On congruences for certain sums of E. Lehmer s type

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1 On congruences for certain sums of E. Lehmer s type Shigeru Kanemitsu, Takako Kuzumaki, Jerzy Urbanowicz To cite this version: Shigeru Kanemitsu, Takako Kuzumaki, Jerzy Urbanowicz. On congruences for certain sums of E. Lehmer s type. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 04, 8. <hal-05408> HAL Id: hal Submitted on Jan 06 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Hardy-Ramanujan Journal 8 (04, -8 submitted /0/04, accepted /06/05, revised 7/04/05 On congruences for certain sums of E. Lehmer s type Shigeru Kanemitsu, Takako Kuzumaki and Jerzy Urbanowicz Abstract. Let n > be an odd natural number and let r ( < r < n be a natural number relatively prime to n. Denote by χ n the principal character modulo n. In Section we prove some new congruences for the sums T r,k (n = [ n r ] i= ( χn(i i (mod k ns+ for s {0,, }, for all divisors r of 4 and for some natural numbers k.we obtain 8 new congruences for T r,k (n, which generalize those obtained in [Ler05], [Leh8] and [Sun08] if n = p is an odd prime. Section 4 is an appendix by the second and third named authors. It contains some new congruences for the sums U r(n = [ n r ] χ n(i i= (mod n n ri s+ for s {0,, } and r 4. The congruences obtained for the sums U r(n extend those proved in [Leh8],[CFZ07] and [CP09] for r {,, 4, 6} and s =. The sums are rational linear combinations of Euler s quotients and in the cases when r {8,, 4}, also of the numbers nϕ(n B nϕ(n,χ ( pnϕ(n, where the generalized Bernoulli numbers B nϕ(n,χ are attached to even quadratic characters χ of conductors dividing 4. Keywords. Congruence, generalized Bernoulli number, special value of L-function, ordinary Bernoulli number, Bernoulli polynomial, Euler number. 00 Mathematics Subject Classification. primary B68; secondary R4, A07.. Notation and introduction Let n > be odd and let χ 0,n (sometimes abbreviated as χ n be the principal Dirichlet character modulo n (with χ 0. designating the constant function χ 0. (x = for all integers x. For r > prime to n denote by q r (n the Euler quotient, i.e., q r (n = rϕ(n. n Here and throughout the paper ϕ is the Euler ϕ-function and B n,χ denotes the n-th generalized Bernoulli number attached to the Dirichlet character χ modulo n defined by the generating function n a= χ(ate at e nt = m=0 B m,χ t m m!. Given the discriminant d of a quadratic field, let χ d denote its quadratic character (Kronecker symbol. We shall denote by χ d,n the character χ d modulo n. It was proved in [Car59] that the numbers B i,χd /i are rational integers unless d = 4 or d = ±p, where p is an odd prime of a special form. If d = 4 and i is odd, then the numbers E i = B i,χ 4 /i are odd integers, called the Euler numbers. If d = ±p, then the numbers B i,χd have p in their denominators and pb i,χd p (mod p ordp(i+. We consider the ordinary Bernoulli numbers B i (i.e., generalized Bernoulli numbers attached to the trivial primitive character χ 0,, except when i = for which B,χ0, = = B and the so-called D-numbers defined in [Kle55] and [Ern79] by D i = B i,χ /i for i odd, having powers of in Passed away on September 6, 0. We thank episciences.org for providing open access hosting of the electronic journal Hardy-Ramanujan Journal The first author was supported by the JSPS, Grant-in-aid for scientific research No Which were omitted in [Leh8], [CFZ07] and [CP09]. E. Lehmer proved her congruences in the case when n = p is an odd prime. The congruences proved in [CFZ07] and [CP09] are for n odd and not divisible by. See also [Ler05].

3 . Notation and introduction their denominators. We also consider the rational integers A i = B i,χ8 /i, F i = B i,χ χ 4 /i and G i = B i,χ χ 8 /i, if i even, and C i = B i,χ 8 /i and H i = B i,χ χ 8 /i if i odd. In this paper we shall consider congruences for the character sums with negative weight T r,k (n = 0<i< n r χ n (i i k modulo powers n s+ for n > odd and s {0,, } where χ n = χ 0,n and r (r 4 and < r < n is coprime to n, and k is subject to the condition k n s ϕ(n. Note that since χ n (i = 0 for (i, n >, the sum is over (i, n =. The central role in this paper is played by an identity proved in [SUZ95, p.76,(6]. Let χ be a Dirichlet character modulo M, N a positive integral multiple of M, and r(> a positive integer prime to N. Then for any integer m 0 we have (m + r m 0<i< N r χ(ii m = B m+,χ r m + χ(r ϕ(r ψ G(r ψ( NB m+,χψ (N, (. where the sum on the right hand side is taken over all Dirichlet characters ψ modulo r. We denote by G(r the group of all such characters; then #G(r = ϕ(r. Here B n,χ (X = n ( n i=0 i Bn i,χ X i denotes the n-th generalized Bernoulli polynomial attached to χ. Since r 4, the group G(r has exponent and all characters modulo r are quadratic. If the character χ modulo M is induced from a character χ modulo some divisor of M then B n,χ = B n, χ ( χ(pp n, (. p M where the product is taken over all primes p dividing M. If (i, n =, then by Euler s theorem we have i ϕ(n (mod n, and more generally, ϕ(n s+ = n s ϕ(n and i ϕ(nns (mod n s+ for s 0. Given r prime to n and integers s 0, k we denote S r,k,s (n = χ n (ii nsϕ(n k. Then we have the congruence 0<i< n r T r,k (n S r,k,s (n (mod n s+, (. which allows us to study T r,k (n through S r,k,s (n. In this paper we specialize to the case that r is a divisor of 4. Then the group G(r = (Z/rZ has exponent, so all the elements ψ are quadratic. The main results of the paper are congruences for the sums T r,k (n modulo n s+ for s {0,, } proved in Section. The congruences will be obtained by applying identity (. to the sums S r,k,s (n. They extend those proved by M. Lerch [Ler05], E. Lehmer [Leh8] and Z.-H. Sun [Sun08] in the case when n = p is an odd prime. In principle, the congruences in this particular case have a different form from those obtained for any natural odd n. Sometimes it is not easy to derive the former congruences from the latter. We shall do it in the second part of the paper. This identity was earlier successfully exploited in [SUZ95], [SUV99] and [FUW97] to solve some other problems. See also the book [UW00] devoted to the identity and related problems.

