Some Arithmetic Functions in Counting Unrooted Topological Maps and Coverings of Surfaces Valery Liskovets Institute of Mathematics National Academy of Sciences Minsk, Belarus liskov@im.bas-net.by International Conference “Embedded Graphs” Euler International Mathematical Institute, St. Petersburg, Russia, October 27 – 31, 2014 Introduction: intention • Exact closed enumeration formulae. • Unrooted maps and related objects, such as non-equivalent coverings of surfaces and subgroups of ﬁnitely gen. groups. • Planar (i.e. spherical) maps and maps on other orientable surfaces. Unweighted, i.e. ”weighted” by 1, rather than 1/|Aut|, etc!.. A brief survey. Phenomenologically. Selectively: initial and characteristic examples, not necessarily latest or most general. Ordered by involved arithmetic functions. Two parts: (I) classical multiplicative functions and (II) a multivariate function introduced recently. In more detail. 1 Introduction: intention • Exact closed enumeration formulae. • Unrooted maps and related objects, such as non-equivalent coverings of surfaces and subgroups of ﬁnitely gen. groups. • Planar (i.e. spherical) maps and maps on other orientable surfaces. • Unweighted, i.e. ”weighted” by 1, rather than 1/|Aut|, etc!.. A brief survey. Phenomenologically. Selectively: initial and characteristic examples, not necessarily latest or most general. Ordered by involved arithmetic functions. Two parts: (I) classical multiplicative functions and (II) a multivariate function introduced recently. In more detail. 1-a Introduction: intention • Exact closed enumeration formulae. • Unrooted maps and related objects, such as non-equivalent coverings of surfaces and subgroups of ﬁnitely gen. groups. • Planar (i.e. spherical) maps and maps on other orientable surfaces. • Unweighted, i.e. ”weighted” by 1, rather than 1/|Aut|, etc!.. • A brief survey. Phenomenologically. Selectively: initial and characteristic examples, not necessarily latest or most general. • Ordered by involved arithmetic functions. Two parts: (I) classical multiplicative functions and (II) a multivariate function introduced recently. In more detail. 1-b Introduction: basic deﬁnitions • Unrooted (maps) = non-isomorphic, = unlabeled, considered up to (sense-preserving) symmetries. Rooting a map: (after Tutte): distinguishing one half-edge (the root) so as to deprive the map all non-trivial automorphisms. The root is known under diverse names, and there are other equivalent deﬁnitions (say, via a distinguished corner) An arithmetic function f (n) is called multiplicative if f (1) = 1 and ( )=1 f (km) = f (k)f (m) whenever GCD k; m . f n is determined by the values () f ( pa ) for all prime p and a 1. 2 Introduction: basic deﬁnitions • Unrooted (maps) = non-isomorphic, = unlabeled, considered up to (sense-preserving) symmetries. • Rooting a map: (after Tutte): distinguishing one half-edge (the root) so as to deprive the map all non-trivial automorphisms. The root is known under diverse names, and there are other equivalent deﬁnitions (say, via a distinguished corner) An arithmetic function f (n) is called multiplicative if f (1) = 1 and ( )=1 f (km) = f (k)f (m) whenever GCD k; m . f n is determined by the values () f ( pa ) for all prime p and a 1. 2-a Introduction: basic deﬁnitions • Unrooted (maps) = non-isomorphic, = unlabeled, considered up to (sense-preserving) symmetries. • Rooting a map: (after Tutte): distinguishing one half-edge (the root) so as to deprive the map all non-trivial automorphisms. The root is known under diverse names, and there are other equivalent deﬁnitions (say, via a distinguished corner) • An arithmetic function f (n) is called multiplicative if f (1) = 1 and f (km) = f (k)f (m) whenever GCD(k, m) = 1. f (n) is determined by the values f (pa) for all prime p and a ≥ 1. 