p − . Proof of theorem 1
To prove Theorem 1, we use the Sato-Tate Conjecture to construct many large intervalson which the proportion of primes p for which λ f ( p ) has leading digit 1 in a given base b ≥ b = 2, we use the leading digits 10because all nonzero real numbers have leading digit 1 in their base 2 expansion.) This showsthat { λ f ( p ) } p ∈ P is not Benford in any base. To do this, we first state a lemma about theSato-Tate measures of certain intervals. Lemma 1.
Fix b ≥ and let c be a sufficiently large positive integer. For d = 1 , , set I d, ( c ) = (cid:91) j ∈ Z (cid:20) b − j db c − , b − j db c − (cid:21) ∩ [0 , . Then µ ST ( I , ( c )) − µ ST ( I , ( c )) > . Proof of Lemma 1. If c >
2, then b < b c − b c − < b c − b c − < b < b c b c − < < b c b c − . Thus I , ( c ) = (cid:91) j ≥− c +1 (cid:20) b − j b c − , b − j b c − (cid:21) = (cid:91) m ≥ (cid:20) b − − m − b − c , b − − m − b − c (cid:21) and I , ( c ) = (cid:20) b c b c − , (cid:21) ∪ (cid:91) j> − c (cid:20) b − j b c − , b − j b c − (cid:21) = (cid:20) − b − c , (cid:21) ∪ (cid:91) m> (cid:20) b − m − b − c , b − m − b − c (cid:21) . For all b ≥
3, we have lim c →∞ (cid:18) µ ST (cid:18)(cid:20) − b − c , (cid:21)(cid:19) − µ ST (cid:18)(cid:20) b − − b − c , b − − b − c (cid:21)(cid:19)(cid:19) = 2 µ ST ([1 / , − µ ST (cid:0)(cid:2) b − , b − (cid:3)(cid:1) > . Thus for all sufficiently large c , we have µ ST (cid:18)(cid:20) − b − c , (cid:21)(cid:19) − µ ST (cid:18)(cid:20) b − − b − c , b − − b − c (cid:21)(cid:19) > imilarly, we have µ ST (cid:18)(cid:20) b − m − b − c , b − m − b − c (cid:21)(cid:19) − µ ST (cid:18)(cid:20) b − − m − b − c , b − − m − b − c (cid:21)(cid:19) > for all m > , and the result follows. (cid:3) Now we may prove Theorem 1.
Proof of Theorem 1.
Let f be a newform as in the statement of the theorem and let b ≥ δ P ( A λ f ( b, S )) does not exist, it suffices to showthat for some fixed β >
1, the value oflim n →∞ { αβ n ≤ p < γβ n : p ∈ A λ f ( b, } { αβ n ≤ p < γβ n } varies with different choices of values of α and γ with α < γ .We start with some preliminary constructions. For S ∈ { , . . . , b − } , define I ,S ( c ) = (cid:91) j ∈ Z (cid:20) Sb − j b c − , ( S + 1) b − j b c − (cid:21) ∩ [0 , ,I ,S ( c ) = (cid:91) j ∈ Z (cid:20) Sb − j b c − , ( S + 1) b − j b c − (cid:21) ∩ [0 , . Fix 0 < (cid:15) < /
40, and let c be a sufficiently large positive integer so that Lemma 1 holdsand µ ST (cid:0) [0 , − ∪ b − S =1 I d,S ( c ) (cid:1) < (cid:15)/ d = 1 ,
2. Let β = b k − , α = ( b c − ) k − , and γ = ( b c − ) k − . We consider the primes p such that(3.1) ( b c − b n ≤ p k − < ( b c − b n . Note that if p is bounded as in (3.1) and | cos θ p | ∈ I ,S , then | λ f ( p ) | ∈ [ Sb n − j , ( S + 1) b n − j )for some j ∈ Z ; that is, its first digits are given by S . By letting c be sufficiently large andsetting S = 1, the Sato-Tate Conjecture implies that(3.2) (cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ { α β n ≤ p < γ β n : p ∈ A λ f ( b, } { α β n ≤ p < γ β n } − µ ST ( I , ( c )) (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) . Similarly, by letting α = ( b c − ) k − and γ = ( b c − ) k − , we find that(3.3) (cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ { α β n ≤ p < γ β n : p ∈ A λ f ( b, } { α β n ≤ p < γ β n } − µ ST ( I , ( c )) (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) . Now, suppose to the contrary that δ P ( A λ f ( b, | µ ST ( I , )( c ) − µ ST ( I , )( c ) | < (cid:15) < , which contradicts Lemma 1. The theorem now follows for bases b ≥
3. For b = 2, one arrivesat the same conclusion as for b ≥ I d, ( c ) for d = 1 , b = 2 are essentially the same. (cid:3) . Proof of Theorem 2
For this section, p will always denote a prime. Let f be a newform satisfying the hypothesesof Theorem 2. Let b ≥ S be a string of digits in base b . By the definitionof a logarithmically Benford sequence and the estimate(4.1) (cid:88) p ≤ x p − ∼ log log x, a proof of Theorem 2 will follow from proving that as x → ∞ ,(4.2) (cid:88) p ≤ xp ∈ A λf ( b,S ) p − ∼ log b (1 + S − ) log log x. This will be a consequence of the following key lemma.