4 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type Two such congruences modulo n were earlier obtained, by using (., in [Cai0] for r =, k = and in [KUW] for r = 4, k =. In the present paper we find 8 new congruences for the sums T r,k (n (mod n s+ for s {0,, }, r 4 and k, in particular for k = or. Most of our congruences for T r,k (n have not been known earlier even in the particular case when n = p is a prime. The machinery introduced in [SUZ95] is much more efficient than the methods exploited in [Ler05], [Leh8] and [Sun08]. In Section 4, which is an appendix by the second and third named authors, we find congruences for the sums U r (n = χ n (i n ri 0<i< n r modulo n s+ for s {0,, } and all r 4 ( < r < n coprime to n. To obtain such congruences it suffices to use appropriate congruences for the sums T r,k (n or, by virtue of (., for the sums S r,k,s (n for k {,, }. The congruences are consequences of those proved in Section and for s = extend those obtained in [CFZ07],[CP09] for r {,, 4, 6}. They have the same form as those proved by E. Lehmer [Leh8] if n = p is an odd prime. The sums are rational linear combinations of Euler s quotients and in the cases when r {8,, 4}, omitted in [Leh8], [CFZ07] and [CP09], also of the numbers nϕ(n B nϕ(n,χ ( pnϕ(n, where B nϕ(n,χ are the generalized Bernoulli numbers attached to even quadratic characters χ of conductors dividing 4. Also some new congruences for s = with an additional summand n B r n ϕ(n ( pn ϕ(n for all r 4 are obtained.. Some auxiliary formulae The idea exploited in [Cai0] and [KUW] to use identity (. to extend classical congruences for the sums T r,k (n seems to be very efficient. This identity allows us to obtain almost automatically many new congruences. Usually the proofs using (. are much easier, more unified and much shorter than those applying other methods. The general scheme of reasoning is uniform. To obtain congruences for the sums T r,k (n modulo n s+ we first determine, using (., the sums S r,k,s (n modulo n s+. We substitute in (. m = n s ϕ(n k and N = M = n, by the definition of S r,k,s (n. We assume that r 4, n > is odd. If r then we have (n, r =. If r, then we additionally assume that n is not divisible by. Note that, since r 4, all generalized Bernoulli numbers occurring in S r,k,s (n are rational. Thus, throughout the paper, we write m = n s ϕ(n k 0. Consequently, we obtain S r,k,s (n = S + S, (.4 where, by (., S = B m+,χ 0,n m + = B m+ m + ( p m (.5 and S = ϕ(r(m + r m ψ G(r ψ( nb m+,χ0,n ψ(n. Note that χ 0,n is even. Thus, if m 0 is even, then B m+ = 0, and so S = 0. If m = 0, then p m = 0, and so S = 0 too. Otherwise, in view of (.5, we have S 0. Furthermore, S = ϕ(r(m + r m ψ G(r ψ( n m+ i=0 ( m + i B i,χ0,n ψn m+ i cf.[suz95, p.74,(6]

5 4. Some auxiliary formulae and B 0,χ0,n ψ = 0 if ψ is not trivial modulo r and otherwise, and hence (recall that (r, n = B 0,χ0,n χ 0,r = ϕ(rn rn n m+ ϕ(r(m + r m ψ G(r ψ( nb 0,χ0,n ψ = n m ϕ(n (m + r m+. S = n m+ ϕ(r(m + r m + ϕ(r(m + r m = nm ϕ(n (m + r m+ + ϕ(r(m + r m ψ G(r m+ i= m i=0 ψ( nb 0,χ0,n ψ ( m + i ( m + i + n m+ i n m i ψ G(r ψ G(r ψ( nb i,χ0,n ψ ψ( nb i+,χ0,n ψ. Consequently, S = Θ s + ϕ(rr m m i=0 ( m n m i U i (r, (.6 i where Θ s = Θ s (n, m, r = nm ϕ(n (m + r m+ (.7 and U i (r = ψ G(r ψ( n B i+,χ 0,n ψ. (.8 i +

6 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 5.A. U i (r for r 4 Let n > be odd and relatively prime to r. Here and subsequently, we set B i = B i ( p i, Ã i = ( n 8 A i ( ( p 8 p i = ( n 8 C i = ( (n (n+5 8 C i ( ( (p (p+5 8 p i = ( (n+(n+ 8 B i+,χ 8 i + B i+,χ8 i + ( ( (p (p+5 8 p i, ( ( p 8 p i, D i = ( ν(n D i ( ( ν(p p i = ( ν(n+ B i+,χ ( ( ν(p p i, i + Ẽ i = ( n Ei ( ( p p i = ( n+ F i = ( n +ν(n F i ( ( p +ν(p p i = ( n +ν(n B i+,χ χ 4 i + B i+,χ 4 i + ( ( p +ν(p p i, G i = ( (n (n+5 8 +ν(n G i ( ( (p (p+5 8 +ν(p p i = ( (n (n+5 8 +ν(n B i+,χ χ 8 i + ( ( p p i, ( ( (p (p+5 8 +ν(p p i, H i = ( n 8 +ν(n H i ( ( p 8 +ν(p p i = ( n ν(n B i+,χ χ 8 i + ( ( p 8 +ν(p p i, where χ (n = ( ν(n, ν(n = 0, resp. if n, resp. (mod. In the following, we compute U i (r for r =,, 4, 6, 8, or 4.. Case r = Then #G( = and G( = {χ 0, }. Then, by definition and identity (., U i ( = { Bi+ i+ ( i, if i is odd; 0, if i is even. (.9. Case r = Then #G( = and G( = {χ 0,, χ }. Then, by definition and identity (., U i ( = { Bi+ i+ ( i, if i is odd; i, if i is even. (.0

7 6. Some auxiliary formulae. Case r = 4 Then #G(4 = and G(4 = {χ 0,4, χ 4 }. Thus, by definition and the same arguments as in the case r = (note that both characters χ and χ 4 are odd, in view of (. we obtain 4. Case r = 6 U i (4 = { Bi+ i+ ( i, if i is odd; if i is even. (. Then #G(6 = and G(6 = {χ 0,6, χ,6 }. Consequently, by (. and the same arguments as in the previous case we obtain 5. Case r = 8 U i (6 = { Bi+ i+ ( i ( i, if i is odd; i ( + i, if i is even. Then #G(8 = 4 and G(8 = {χ 0,8, χ 4,8, χ 8, χ 8 }. Therefore, in view of (., (. 6. Case r = U i (8 = { Bi+ i+ ( i + Ãi, if i is odd; + C i, if i is even. (. Then #G( = 4 and G( = {χ 0,, χ,, χ 4,, χ ( ( 4 }. Consequently, by definition and (., 7. Case r = 4 U i ( = { Bi+ i+ ( i ( i + F i, if i is odd; i ( + i + Ẽi( + i, if i is even. (.4 Then #G(4 = 8 and G(4 = {χ 0,4, χ,4, χ 4,4, χ ( ( 4,4, χ ( ( 8, χ ( 8, χ 8,4, χ 8,4 }. Consequently, in view of (., U i (4 = { Bi+ i+ ( i ( i + F i + G i + Ãi( + i, if i is odd; i ( + i + Ẽ( + i + H i + C i ( i, if i is even..b. The sums S r,k,s (n (mod n s+ for m > s, r 4, s (.5 The generalized Bernoulli numbers attached to Dirichlet characters modulo r, with r 4, are rational numbers. In what follows we consider congruences for S r,k,s (n modulo n s+ for n > odd and s {0,, }. We assume that n is not divisible by if r ; then r and ϕ(r are coprime to n. It is shown in the previous section that the numbers U i (r are linear combinations of the numbers à i, C i, D i, Ẽi, F i, G i, H i and the quotients B i+. Denote by U odd i (r, resp. Ui even (r the sum U i (r i + taken over odd, resp. even characters ψ modulo r. Note that U i (r = Ui odd (r + Ui even (r and (r = 0, U even (r = 0 if i is odd or even, respectively. U odd i i First we recall some divisibility properties of the quotients B i+,χ for primitive Dirichlet characters i + χ of conductors f χ nr. These quotients, multiplied by some Euler factors, are summands of U i. We start with some elementary lemmas on the quotients B i+ of the ordinary Bernoulli numbers. Lemma i +. is called the von Staudt and Clausen theorem. Lemma. due to L. Carlitz is its generalization.