2-b Enumeration: unrooted/unlabeled Unrooted maps/unlabeled objects. Typical way: enumeration in terms of rooted/labeled ones. Burnside/Redﬁeld/P´ olya. Recall: Burnside’s Lemma (the name is improper, but. . . ). ( ∑ 1 |G\\Y | = |Y | + |Yg | |G| g∈G, g̸=1 ) G\\Y : the set of orbits (“unlabelled” objects) of a ﬁnite group G in its action on a set Y . Yg : the set of objects in Y (“labelled” ) ﬁxed by g. • Reduction to separate elements of the group. Particularly eﬃcient when the group G acts semi-regularly. Just for maps, coverings,. . . [ cycles of equal length in every g 2 G] 3 Enumeration: unrooted/unlabeled Unrooted maps/unlabeled objects. Typical way: enumeration in terms of rooted/labeled ones. Burnside/Redﬁeld/P´ olya. Recall: Burnside’s Lemma (the name is improper, but. . . ). ( ∑ 1 |G\\Y | = |Y | + |Yg | |G| g∈G, g̸=1 ) G\\Y : the set of orbits (“unlabelled” objects) of a ﬁnite group G in its action on a set Y . Yg : the set of objects in Y (“labelled” ) ﬁxed by g. • Reduction to separate elements of the group. • Particularly eﬃcient when the group G acts semi-regularly∗. Just for maps, coverings,. . . [∗ cycles of equal length in every g ∈ G] 3-a Enumeration: rooted/labeled • Rooted maps (equivalent to ‘labeled’ ones in terms of enumerative combinatorics). Much easier to enumerate. • Very speciﬁc and well developed techniques. In many important cases formulae are remarkably simple. Example: A0 n :=#(rooted n-edged planar maps). Maps without restrictions: multiple edges and loops are allowed. [W.Tutte, 1963]: () n(2n)! 2 3 0 A (n ) = n!(n + 2)! Related rooted objects (other than maps) are enumerated eﬀectively as well, although often with not so simple formulae. Diverse techniques. 4 Enumeration: rooted/labeled • Rooted maps (equivalent to ‘labeled’ ones in terms of enumerative combinatorics). Much easier to enumerate. • Very speciﬁc and well developed techniques. In many important cases formulae are remarkably simple. • Example: A′(n):=#(rooted n-edged planar maps). Maps without restrictions: multiple edges and loops are allowed. [W.Tutte, 1963]: 2 · 3n(2n)! ′ A (n) = n! (n + 2)! Related rooted objects (other than maps) are enumerated eﬀectively as well, although often with not so simple formulae. Diverse techniques. 4-a Enumeration: rooted/labeled • Rooted maps (equivalent to ‘labeled’ ones in terms of enumerative combinatorics). Much easier to enumerate. • Very speciﬁc and well developed techniques. In many important cases formulae are remarkably simple. • Example: A′(n):=#(rooted n-edged planar maps). Maps without restrictions: multiple edges and loops are allowed. [W.Tutte, 1963]: 2 · 3n(2n)! ′ A (n) = n! (n + 2)! • Related rooted objects (other than maps) are enumerated eﬀectively as well, although often with not so simple formulae. Diverse techniques. 4-b I. CLASSICAL MULTIPLICATIVE FUNCTIONS Euler totient function: necklaces, plane trees, etc. ∏ ϕ(n):=#(units of Cn). ϕ(n) = n • As a multiplicative function: (1 − p−1). p|n prime ϕ(pa) = pa−1(p − 1) p prime, a ≥ 1 Trivially arises for counting objects up to the action of cyclic groups Cn. Lk (n):=#(necklaces E.g., with n beads of k types up to rotations). X 1 n m L (n ) = k k n mjn m Plane trees. Similarly. Less trivial. Chord diagrams. Similarly. Even less trivial in general. . . 