Lemma 2.
Let f ( z ) = (cid:80) ∞ n =1 λ f ( n ) q n ∈ S new k (Γ ( N )) be a newform of even weight k ≥ without complex multiplication. Let b ≥ be a given base, let S be an initial string of digitsin base b , and let (cid:96) > max { S, } be an integer. As x → ∞ , we have (1 + o (1))(log b (1 + S − ) − log(1 + (cid:96) − )) log log x ≤ (cid:88) p ≤ xp ∈ A λf ( b,S ) | cos θ p | >(cid:96) − p − ≤ (1 + o (1))(log b (1 + S − ) + log(1 + (cid:96) − )) log log x + 2 log log (cid:96). Proof.
We prove the upper bound; the lower bound is proven similarly. Writing λ f ( p ) =2 p k − cos θ p , we first observe that (cid:88) p ≤ xp ∈ A λf ( b,S ) | cos θ p | >(cid:96) − p − = ∞ (cid:88) t = −∞ (cid:96) − (cid:88) i = (cid:96) (cid:88) p ≤ xS · b t ≤| λ f ( p ) | < ( S +1) · b ti(cid:96) < | cos θ p |≤ i +1 (cid:96) p − ≤ ∞ (cid:88) t = −∞ (cid:96) − (cid:88) i = (cid:96) (cid:88) p ≤ x (cid:16) S(cid:96) i +1 b t (cid:17) k − ≤ p ≤ (cid:18) ( S +1) (cid:96) i b t (cid:19) k − i(cid:96) < | cos θ p |≤ i +1 (cid:96) p − . To bound the contribution when t <
0, all of the primes in the sum are at most ( ( S +1) (cid:96)b ) k − ;since (cid:96) > max { S, } , we have − (cid:88) t = −∞ (cid:96) − (cid:88) i = (cid:96) (cid:88) p ≤ x (cid:16) S(cid:96) i +1 b t (cid:17) k − ≤ p ≤ (cid:18) ( S +1) (cid:96) i b t (cid:19) k − i(cid:96) < | cos θ p |≤ i +1 (cid:96) p − ≤ (cid:88) p ≤ ( ( S +1) (cid:96)b ) k − p − ≤ (cid:96). o bound the contribution when t ≥
0, fix (cid:96) ≤ i ≤ (cid:96) −
1. Recall that we may write λ f ( p ) = 2 p k − cos θ p . If ( ( S +1) (cid:96) i b t ) k − ≤ x , then Corollary 1 implies that (cid:88) p ≤ xS · b t ≤| λ f ( p ) | < ( S +1) · b ti(cid:96) < | cos θ p |≤ i +1 (cid:96) p − ≤ (cid:88) (cid:16) S(cid:96) i +1 b t (cid:17) k − ≤ p ≤ (cid:18) ( S +1) (cid:96) i b t (cid:19) k − i(cid:96) < | cos θ p |≤ i +1 (cid:96) p − = 2(1 + o (1)) µ ST (cid:18)(cid:20) i(cid:96) , i + 1 (cid:96) (cid:21)(cid:19) (cid:88) (cid:16) S(cid:96) i +1 b