8 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 7 Lemma.. (See [Wash97, Theorem 5.0] or [IR90, Corollary to Theorem, p. ]. Let k be an even natural number and let p be a prime number. Then B k contains p in its denominator if and only if p k and pb k (mod p. Lemma.. (See [Car] If p ν (p k, ν 0 then pb k p (mod p ν+. Lemma.. (See [Ern79, Proposition 5..4, p. 8]. If p k then the quotients B k k p-integral. are Since conductors of non-trivial characters occurring in U i (r are coprime to n, they are not powers of a prime divisor of n. In such cases we have a useful lemma: Lemma.4. (See [Ern79, Theorem.5]. Let χ be a primitive Dirichlet character with conductor f χ. If f χ is not a power of a given prime number p, then the quotients B n,χ (n are p-integral. n We set NT Ui even (r = Ui even (r B i+ i+ p r ( pi. By Lemma.4 we obtain: Lemma.5. Let r be coprime to p for a given prime number p n. Then the numbers Ui odd (r for i even and the numbers NT Ui even (r for i odd are p-integral. Assume that m = n s ϕ(n k > s for s {0,, }. Since for odd n >, ϕ(n is even, m and k are of the same parity. We divide each of the cases s = 0, or. Our purpose is to obtain some congruences for the sums S r,k,s (n modulo n for s {0,, }, and next using congruence (. to obtain congruences for the sums T r,k (n. We prove that the latter sums are congruent modulo n s+ to linear combinations of the quotients B m /m and some of the numbers Ãm, C m, C m, D m, D m, Ẽm, Ẽm, F m, G m, H m, H m if k is even, and of the quotients B m /(m, B m+ /(m+ and some of the numbers Ãm, Ãm, C m, D m, Ẽm, F m, F m, G m, G m, H m if k is odd 4. We start with the study of the case s =. Next, similarly, we derive the remaining congruences modulo n and modulo n. First we show when the numbers Θ s (defined in (.7 are congruent to 0 modulo n s+. Lemma.6. Let n > be odd and let < r n be coprime to n. Assume that m > s and p n is a prime. Then the numbers Θ s in (.7 are p-integral and Θ s = n m ϕ(n (m + r m+ 0 (mod ns+ except when s =, n, ϕ(n and m = 5. Proof. First we prove that the numbers Θ s are p-integral for m s +. It suffices to show that mord p (n ord p (m + 0. Let us define the function g(x = x log p (x +, which is increasing for x. Since log p (m + ord p (m + and ord p (n we obtain that mord p (n ord p (m + m log p (m + = g(m g(s + > 0 because g( = log p (5 > 0, g( = log p (4 > 0 and g( = log p ( > 0 for any prime p. 4 As well as of Euler s quotients q (n or q (n if k =. 5 Then Θ = n ϕ(n/r and the exceptional n s have the form n = u i= pe i i where p i (mod for i =,..., u. Moreover k = nϕ(n is even. Obviously, if k and (k, n =, then the congruence Θ 0(mod n s+ is true because m + and n are coprime. We leave it to the reader to verify that the congruence holds if k =.

9 8. Some auxiliary formulae Let us consider the functions f s (x = x s log p (x+ for x, which are increasing for x 6. Note that the congruence Θ s 0(mod n s+ for m > s holds if and only if (m sord p (n + ord p (ϕ(n ord p (m + > 0 for every p n. In view of log p (m + ord p (m + and ord p (n the above follows from the inequality f s (m > 0 for m if s =,, and for m if s = 0 because (m sord p (n ord p (m + (m s log p (m + = f s (m and f s (m f ( = log p (4 > 0 if s =, f s (m f ( = log p (4 > 0 if s = and f s (m f 0 ( = log p ( > 0 if s = 0 for every p n. This gives the congruence Θ s 0(mod n s+ for s = 0, and m > s and s = and m. In the case when s = and m = we have f ( = log p ( > 0 if p 5, and so the congruence holds for n. We are left with the task of checking when the congruence holds for s =, m = and n. Then it is easily seen that the congruence Θ = n ϕ(n r 0(mod n holds if and only if ord (ϕ(n. This does not hold if and only if s =, n, ϕ(n, m =, as claimed..b.a. The case when s = Assume that m = n ϕ(n k and k < n ϕ(n (m >. Then, by Lemma.6, Θ 0 (mod n. Case (i: If k is even, then m + = n ϕ(n k + is odd. Consequently S = 0 in (.4. Thus, combining (.4 and (.6 gives S r,k, (n = Θ + S S (mod n, and S r,k, (n S ϕ(rr m ( ( m + Um odd (r + mnum (r even (.6 ( m n U even m (r (mod n n U odd m (r + because for every prime number p n by Lemma.5, the summands Um odd (r, ( n and n U even m (r 7 are p-integral. ( m n U odd m (r, mnum (r even Case (ii: If k is odd, then m + is even and S 0. Moreover, by Lemma.6, Θ 0(mod n. Thus, by (.4, (.5, (.6, we obtain since, by Lemmas.4 or.5, S r,k, (n B m+ m + + ϕ(rr even (U m m +mnu odd m (r + (r ( m n U even m (r (mod n ( m n U odd m (r 8 is p-integral for any p n and divisible by n. 6 The functions g(x and f s(x are increasing since g (x = f s(x = 7 With m, m even and m, m odd. 8 With m even. > 0 for x. (x+ log p

10 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 9 Consequently, if k is odd and r, ϕ(r are relatively prime to n, we find, by Lemma.4, that S r,k, (n B m+ + m + ϕ(rr m ( q m (.7 q r ( ( m + ϕ(rr m NT Um even (r + mnum (r odd + n U even m (r (mod n. ( m n U even m (r and mnum odd (r9 Note that for p n, by Lemma.5, the summands NT Um even (r, are p-integral. Moreover, if p n and p m +, i.e., p is in the denominator of B m+, then by the little Fermat theorem, we have q m q (mod p ordp(m++ and r m r (mod p ordp(m++ (recall that r is coprime to n, and + ϕ(rr m ( q m + r ( q = 0 (mod p ordp(m++. ϕ(r q r Hence and from Lemma., it follows that for p n the first summand of the right hand side of (.7 is p-integral in the case when p m +. If p m +, then the same conclusion follows from Lemma...B.b. The case when s = Assume that m = nϕ(n k and k < nϕ(n (m >. Then, by Lemma.6, Θ 0 (mod n if m >. If m = and r 8, then the congruence holds if n is not divisible by or divisible by 9. If m = and n, then it is true for ϕ(n. Case (i: If k is even, then analysis similar to that in the proof of (.6 shows that S r,k, (n ϕ(rr odd (U m m q r (r + mnum (r even (mod n (.8 ( m n U odd m (r + if m > or m = and n is not exceptional in the sense of Lemma.6 since ( m n Um (r even is divisible by n. If m = and n is exceptional, i.e. n and ϕ(n, then we should add to the right hand side of (.8 the correction Θ = n ϕ(n/r, but we prefer to exclude the case when m =, i.e., k = nϕ(n. Case (ii: If k is odd, then by Lemma.6 we have Θ 0(mod n and a similar argument to that in the proof of (.7 shows that S r,k, (n (.9 B m+ + m + ϕ(rr m ( q m q r ( ( m + ϕ(rr m NT Um even (r + mnum (r odd + n U even m (r (mod n. 9 With m, m odd and m even.