5 I. CLASSICAL MULTIPLICATIVE FUNCTIONS Euler totient function: necklaces, plane trees, etc. ∏ ϕ(n):=#(units of Cn). ϕ(n) = n • As a multiplicative function: (1 − p−1). p|n prime ϕ(pa) = pa−1(p − 1) p prime, a ≥ 1 • Trivially arises for counting objects up to the action of cyclic groups Cn. E.g., Lk (n):=#(necklaces with n beads of k types up to rotations). ( ) 1 ∑ n m Lk (n) = k ϕ n m|n m Plane trees. Similarly. Less trivial. Chord diagrams. Similarly. Even less trivial in general. . . 5-a I. CLASSICAL MULTIPLICATIVE FUNCTIONS Euler totient function: necklaces, plane trees, etc. ∏ ϕ(n):=#(units of Cn). ϕ(n) = n • As a multiplicative function: (1 − p−1). p|n prime ϕ(pa) = pa−1(p − 1) p prime, a ≥ 1 • Trivially arises for counting objects up to the action of cyclic groups Cn. E.g., Lk (n):=#(necklaces with n beads of k types up to rotations). ( ) 1 ∑ n m Lk (n) = k ϕ n m|n m • Plane trees. Similarly. Less trivial. • Chord diagrams. Similarly. Even less trivial in general. . . 5-b Euler totient function: unrooted arbitrary planar maps A+(n):=#(unrooted planar maps with n edges). Sense-preserving: +. Theorem [VL, 1981]. ( ) n+3 n−1 ′ ] [ A , n odd ( ) ) ( ∑ m+2 n 1 4 2 A′(m) + A′(n)+ ϕ A+(n) = ( ) n−1 n−2 2 2n m m<n ′ A , n even m|n 4 2 The ﬁrst result of this form. Most signiﬁcant (and unexpected!) was the very existence of such a simple closed formula. A0(n):=#(all rooted planar maps). Tutte’s formula, repeatedly: 0 A (n) = 2 3 n (2n)! n! (n + 2)! = 2 3 n (n + 1)(n + 2) 2n n ; n0 A technique based on quotient maps (= orbifolds) with respect to rotations of order `: spheres with two `-poles. ` : boring technical complications (one or two half-edges). Reproved and generalized later. Rather elementary presently. =2 6 Euler totient function: unrooted arbitrary planar maps A+(n):=#(unrooted planar maps with n edges). Sense-preserving: +. Theorem [VL, 1981]. ( ) n+3 n−1 ′ ] [ A , n odd ( ) ) ( ∑ m+2 n 1 4 2 A′(m) + A′(n)+ ϕ A+(n) = ( ) n−1 n−2 2 2n m m<n ′ A , n even m|n 4 2 • The ﬁrst result of this form. Most signiﬁcant (and unexpected!) was the very existence of such a simple closed formula. A0(n):=#(all rooted planar maps). Tutte’s formula, repeatedly: 0 A (n) = 2 3 n (2n)! n! (n + 2)! = 2 3 n (n + 1)(n + 2) 2n n ; n0 A technique based on quotient maps (= orbifolds) with respect to rotations of order `: spheres with two `-poles. ` : boring technical complications (one or two half-edges). Reproved and generalized later. Rather elementary presently. =2 6-a Euler totient function: unrooted arbitrary planar maps A+(n):=#(unrooted planar maps with n edges). Sense-preserving: +. Theorem [VL, 1981]. ( ) n+3 n−1 ′ ] [ A , n odd ( ) ) ( ∑ m+2 n 1 4 2 A′(m) + A′(n)+ ϕ A+(n) = ( ) n−1 n−2 2 2n m m<n ′ A , n even m|n 4 2 • The ﬁrst result of this form. Most signiﬁcant (and unexpected!) was the very existence of such a simple closed formula. • A′(n):=#(all rooted planar maps). Tutte’s formula, repeatedly: (2n) 2 · 3n 2 · 3n (2n)! = , A (n) = n! (n + 2)! (n + 1)(n + 2) n ′ n≥0 A technique based on quotient maps (= orbifolds) with respect to rotations of order `: spheres with two `-poles. ` : boring technical complications (one or two half-edges). Reproved and generalized later. Rather elementary presently. =2 6-b Euler totient function: unrooted arbitrary planar maps A+(n):=#(unrooted planar maps with n edges). Sense-preserving: +. Theorem [VL, 1981]. ( ) n+3 n−1 ′ ] [ A , n odd ( ) ) ( ∑ m+2 n 1 4 2 A′(m) + A′(n)+ ϕ A+(n) = ( ) n−1 n−2 2 2n m m<n ′ A , n even m|n 4 2 • The ﬁrst result of this form. Most signiﬁcant (and unexpected!) was the very existence of such a simple closed formula. • A′(n):=#(all rooted planar maps). Tutte’s formula, repeatedly: (2n) 2 · 3n 2 · 3n (2n)! = , A (n) = n! (n + 2)! (n + 1)(n + 2) n ′ n≥0 • A technique based on quotient maps (= orbifolds) with respect to rotations of order ℓ: spheres with two ℓ-poles. ℓ = 2: boring technical complications (one or two half-edges). • Reproved and generalized later. Rather elementary presently. 