t (cid:17) k − ≤ p ≤ (cid:16) ( S +1) (cid:96) i b t (cid:17) k − p − = 2(1 + o (1)) µ ST (cid:18)(cid:20) i(cid:96) , i + 1 (cid:96) (cid:21)(cid:19) log (cid:32) log b ( (cid:96) i ) + log b ( S + 1) + t log b ( (cid:96) i +1 ) + log b ( S ) + t (cid:33) . Setting B = log b ( (cid:96) i +1 ) + log b ( S ) and B = log b ( (cid:96) i ) + log b ( S + 1), we have for any large N that N (cid:88) t =0 (cid:88) (cid:16) S(cid:96) i +1 b t (cid:17) k − ≤ p ≤ (cid:16) ( S +1) (cid:96) i b t (cid:17) k − p − ∼ log N (cid:89) t =0 B + tB + t . Switching the order of summation, we obtain the inequality ∞ (cid:88) t =0 (cid:96) − (cid:88) i = (cid:96) (cid:88) p ≤ xS · b t ≤| λ f ( p ) | < ( S +1) · b ti(cid:96) < | cos θ p |≤ i +1 (cid:96) p − ≤ (cid:96) − (cid:88) i = (cid:96) µ (cid:18)(cid:20) i(cid:96) , i + 1 (cid:96) (cid:21)(cid:19) (1 + o (1)) ∞ (cid:88) t =0 (cid:88) (cid:16) S(cid:96) i +1 b t (cid:17) k − ≤ p ≤ min (cid:40)(cid:16) ( S +1) (cid:96) i b t (cid:17) k − ,x (cid:41) p − ≤ (cid:96) − (cid:88) i = (cid:96) µ (cid:18)(cid:20) i(cid:96) , i + 1 (cid:96) (cid:21)(cid:19) (1 + o (1)) log (cid:89) ≤ t ≤ log b (cid:32) ix k − S +1) (cid:96) (cid:33) B + tB + t + 2 log log (cid:96). Using Euler’s formula for the Gamma functionΓ( z ) = lim n →∞ n ! n z (cid:81) ni =0 ( z + i ) , e find that the contribution from t ≥ (cid:96) − (cid:88) i = (cid:96) µ (cid:18)(cid:20) i(cid:96) , i + 1 (cid:96) (cid:21)(cid:19) (1 + o (1)) (cid:18) log Γ( B )Γ( B ) + ( B − B ) log log x (cid:19) ≤ (1 + o (1)) (cid:0) log b (cid:0) (cid:96) − (cid:1) + log b (cid:0) S − (cid:1)(cid:1) log log x. This proves the claimed upper bound. Using the inequality (cid:88) p ≤ xS · b t ≤ p k − | cos θ p | < ( S +1) · b ti(cid:96) < | cos θ p |≤ i +1 (cid:96) p − ≥ (cid:88) ( S(cid:96)i b t ) k − ≤ p ≤ ( ( S +1) (cid:96)i +1 b t ) k − i(cid:96) < | cos θ p |≤ i +1 (cid:96) p − for (cid:96) ≤ i ≤ (cid:96) −
1, the lower bound is proven similarly. (cid:3)
Proof of Theorem 2.