11 0. The main results of the paper.b.c. The case when s = 0 Assume that m = ϕ(n k and k < ϕ(n. Then, by Lemma.6, Θ 0 0 (mod n. Case (i: If k is even, then in the same way as in the proof of (.8 we obtain S r,k,0 (n ( ϕ(rr m Um odd (r + mnum (r even (mod n. (.0 Case (ii: If k is odd, then by a similar argument to that in the proof of (.9 we find S r,k,0 (n B m+ + m + ϕ(rr m ( q m even + NT U ϕ(rrm m (r (mod n (. because mnu odd m (r + and 5. q r ( m n U even m (r is divisible by n, which is an easy con sequence of Lemmas. The main results of the paper In this section we compute the sums T r,k (n (mod n s+ for s {0,, } and all r 4, using congruence (. and congruences for the sums S r,k,s (n, namely congruences (.6 and (.7 if s =, (.8 and (.9 if s =, and (.0 and (. if s = 0. We divide each of the three cases s = 0, or into seven subcases: r =,, 4, 6, 8,, 4, obtaining congruences for T r,k (n for k < n s ϕ(n s. In the second part of the paper we shall derive from obtained congruences some congruences in the case when n = p is an odd prime. Some of such congruences were proved by M. Lerch [Ler05], E. Lehmer [Leh8] and Z.-H. Sun [Sun08], but most of them were not earlier known. We substitute formulae (.9-5 into congruences (.6, (.8 and (.0 if k is even and congruences (.7, (.9 and (. if k is odd. Consequently, after some calculations, we obtain Theorems and Corollaries. In the theorems below, given any k and ρ Z, we write I(k, ρ = {n > : n and p n if p k + ρ} 0 for example I(, = {n > :, n}, I(, = I(, = {n > :,, 5 n} or I(5, = I(4, = {n > :,, 7 n} and Q (n = q (n + nq (n n q (n, Q (n = q (n + 4 nq (n n q (n. The sums T r, (n presented in Corollaries below are congruent to linear combinations of Euler s quotients ÊQ r(n plus some generalized Bernoulli numbers where ÊQ (n = Q (n, ÊQ (n = Q (n, ÊQ 4(n = Q (n, ÊQ 6(n = Q (n+q (n, ÊQ 8(n = Q (n, ÊQ (n = Q (n+q (n and ÊQ 4(n = Q (n+q (n. For i =, set Q i (n = Q i(n(mod n and Q i (n = Q i(n (mod n.. Case r = 0 Note that if k and ρ are of the same parity and n I(k, ρ, then n.

12 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type Theorem.. Given an odd n > and k < n s ϕ(n s, write m = n s ϕ(n k. Then: (i In the case s = (k+ n B m + ( k + ( k+ n Bm (mod n for k even, 4 T,k (n k ( m+ B m+ m + k 8 (k+ n Bm (mod n for k odd, in particular, k is even and n I(k,, then T,k (n (k+ n B m (mod n. (ii In the case s =, (cf. [Sun08] if k is odd and n = p is an odd prime number T,k (n (k+ n B m (mod n for k even, k ( m+ B m+ m + k 8 (k+ n Bm (mod n for k odd, in particular, k is odd and n I(k,, then T,k (n k ( m+ B m+ m + (mod n. (iii In the case s = 0 T,k (n (k+ n B m (mod n for k even, k ( m+ B m+ m + (mod n for k odd, in particular, k is even and n I(k, 0, then T,k (n 0 (mod n. Proof. If k is even, resp. odd, then it suffices to apply congruence (.6, (,8, (,0 resp. (.7, (.9, (.. Substituting (.9 into these congruences gives the theorem immediately. Corollary.. Let n > be odd. Then: (i (cf. [Sun08], [Cai0] and [Leh8] if n = p is an odd prime (ii T, (n Q (n 7 8 n Bn ϕ(n (mod n, T, (n Q (n (mod n if n, T, (n Q (n (mod n. T, (n 7 n B n ϕ(n + 8 n Bn ϕ(n 4 (mod n, T, (n 7 n B n ϕ(n (mod n if, 5 n, T, (n 7 n B nϕ(n (mod n, T, (n 0 (mod n if n.

13 . The main results of the paper Proof. (ithis is a particular case of Theorems. for k =. Then m + = n s ϕ(n and, by ϕ(n = nq (n +, we have B n s ϕ(n ( m+ B m+ m + = ( ( + nq (n ns n s ϕ(n because α Z, s and = (Q (n + αn n B n s ϕ(n ϕ(n Q (n (mod n s+ n B n s ϕ(n ϕ(n (mod n s+ (. Indeed, if p 0 n is a prime, then (p 0 p (s+ordp 0 (n 0 n s ϕ(n and, by Lemma., n B n s ϕ(n ϕ(n n(p 0 p 0 ϕ(n ( p = (mod p (s+ordp0 (n 0.,p p 0 This completes the proof of (. and of Corollary. (i. (ii is an immediate consequence of Theorems. for k =.. Case r = Theorem.. Given an odd n > not divisible by and k < n s ϕ(n s, write m = n s ϕ(n k. Then: (i In the case s = T,k (n k D m + 6 (k+ n B m + k ( k + n Dm (mod n for k even, n I(k,, k ( m+ B m+ m + k kn D m k 6 (k+ n Bm (mod n for k odd. (ii In the case s = k D m + T,k (n 6 (k+ n B m (mod n for k even, k ( m+ B m+ m + k kn D m (mod n for k odd, n I(k,. (iii In the case s = 0, (cf. [Sun08] if n = p is a prime k D m (mod n for k even, n I(k, 0, T,k (n k ( m+ B m+ (mod n for k odd. m + Proof. For k even, resp. odd we combine formula (.0 with congruence (.6,(.8,(.0, resp. (.7,(.9,(.. Hence the theorem follows at once. Corollary.4. Let n > be odd and not divisible by. Then: (i (cf. [Sun08] if n = p is a prime T, (n Q (n n D n ϕ(n 8 n Bn ϕ(n (mod n, T, (n Q (n n D nϕ(n (mod n, T, (n Q (n (mod n.

14 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type (ii T, (n D n ϕ(n + n B n ϕ(n + 9 n Dn ϕ(n 4 (mod n if 5 n, T, (n D nϕ(n + n B nϕ(n (mod n, T, (n D ϕ(n (mod n. Proof. (i This is a particular case of Theorems. for k =. Then m + = n s ϕ(n and, by ϕ(n = nq (n + and (., we obtain ( m+ B m+ m + = ( ( + nq (n ns B n s ϕ(n n s ϕ(n (Q (n + βn n B n s ϕ(n ϕ(n because β Z and s. The rest of the proof is straightforward. (ii This is a particular case of Theorems. for k =. Q (n (mod n s+. Case r = 4 Theorem.5. Given an odd n > and k < n s ϕ(n s, write m = n s ϕ(n k. Then: (i In the case s= ( k Ẽ m + k ( k+ n B k + m + k n Ẽ m (mod n k even, n I(k,, T 4,k (n k ( m m+ B m+ m + k knẽm k 4 k( k+ n Bm (mod n for k odd. (ii In the case s = (cf. [Sun08] if k is odd and n = p is an odd prime Ẽ m + k ( k+ n B m (mod n for k is even, T 4,k (n ( m m+ B m+ m + k knẽm (mod n for k odd, n I(k,. (iii (cf. [Sun08] if n = p is an odd prime Ẽ m (mod n for k even, n I(k, 0, T 4,k (n ( m m+ B m+ m + (mod n for k odd. Proof. This is an immediate consequence of (.6-. We apply formula (.. Corollary.6. Let n > be odd. Then: (i (cf. [Sun08] if n = p is an odd prime T 4, (n Q (n nẽn ϕ(n 7 8 n Bn ϕ(n (mod n, T 4, (n Q (n nẽnϕ(n (mod n if n, T 4, (n Q (n (mod n. Theorem.(i is also true for k = nϕ(n if we assume that n is not exceptional in the sense of Lemma.6; for exceptional n we should add the correction Θ = 9 n ϕ(n to the right hand side of the congruence.