6-c Euler totient function: unrooted non-separable planar maps B +(n):=#(unrooted non-separable (= 2-connected) planar maps). Theorem [VL–T.Walsh, 1983]. ( ) n+1 n+1 ′ [ ] ( )( B , n odd ) ∑ 1 3m−1 n 4 2 B +(n) = B ′(n)+ B ′(m) + ϕ ( ) 3n−4 ′ n 2n m 2 m<n B , n even m|n 16 2 A much more unexpected reductive formula because quotient maps (up to 2 poles) are not necessarily non-separable. B 0(n):=#(rooted non-separable planar maps). [W.Tutte, 1963]: 3n 4 2(3 n 3)! 0 = ; n1 B (n ) = n!(2n 1)! 3(3n 2)(3n 1) n Further. Eulerian planar maps, loopless,. . . : similar formulae. Again: despite that quotient maps do not generally preserve the underlying property (even valency,. . . ). Partially explained. 7 Euler totient function: unrooted non-separable planar maps B +(n):=#(unrooted non-separable (= 2-connected) planar maps). Theorem [VL–T.Walsh, 1983]. ( ) n+1 n+1 ′ [ ] ( )( B , n odd ) ∑ 1 3m−1 n 4 2 B +(n) = B ′(n)+ B ′(m) + ϕ ( ) 3n−4 ′ n 2n m 2 m<n B , n even m|n 16 2 • A much more unexpected reductive formula because quotient maps (up to 2 poles) are not necessarily non-separable. B 0(n):=#(rooted non-separable planar maps). [W.Tutte, 1963]: 3n 4 2(3 n 3)! 0 = ; n1 B (n ) = n!(2n 1)! 3(3n 2)(3n 1) n Further. Eulerian planar maps, loopless,. . . : similar formulae. Again: despite that quotient maps do not generally preserve the underlying property (even valency,. . . ). Partially explained. 7-a Euler totient function: unrooted non-separable planar maps B +(n):=#(unrooted non-separable (= 2-connected) planar maps). Theorem [VL–T.Walsh, 1983]. ( ) n+1 n+1 ′ [ ] ( )( B , n odd ) ∑ 1 3m−1 n 4 2 B +(n) = B ′(n)+ B ′(m) + ϕ ( ) 3n−4 ′ n 2n m 2 m<n B , n even m|n 16 2 • A much more unexpected reductive formula because quotient maps (up to 2 poles) are not necessarily non-separable. • B ′(n):=#(rooted non-separable planar maps). [W.Tutte, 1963]: 2(3n − 3)! = B ′(n) = n! (2n − 1)! (3n) 4 , 3(3n − 2)(3n − 1) n n≥1 Further. Eulerian planar maps, loopless,. . . : similar formulae. Again: despite that quotient maps do not generally preserve the underlying property (even valency,. . . ). Partially explained. 7-b Euler totient function: unrooted non-separable planar maps B +(n):=#(unrooted non-separable (= 2-connected) planar maps). Theorem [VL–T.Walsh, 1983]. ( ) n+1 n+1 ′ [ ] ( )( B , n odd ) ∑ 1 3m−1 n 4 2 B +(n) = B ′(n)+ B ′(m) + ϕ ( ) 3n−4 ′ n 2n m 2 m<n B , n even m|n 16 2 • A much more unexpected reductive formula because quotient maps (up to 2 poles) are not necessarily non-separable. • B ′(n):=#(rooted non-separable planar maps). [W.Tutte, 1963]: 2(3n − 3)! = B ′(n) = n! (2n − 1)! (3n) 4 , 3(3n − 2)(3n − 1) n n≥1 • Further. Eulerian planar maps, loopless,. . . : similar formulae. Again: despite that quotient maps do not generally preserve the underlying property (even valency,. . . ). Partially explained. 