Let 0 < (cid:15) < log b (2), let (cid:96) > max { b (cid:15) − , S, } be an integer, and let x > exp((log (cid:96) ) /(cid:15) ). If we write λ f ( p ) = 2 p k − cos θ p with θ p ∈ [0 , π ], then(4.3) (cid:88) p ≤ xp ∈ A f ( b,S ) p − = (cid:88) p ≤ xp ∈ A λf ( b,S ) | cos θ p |≤ (cid:96) − p − + (cid:88) p ≤ xp ∈ A λf ( b,S ) | cos θ p | >(cid:96) − p − . By Corollary 1, the first term is at most(4.4) (cid:88) p ≤ x ≤| cos θ p |≤ (cid:96) − p − = (2 + o (1)) µ ST (cid:0)(cid:2) , (cid:96) − (cid:3)(cid:1) log log x. Using Lemma 2, we now have (1 + o (1))(log b (1 + S − ) − log b (1 + (cid:96) − )) log log x ≤ (cid:88) p ≤ xp ∈ A f ( b,S ) p − ≤ (1 + o (1))(log b (1 + S − ) + log b (1 + (cid:96) − ) + 2 µ ST ([0 , (cid:96) − ])) log log x + 2 log log (cid:96). Thus (1 + o (1))(log b (1 + S − ) − (cid:15) ) log log x ≤ (cid:88) p ≤ xp ∈ A f ( b,S ) p − ≤ (1 + o (1))(log b (1 + S − ) + 9 log( b ) (cid:15) ) log log x. Letting (cid:15) →
0, we obtain (4.2), as desired. (cid:3)
References [1] Theresa C. Anderson, Larry Rolen, and Ruth Stoehr,
Benford’s law for coefficients of modular forms andpartition functions , Proc. Amer. Math. Soc. (2011), no. 5, 1533–1541. MR2763743 (2012f:11087)[2] Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor,
A family of Calabi-Yau va-rieties and potential automorphy II , Publ. Res. Inst. Math. Sci. (2011), no. 1, 29–98. MR2827723(2012m:11069)
3] Frank Benford,
The law of anomalous numbers , Proc. Amer. Philos. Soc. (1938), no. 4, 551–572.[4] Arno Berger, Leonid A. Bunimovich, and Theodore P. Hill, One-dimensional dynamical systems andBenford’s law , Trans. Amer. Math. Soc. (2005), no. 1, 197–219. MR2098092 (2005m:37017)[5] Persi Diaconis,
The distribution of leading digits and uniform distribution mod 1, Ann. Probability (1977), no. 1, 72–81. MR0422186 (54 Note on the initial digit problem , J. Fib. Quart. (1969), no. 5, 474–475.[7] Theodore P. Hill, The significant-digit phenomenon , Amer. Math. Monthly (1995), no. 4, 322–327.MR1328015 (96f:11101)[8] ,
A statistical derivation of the significant-digit law , Statist. Sci. (1995), no. 4, 354–363.MR1421567 (98a:60021)[9] Alex V. Kontorovich and Steven J. Miller, Benford’s law, values of L -functions and the x + 1 problem ,Acta Arith. (2005), no. 3, 269–297. MR2188844 (2007c:11085)[10] Jeffrey C. Lagarias and K. Soundararajan, Benford’s law for the x + 1 function , J. London Math. Soc.(2) (2006), no. 2, 289–303. MR2269630 (2007h:37007)[11] Steven J. Miller and Ramin Takloo-Bighash, An invitation to modern number theory , Princeton Uni-versity Press, Princeton, NJ, 2006. With a foreword by Peter Sarnak. MR2208019 (2006k:11002)[12] J¨urgen Neukirch,
Algebraic number theory , Grundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated fromthe 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.MR1697859 (2000m:11104)[13] Simon Newcomb,
Note on the Frequency of Use of the Different Digits in Natural Numbers , Amer. J.Math. (1881), no. 1-4, 39–40. MR1505286[14] Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series , CBMSRegional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Math-ematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.MR2020489 (2005c:11053)[15] Ralph A. Raimi,
The first digit problem , Amer. Math. Monthly (1976), no. 7, 521–538. MR0410850(53 A course in arithmetic , Springer-Verlag, New York-Heidelberg, 1973. Translated from theFrench, Graduate Texts in Mathematics, No. 7. MR0344216 (49
Introduction to analytic and probabilistic number theory , Cambridge Studies inAdvanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995. Translated from thesecond French edition (1995) by C. B. Thomas. MR1342300 (97e:11005b)[18] R. E. Whitney,
Initial digits for the sequence of primes , Amer. Math. Monthly (1972), 150–152.MR0304337 (46 Marie Jameson, Department of Mathematics, University of Tennessee, Knoxville, TN37996
E-mail address : [email protected] Jesse Thorner, Department of Mathematics and Computer Science, Emory University,Atlanta, Georgia 30322
E-mail address : [email protected] Lynnelle Ye, Department of Mathematics, Harvard University, Cambridge, MA 02138
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