15 4. The main results of the paper (ii (cf. [KUW] T 4, (n 4Ẽn ϕ(n + 7n B n ϕ(n + n Ẽ n ϕ(n 4 (mod n if, 5 n, T 4, (n 4Ẽnϕ(n + 7n B nϕ(n (mod n, T 4, (n 4Ẽϕ(n (mod n if n. Proof. (i This is a particular case of Theorems.5 for k =. Then m + = n s ϕ(n and, by ϕ(n = nq (n + and (., we have B n s ϕ(n ( m m+ B m+ m + = (( ( + nq (n ns + ( ( + nq (n ns n s ϕ(n Q (n + Q (n + γn n B n s ϕ(n ϕ(n Q (n (mod n s+ because γ Z and s. This gives the theorem at once since the rest of the proof is straightforward. (ii This is a particular case of Theorems.5 in case k =. 4. Case r = 6 Theorem.7. Given an odd n > 5 not divisible by and k < n s ϕ(n s, write m = n s ϕ(n k. Then: (i In the case s = k (k + D m + (k+ ( k+ n B m + k ( k+ T 6,k (n 8 ( k+ n Dm (mod n for k even, n I(k,, k k ( m m 6 m B m+ m + k 4 (k+ + kn D m k 44 (k+ ( k+ n Bm (mod n for k odd. (ii In the case s = k (k + D m + (k+ ( k+ n B m (mod n for k even, T 6,k (n k k ( m m 6 m B m+ m + k 4 (k+ + kn D m (mod n for k odd, n I(k,. (iii In the case s = 0 (cf. [Sun08] if n = p is an odd prime k T 6,k (n (k + D m (mod n for k even, n I(k, 0, k k ( m m 6 m B m+ (mod n for k odd. m + Proof. This is an immediate consequence of congruences (.6,(.8,(.0 if k is even or (.7,(.9,(. if k is odd and formula (..

16 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 5 Corollary.8. Let n > 5 be odd and not divisible by. Then: (i (ii T 6, (n Q (n + Q (n 5 4 n D n ϕ6(n 9 7 n Bn ϕ(n (mod n, T 6, (n Q (n + Q (n 5 4 n D nϕ(n (mod n, T 6, (n Q (n + Q (n (mod n. T 6, (n 5 D n ϕ(n n B n ϕ(n n Dn ϕ(n 4 (mod n, T 6, (n 5 D nϕ(n n B nϕ(n (mod n, T 6, (n 5 D ϕ(n (mod n. Proof. (i This is a particular case of Theorems.7 for k =. Then m + = n s ϕ(n and, in view of ϕ(n = nq (n +, ϕ(n = nq (n + and (., we find that ( m m 6 m B m+ m + = (( ( + nq (n ns + ( ( + nq (n ns B n s ϕ(n + ( ( + nq (n ns ( + nq (n ns n s ϕ(n ( Q (n + 4 Q (n + Q (n + Q (n + λn n B n s ϕ(n ϕ(n Q (n + Q (n (mod n s+ because λ Z and s. This gives the theorem. (ii The theorem follows easily from Theorems.7 for k =. 5. Case r = 8 Theorem.9. Given an odd n > 7 and k < n s ϕ(n s, write m = n s ϕ(n k. Then: (i In the case s = T 8,k (n k Ẽ m + k Cm + k ( k+ n B m k knãm ( ( k + k + + k n Ẽ m + k n Cm (mod n for k even, n I(k,, T 8,k (n k ( m m+ B m+ m + + k à m k knẽm k kn C m k 5 kn ( k+ B m + k ( k + (mod n n à m for k odd. Theorem.9 (ii is also true for k = nϕ(n if we assume that n is not exceptional in the sense of Lemma.6; for exceptional n we should add the correction Θ = 56 n ϕ(n to the right hand side of the congruence.

17 6. The main results of the paper (ii In the case s = k Ẽ m + k Cm + k ( k+ n B m k knãm (mod n for k is even, T 8,k (n k ( m m+ B m+ m + + k à m k knẽm k kn C m (mod n for k odd, n I(k,. (iii In the case s = 0 Ẽ m + k Cm (mod n for k even, n I(k, 0, T 8,k (n ( m m+ B m+ m + + k à m (mod n for k odd. Proof. This follows from congruence (.6,(.8,(.0, resp. (.7,(.9,(. for k even, resp. odd and formula (.. Corollary.0. Let n > 7 be odd. Then: (i (ii T 8, (n Q (n + Ãn ϕ(n nẽn ϕ(n n C n ϕ(n 7 8 n Bn ϕ(n + n à n ϕ(n (mod n, T 8, (n Q (n + Ãnϕ(n nẽnϕ(n n C nϕ(n (mod n if n, T 8, (n Q (n + Ãϕ(n (mod n. T 8, (n 8Ẽn ϕ(n + 6 C n ϕ(n + 4n B n ϕ(n nãn ϕ(n + 4n Ẽ n ϕ(n n Cn ϕ(n 4 (mod n if, 5 n, T 8, (n 8Ẽnϕ(n + 6 C nϕ(n + 4n B nϕ(n nãnϕ(n (mod n, T 8, (n 8Ẽϕ(n + 6 C ϕ(n (mod n if n. Proof. (i This is a particular case of Theorems.9 for k =. Then m + = n s ϕ(n and, by virtue of ϕ(n = nq (n + and (., we obtain B n s ϕ(n ( m m+ B m+ m + = ( ( + nq (n ns ( + nq (n ns n s ϕ(n ( Q (n + Q (n + ξn n B n s ϕ(n ϕ(n Q (n (mod n s+ because ξ Z and s. This gives the theorem at once. (ii It is trivial. 6. Case r = Theorem.. Given an odd n > not divisible by and k < n s ϕ(n s, write m = n s ϕ(n k. Then:

18 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 7 (i In the case s = T,k (n k k ( k + D m + k ( k + Ẽm + k (k+ ( k+ n B m k k kn F m ( k + + k 4 k ( k+ + n Dm ( + k k + ( k+ + n Ẽ m (mod n for k even, n I(k,, 9 T,k (n k k ( m m + 6 m 4 m B m+ m + + k k Fm (ii In the case s = k k ( k+ + kn D m k ( k+ + knẽm k 5 9 (k+ ( k+ kn Bm ( k + + k k n Fm (mod n for k odd. T,k (n k k ( k + D m + k ( k + Ẽm + k (k+ ( k+ n B m k k kn F m (mod n for k even, k k ( m m + 6 m 4 m B m+ m + + k k Fm k k ( k+ + kn D m k ( k+ + knẽm (mod n for k odd, n I(k,. (iii In the case s = 0 k k ( k + D m + k ( k + Ẽm (mod n for k even, n I(k,, T,k (n k k ( m m + 6 m 4 m B m+ m + + k k Fm (mod n for k odd. Proof. Apply congruences (.6,(.8,(.0, resp. (.7,(.9,(. and formula (.4. Corollary.. Let n > be odd not divisible by. Then: (i T, (n Q (n + Q (n + F n ϕ(n 5 4 n D n ϕ(n 5 nẽn ϕ(n 9 7 n Bn ϕ(n + n Fn ϕ(n (mod n, T, (n Q (n + Q (n + F nϕ(n 5 4 n D nϕ(n 5 nẽnϕ(n (mod n, T, (n Q (n + Q (n + F ϕ(n (mod n.

19 8. The main results of the paper (ii T, (n 5 D n ϕ(n + 0Ẽn ϕ(n + 9 n B n ϕ(n 7n F n ϕ(n D n ϕ(n n Ẽ n ϕ(n 4 (mod n for5 n, T, (n 5 D nϕ(n + 0Ẽnϕ(n + 9 n B nϕ(n 7n F nϕ(n (mod n, T, (n 5 D ϕ(n + 0Ẽϕ(n (mod n. Proof. (i This is a particular case of Theorems. for k =. Then m + = n s ϕ(n and, by virtue of ϕ(n = nq (n +, ϕ(n = nq (n + and (., we have ( m m + 6 m 4 m B m+ ( m + = ( ( + nq (n ns + ( ( + nq (n ns ( ( + nq (n ns ( ( + nq (n ns + ( ( + nq (n ns ( + nq (n ns Bn s ϕ(n n s ϕ(n ( 4 Q (n + Q (n 4 Q (n Q (n + Q (n + n Q (n + ηn Bn s ϕ(n ϕ(n Q (n + Q (n (mod n s+ because η Z and s. The rest of the proof is straightforward. 7. Case r = 4 Theorem.. Given an odd n > not divisible by and k < n s ϕ(n s, write m = n s ϕ(n k. Then: (i In the case s = T 4,k (n k k ( k + D m + k 4 ( k + Ẽm + k k Hm + k ( k C m + k 4 ( k+ ( k+ n B m k k kn F m k k kn G m k ( k+ + knãm ( ( k + + k 5 ( k k+ + n Dm + k 4 k + 9 ( ( k + + k n k Hm + k k + 9 (mod n for k even n I(k,, ( k+ n Cm ( k+ + n Ẽ m

20 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 9 T 4,k (n k k ( m m + 6 m 8 4 m B m+ m + + k k Fm (ii In the case s = + k k Gm + k ( k + Ãm k 4 k ( k+ + kn D m k 4 ( k+ + knẽm k k kn H m k ( k 6 k + ( k+ ( k+ kn Bm k k 9 ( n Gm + k k + + k k ( k + (mod n 9 for k odd. T 4,k (n k k ( k + D m + k 4 ( k + Ẽm + k k Hm ( k+ kn C m n Fm ( k+ + n à m + k ( k C m + k 4 ( k+ ( k+ n B m k k kn F m k k kn G m k ( k+ + knãm (mod n for k even, T 4,k (n k k ( m m + 6 m 8 4 m B m+ m + + k k Fm + k k Gm + k ( k + Ãm k 4 k ( k+ + kn D m k 4 ( k+ + knẽm k k kn H m k ( k+ kn C m (mod n for k odd, n I(k,. (iii In the case s = 0 k k ( k + D m + k 4 ( k + Ẽm + k k Hm + k ( k C m (mod n for k even n I(k, 0, T 4,k (n k ( m m + 6 m 8 4 m B m+ m+ + k k Fm + k k Gm + k ( k + Ãm (mod n for k odd. Proof. This follows from congruences (.6,(.8,(.0, resp. (.7,(.9,(. if k is even, resp. odd with the use of (.5. Corollary.4. Let n > be odd and not divisible by. Then: (i T 4, (n Q (n + Q (n + F n ϕ(n + G n ϕ(n + 4Ãn ϕ(n 5 4 n D n ϕ(n 5 nẽn ϕ(n n H n ϕ(n 8 n C n ϕ(n 9 7 n Bn ϕ(n + n Fn ϕ(n + n Gn ϕ(n n à n ϕ(n (mod n, T 4, (n Q (n + Q (n + F nϕ(n + G nϕ(n + 4Ãnϕ(n 5 4 n D nϕ(n 5 nẽnϕ(n n H nϕ(n 8 n C nϕ(n (mod n, T 4, (n Q (n + Q (n + F ϕ(n + G ϕ(n + 4Ãϕ(n (mod n.

21 0 4. Further congruences of E. Lehmer s type (by T. Kuzumaki and J. Urbanowicz (ii T 4, (n 0 D n ϕ(n + 40Ẽn ϕ(n + 7 H n ϕ(n + 64 C n ϕ(n + 8 n B n ϕ(n 6n F n ϕ(n 44n G n ϕ(n 448 nãn ϕ(n + 5 n Dn ϕ(n n Ẽ n ϕ(n 4 + 6n Hn ϕ(n n Cn ϕ(n 4 (mod n if 5 n, T 4, (n 0 D nϕ(n + 40Ẽnϕ(n + 7 H nϕ(n + 64 C nϕ(n + 8 n B nϕ(n 6n F nϕ(n 44n G nϕ(n 448 nãnϕ(n (mod n, T 4, (n 0 D ϕ(n + 40Ẽϕ(n + 7 H ϕ(n + 64 C ϕ(n (mod n. Proof. (i This is a particular case of Theorems. for k =. Then m + = n s ϕ(n and, in view of ϕ(n = nq (n +, ϕ(n = nq (n + and (., we have ( m m + 6 m 8 4 m B m+ ( m + = ( ( + nq (n ns + ( ( + nq (n ns ( ( + nq (n ns ( ( + nq (n ns + ( ( + nq (n ns ( + nq (n ns Bn s ϕ(n n s ϕ(n ( 4 Q (n + Q (n 4 Q (n Q (n + Q (n + n Q (n + ωn Bn s ϕ(n ϕ(n Q (n + Q (n (mod n s+ because ω Z and s. This proves the theorem. (ii It is trivial. 4. Further congruences of E. Lehmer s type (by T. Kuzumaki and J. Urbanowicz In Theorem 4. below we find some congruences for U r (n modulo n s+ for s {0,, } in each of the seven cases r =,, 4, 6, 8, or 4. Some of these congruences for s {0, } and r {,, 4, 6} were proved in [CFZ07] and [CP09]. The remaining ones are new. Three of them for s = and r {8,, 4} were omitted both in [Leh8] and in [CFZ07], [CP09]. Write ρ i (r = δ ordi (r,0 (i =, where, as usual, δ X,Y denotes the Kronecker delta function. Given odd n > r, we set EQ r (n =α (rq (r + α (rq (r + β (rnq (n + β (rnq (n (4. + γ (rn q (n + γ (rn q (n,

22 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type where and ( ordi (r α i (r = ρ i (r r ( β i (r = ρ i (r ord (r r ( ordi (r γ i (r = ρ i (r r + iϕ(r ρ 5 i(r 6ϕ(r iϕ(r + ρ 5 i(r ϕ(r + iϕ(r ρ 5 i(r 8ϕ(r B r (n = n r B n ϕ(n. Set EQ r(n = α (rq (r+α (rq (r and EQ r = α (rq (r+α (rq (r+β (rnq (n+β (rnq (n. Obviously, we have EQ r (n EQ r(n (mod n and EQ r (n EQ r(n (mod n. Note that B r (n 0 (mod n, and B r (n 0 (mod n if n is not divisible by. It was shown in Section that the sums T r, (n are congruent to linear combinations of Euler s quotients ÊQ r(n plus some generalized Bernoulli numbers. In view of Proposition 4. below we have EQ r (n = r ÊQ r(n. Theorem 4.. Assume that s {0,, } and r 4. Let n > r be odd and not divisible by if s = or r. Then, in the above notation: EQ r (n + B r (n (mod n s+ for r 6, EQ r (n + B r (n U r (n 4Ãn s ϕ(n (mod n s+ for r = 8, EQ r (n + B r (n 4 F n s ϕ(n (mod n s+ for r =, EQ r (n + B r (n 6Ãn s ϕ(n 8 F n s ϕ(n 8 G n s ϕ(n (mod n s+ for r = 4. Here EQ r (n EQ r(n (mod n, EQ r (n EQ (n (mod n, B r (n 0 (mod n if n is not divisible by and B r (n 0 (mod n. 4.A. Some useful observations We deduce Theorem 4. from Propositions 4.,4. and congruences for the sums T r,k (n given in Section. First we find some useful congruences modulo powers of n between the sums U r (n and some linear combinations of T r, (n, T r, (n and T r, (n. Proposition 4.. (cf. [Leh8]. Assume that n > is odd and r ( < r < n is coprime to n. Then: r T r,(n n r T r,(n n r T r,(n (mod n, U r (n r T r,(n n r T r,(n (mod n, r T r,(n (mod n. Proof. Obviously, (n, i = if and only if (n ri, n =. Consequently, U r (n χ n (i(n ri ns ϕ(n 0<i< n r = 0<i< n r χ n (i n s ϕ(n j=0 ( n s ϕ(n j,, n j ( ri ns ϕ(n j (mod n s+

23 4. Further congruences of E. Lehmer s type (by T. Kuzumaki and J. Urbanowicz ( n and hence, since r nsϕ(n j r j (mod n s+ s ϕ(n and n n (mod n, r S r,,(n n r S r,,(n n r S r,,(n (mod n, U r (n r S r,,(n n r S r,,(n (mod n, r S r,,0(n (mod n. Now Proposition 4. follows from (. at once. In Section some formulae for ÊQ r(n are determined. Since, by Proposition 4., we have EQ r (n = r ÊQ r(n, the formulae imply corresponding formulae for EQ r (n. In the next proposition, we present the formulae in a slightly different form. Proposition 4.. In the above notation, if r 4, then (4. holds. Proof. Following (.7 and Proposition 4. we know that EQ r (n = r ÊQ r(n B m+ + r(m + ϕ(rr m ( q m (mod n s+ where m = n s ϕ(n. Consequently, q r where EQ r (n X B m+ r m+ (m + (mod ns+ (4.4 X = r m ϕ(r ( q m. Thus, in view of (4.4 and (., to obtain (4. it suffices to determine X (mod n s+4. Indeed, we have X = r (rϕ(n ns ( ρ ( (r ϕ(r (ϕ(n ns ρ (r (ϕ(n ns, and by virtue of i ϕ(n = + nq i (n (i =, Thus, X r + r X = r ( + nq (n ord (rn s ( + nq (n ord (rn s ϕ(r + ρ (r ϕ(r ( + nq (n ns i=, ϕ(r + + ρ (r ϕ(r ( + nq (n ns ρ (rρ (r ( + nq (n ns ( + nq (n ns. 6ϕ(r ( n s+ ord i (rq i (n ns+ ord i (rnqi (n + ns+ ord i (rn qi (n i=, ρ (rρ (r 6ϕ(r ( ρ i (r + n s+ q i (n iϕ(r ns+ nqi (n + ns+ n qi (n ρ (rρ (r 6ϕ(r i=, q r ( n s+ q i (n ns+ nqi (n + ns+ n qi (n (mod n s+4,

24 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type and so, X Y + r ns+ + i=, i=, ρ i (r iϕ(r ns+ ρ (rρ (r n s+ 6ϕ(r i=, ( ord i (rq i (n ord i(rnqi (n + ord i(rn qi (n ( q i (n nq i (n + n qi (n ( q i (n nq i (n + n qi (n (mod n s+4, where Y = r ϕ(r + ρ (r ϕ(r + ρ (r ϕ(r ρ (rρ (r. 6ϕ(r An easy verification shows that Y = 0. To check it we consider the cases. If ρ (r = 0 and ρ (r = ; then r = and obviously Y = 0. If ρ (r = and ρ (r = 0; then r =, 4, 8 and we have Y = r ϕ(r = 0 since r = ϕ(r for these r. Finally, if ρ (r = ρ (r = ; then r = 6,, 4 and Y = r = 0 since r = ϕ(r in these cases. This completes the proof of Proposition 4.. ϕ(r 4.B. Proof of Theorem 4. The proof of Theorem 4. falls naturally into seven cases r =,, 4, 6, 8, or 4. In view of Proposition 4., in each of the cases, it suffices to determine: (i the sums T r, (n (mod n s+ for s {0,, }, which are determined in (i of Corollaries.,.4,.6,.8,.0,. or.4; (ii the congruences for nt r, (n (mod n s+ for s {, }, which follow immediately from parts (ii of Corollaries.,.4,.6,.8,.0,. or.4; (iii the congruences for n T r, (n (mod n, which follow easily from parts (i of Theorems.,.,.5,.7,.9,. or. for k = 4. Set Q i (n Q i(n (mod n and Q i (n Q i(n (mod n (i =,. We consider the cases:. If r =, Theorem 4. is a consequence of Proposition 4., Theorems. and Corollary.; then for n > odd and s = we have T, (n Q (n 7 8 n Bn ϕ(n (mod n, nt, (n 7 n Bn ϕ(n (mod n, n T, (n n Bn ϕ(n (mod n. The first of these congruences is the same as that Section and the second one is an immediate consequence of that in Section. The third congruence follows immediately from Theorem. for k = ; then n T, (n 6n Bn ϕ(n n ϕ(n (mod n. 4 More precisely, we need to determine T r,(n, nt r,(n, n T r,(n(mod n if s =, T r,(n, nt r,(n (mod n if s = and T r,(n (mod n if s = 0.

25 4 4. Further congruences of E. Lehmer s type (by T. Kuzumaki and J. Urbanowicz On the other hand, which completes the proof in this case. For s = n Bn ϕ(n n ϕ(n n Bn ϕ(n (mod n, (4.5 T, (n Q (n 7 8 n Bnϕ(n (mod n, nt, (n 7 n Bnϕ(n (mod n. If we assume that n, then B nϕ(n is p-integral for any p n and so as claimed. If s = 0 T, (n Q (n (mod n. T, (n Q (n (mod n, nt, (n 0 (mod n. If r =, Theorem 4. is an immediate consequence of Proposition 4., Theorems. and Corollary.4; then for odd n >, n and s = we have T, (n Q (n n D n ϕ(n 8 n Bn ϕ(n (mod n, nt, (n n D n ϕ(n + n Bn ϕ(n (mod n, n T, (n 6n Bn ϕ(n (mod n. Again the first congruence is the same as that in Section and the second one is an easy consequence of that in that section. The third congruence follows from Theorem. (i for k = and (4.5; then n T, (n n Bn ϕ(n n ϕ(n (mod n. For s = T, (n Q (n n D nϕ(n (mod n, nt, (n n D nϕ(n (mod n. Likewise, if s = 0, T, (n Q (n (mod n.. If r = 4, Theorem 4. follows from Proposition 4. and Theorems.5 and Corollary.6; then for n > odd and s = we have T 4, (n Q (n nẽn ϕ(n 7 8 n Bn ϕ(n (mod n, nt 4, (n 4nẼn ϕ(n + 7n Bn ϕ(n (mod n, n T 4, (n 7 n Bn ϕ(n (mod n. The first congruence is the same as that in Section and the second one is an immediate consequence of that in that section. The third congruence follows immediately from Theorem.5 for k = and (4.5; then n T 4, (n 7n Bn ϕ(n n ϕ(n (mod n.

26 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 5 For s = and so T 4, (n Q (n nẽnϕ(n 7 8 n Bnϕ(n (mod n, nt 4, (n 4nẼnϕ(n + 7n Bnϕ(n (mod n T 4, (n Q (n nẽnϕ(n (mod n, nt 4, (n 4nẼnϕ(n (mod n if n. If s = 0 T 4, (n Q (n (mod n. 4. If r = 6, Theorem 4. is an immediate consequence of Proposition 4., Theorems.7 and Corollary.8; then for odd n > 5, n and s = we have T 6, (n Q (n + Q (n 5 4 n D n ϕ(n 9 7 n Bn ϕ(n (mod n, nt 6, (n 5 n D n ϕ(n n Bn ϕ(n (mod n, n T 6, (n 45n Bn ϕ(n (mod n. The first congruence is the same as that in Section and the second one is an immediate consequence of that in that section. The third congruence follows from (4.5 and the congruence n T 6, (n 90n Bn ϕ(n n ϕ(n (mod n. For s = If s = 0 T 6, (n Q (n + Q (n (mod n. T 6, (n Q (n + Q (n 5 4 n D nϕ(n (mod n, nt 6, (n 5 n D nϕ(n (mod n. 5. If r = 8, Theorem 4. follows from Proposition 4., Theorems.9 and Corollary.0 ; then for n > 7 odd and s = we have T 8, (n Q (n + Ãn ϕ(n nẽn ϕ(n n C n ϕ(n 7 8 n Bn ϕ(n + n à n ϕ(n (mod n, nt 8, (n 8nẼn ϕ(n + 6n C n ϕ(n + 4n Bn ϕ(n n à n ϕ(n (mod n, n T 8, (n n Bn ϕ(n + 8n à n ϕ(n (mod n. The first congruence is the same as that in Section, the second one follows from that in Section and the third one is an immediate consequence of Theorem.9 for k = and (4.5; then n T 8, (n n Bn ϕ(n n ϕ(n + 8n à n ϕ(n (mod n. For s = T 8, (n Q (n + Ãnϕ(n nẽnϕ(n n C nϕ(n 7 8 n Bnϕ(n (mod n, nt 8, (n 8nẼnϕ(n + 6n C nϕ(n + 4n Bnϕ(n (mod n

27 6 4. Further congruences of E. Lehmer s type (by T. Kuzumaki and J. Urbanowicz and so T 8, (n Q (n + Ãnϕ(n nẽnϕ(n n C nϕ(n (mod n, nt 8, (n 8nẼnϕ(n + 6n C nϕ(n (mod n if n. If s = 0, T 8, (n Q (n + Ãϕ(n (mod n. 6. If r =, Theorem 4. follows at once from Proposition 4., Theorems. and Corollary.; then for n > odd and s = we have T, (n Q (n + Q (n + F n ϕ(n 5 4 n D n ϕ(n 5 nẽn ϕ(n 9 7 n Bn ϕ(n + n Fn ϕ(n (mod n, nt, (n 5n D n ϕ(n + 0nẼn ϕ(n + 9 n Bn ϕ(n 7n Fn ϕ(n (mod n, n T, (n 6 n Bn ϕ(n + 4n Fn ϕ(n (mod n. The first congruence is the same as that in Section, the second one is implied by that in that section and the third one follows from Theorem. for k = and (4.5; then For s = n T, (n 6n Bn ϕ(n n ϕ(n + 4n Fn ϕ(n (mod n. T, (n Q (n + Q (n + F nϕ(n 5 4 n D nϕ(n 5 nẽnϕ(n (mod n, nt, (n 5n D nϕ(n + 0nẼnϕ(n (mod n. If s = 0, T, (n Q (n + Q (n + F ϕ(n (mod n. 7. If r = 4, Theorem 4. follows from Proposition 4., Theorems. and Corollary.4; then for n > odd and s = we have T 4, (n Q (n + Q (n + F n ϕ(n + G n ϕ(n + 4Ãn ϕ(n 5 4 n D n ϕ(n 5 nẽn ϕ(n n H n ϕ(n 8 n C n ϕ(n 9 7 n Bn ϕ(n + n Fn ϕ(n + n Gn ϕ(n n à n ϕ(n (mod n, nt 4, (n 0n D n ϕ(n + 40nẼn ϕ(n + 7n H n ϕ(n + 64n C n ϕ(n + 8 n Bn ϕ(n 6n Fn ϕ(n 44n Gn ϕ(n 448 n à n ϕ(n (mod n, n T 4, (n 455 n Bn ϕ(n + 78n Fn ϕ(n + 78n Gn ϕ(n + 79n à n ϕ(n (mod n. Again the first congruence is the same as that in Section, the second one follows immediately from that in that section and the third one follows from Theorem. for k = and(4.5; then n T 4, (n 455n Bn ϕ(n n ϕ(n + 78n Fn ϕ(n + 78n Gn ϕ(n + 79n à n ϕ(n (mod n.

28 Kanemitsu, Kuzumaki and Urbanowicz, On congruences for certain sums of E. Lehmer s type 7 For s = For s = 0, T 4, (n Q (n + Q (n + F nϕ(n + G nϕ(n + 4Ãnϕ(n 5 4 n D nϕ(n 5 nẽnϕ(n n H nϕ(n 8 n C nϕ(n (mod n, nt 4, (n 0n D nϕ(n + 40nẼnϕ(n + 7n H nϕ(n + 64n C nϕ(n (mod n. T 4, (n Q (n + Q (n + F ϕ(n + G ϕ(n + 4Ãϕ(n (mod n. This completes the proof of Theorem Concluding remarks Let p be a prime number and let r be a natural number such that < r < p. Assume that s {0,, } and r 4. In the next part of the paper we are going to prove some new congruences for the sums T r,k (p = [ p r ] i= (/ik modulo p s+ for k, in particular for k = or in all the cases. Similarly we would like to derive some new congruences for the sums U r (p = [ p r ] i= p ri modulo ps+. We shall use the congruences proved in the present paper in the case when n = p is an odd prime as well as Kummer s congruences for the generalized Bernoulli numbers. 6. Acknowledgements The authors gratefully acknowledge a discussion with Professor Nianliang Wang concerning an earlier version of Section of the paper. We are also grateful to the referee for valuable comments and helpful suggestions. References [Cai0] T.X. Cai, A congruence involving the quotients of Euler and its appli- cations (I, Acta Arithmetica 0 (00, -0. [CFZ07] T.X. Cai, X.D. Fu and X. Zhou, A congruence involving the quotients of Euler and its applications (II, Acta Arithmetica 0 (007, 0-4. [CP09] H.-Q. Cao and H. Pan, Note on some congruences of Lehmer, J. Number Theory 9 (009, [Car59] L. Carlitz, Arithmetic properties of generalized Bernoulli numbers, J. Reine Angew. Math. 0 (959, [Car] L. Carlitz, Some congruences for the Bernoulli numbers, Amer. J. Math., 75 (95, 6-7. [Ern79] [FUW97] [IR90] [KUW] [Kle55] [Leh8] R. Ernvall, Generalized Bernoulli numbers, generalized irregular primes, and class number, Ann. Univ. Turku, Ser. A, 78 (979, 7 pp. G.J. Fox, J. Urbanowicz and K.S. Williams, Gauss congruence from Dirichlet s class number formula and generalizations, in: K. Gy ry, H. Iwaniec and J. Urbanowicz, editors, Number Theory in Progress (Za- kopane, 997, vol. II, Walter de Gruyter, Berlin-New York, 999, K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, nd edition, Springer-Verlag, New York, 990. S. Kanemitsu, J. Urbanowicz and N.-L. Wang, On some new congruences for generalized Bernoulli numbers, 4 pp., accepted for publication in Acta Arithmetica, 0. H. Kleboth, Untersuchung über Klassenzahl und Reziprozitätsgesetz im Körper der 6l-ten Einheitswurzeln und die Diophantische Gleichungo x l + l y l = z l für eine Primzahl l grosser als, Dissertation, Universität Zürich, 955, 7 pp. E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Annals of Math. 9 (98, [Ler05] M. Lerch, Zur Theorie des Fermatschen Quotienten ap p = q(a Math. Annalen 60 (905,

29 8 6. Acknowledgements [SUV99] A. Schinzel, J. Urbanowicz and P. Van Wamelen, Class numbers and short sums of Kronecker symbols, J. Number Theory 78 (999, [Sun08] Z.-H. Sun, Congruences involving Bernoulli and Euler numbers, J. Number Theory 8 (008, 80-. [SUZ95] J. Szmidt, J. Urbanowicz and D. Zagier, Congruences among generalized Bernoulli numbers, Acta Arithmetica 7 (995, [UW00] J. Urbanowicz and K.S. Williams, Congruences for L-functions, Kluwer Academic Publishers, Dordrecht/Boston/London, 000. [Wash97] L.C. Washington, Introduction to Cyclotomic Fields, second edition, Springer-Verlag, New York, 997. Shigeru Kanemitsu Graduate School of Advanced Technology Kinki University Iizuka Fukuoka, Japan kanemitu@fuk.kindai.ac.jp Takako Kuzumaki Applied Physics Course Faculty of Engineering Gifu University, Gifu Japan kuzumaki@gifu-u.ac.jp Jerzy Urbanowicz Institute of Mathematics Polish Academy of Sciences and Institute of Computer Science Polish Academy of Sciences Warsaw, Poland