7-c M¨ obius function: subgroups of the free group µ(n): the M¨ obius inversion. (Recall: µ(n) is the multiplicative function determined by µ(p) := −1, µ(pa) := 0 for p prime and a > 1.) Fr := the free group of rank r ≥ 2, NFr (n):=#(conjugacy classes of subgroups of index n in Fr ). Also: =#(transitive permutation r-tuples up to joint conjugacy); =#(non-equivalent n-fold coverings of a bordered surface). Theorem [VL, 1971]. X X n r 1 d MFr (m) NFr (n) = n md n ( mjn 1)m+1 dj m MFr (n):=#(n-index subgroups of Fr ). [M.Hall, 1949]: MFr (n) = n n!r 1 nX1 t=1 (n t)!r 1 MFr (t); n > 1; and MFr (1) = 1: Again: unexpectedly simple formula. Reproved subsequently: R.Stanley, G.Jones,. . . Admits a smarter representation: later. 8 M¨ obius function: subgroups of the free group µ(n): the M¨ obius inversion. (Recall: µ(n) is the multiplicative function determined by µ(p) := −1, µ(pa) := 0 for p prime and a > 1.) Fr := the free group of rank r ≥ 2, NFr (n):=#(conjugacy classes of subgroups of index n in Fr ). Also: =#(transitive permutation r-tuples up to joint conjugacy); =#(non-equivalent n-fold coverings of a bordered surface). Theorem [VL, 1971]. ( ) ∑ n 1 ∑ µ d(r−1)m+1 NFr (n) = MFr (m) n m|n md d| n m • MFr (n):=#(n-index subgroups of Fr ). [M.Hall, 1949]: MFr (n) = n · n!r−1 − n−1 ∑ (n − t)!r−1MFr (t), n > 1, and MFr (1) = 1. t=1 Again: unexpectedly simple formula. Reproved subsequently: R.Stanley, G.Jones,. . . Admits a smarter representation: later. 8-a M¨ obius function: subgroups of the free group µ(n): the M¨ obius inversion. (Recall: µ(n) is the multiplicative function determined by µ(p) := −1, µ(pa) := 0 for p prime and a > 1.) Fr := the free group of rank r ≥ 2, NFr (n):=#(conjugacy classes of subgroups of index n in Fr ). Also: =#(transitive permutation r-tuples up to joint conjugacy); =#(non-equivalent n-fold coverings of a bordered surface). Theorem [VL, 1971]. ( ) ∑ n 1 ∑ µ d(r−1)m+1 NFr (n) = MFr (m) n m|n md d| n m • MFr (n):=#(n-index subgroups of Fr ). [M.Hall, 1949]: MFr (n) = n · n!r−1 − n−1 ∑ (n − t)!r−1MFr (t), n > 1, and MFr (1) = 1. t=1 • Again: unexpectedly simple formula. Reproved subsequently: R.Stanley, G.Jones,. . . Admits a smarter representation: later. 8-b M¨ obius function: smooth coverings of surfaces Sγ : a closed orientable surface of genus γ. NSγ (n):=#(non-equivalent smooth n-sheeted coverings over Sγ ). Theorem [A.Mednykh, 1982] (the Hurwitz problem for smooth cov.). X X 1 n NS (n) = M S (m) d n md n (2 mjn 2)m+2 dj m MS (n):=#(n-index subgroups in the fundamental group of S ). M S (n ) is expressed in terms of irreducible characters of the symmetric group Sn [AM, 1982]. . . Subsequently has been generalized to non-orientable surfaces and branched coverings (less transparently). A lot of results. Proof: initially heavy and rather artiﬁcial. Presently this is a particular case of a much more general and clear result! 9 M¨ obius function: smooth coverings of surfaces Sγ : a closed orientable surface of genus γ. NSγ (n):=#(non-equivalent smooth n-sheeted coverings over Sγ ). Theorem [A.Mednykh, 1982] (the Hurwitz problem for smooth cov.). ( ) ∑ 1 ∑ n NSγ (n) = MSγ (m) µ d(2γ−2)m+2 n m|n md d| n m • MSγ (n):=#(n-index subgroups in the fundamental group of Sγ ). MSγ (n) is expressed in terms of irreducible characters of the symmetric group Sn [AM, 1982]. . . Subsequently has been generalized to non-orientable surfaces and branched coverings (less transparently). A lot of results. Proof: initially heavy and rather artiﬁcial. Presently this is a particular case of a much more general and clear result! 9-a M¨ obius function: smooth coverings of surfaces Sγ : a closed orientable surface of genus γ. NSγ (n):=#(non-equivalent smooth n-sheeted coverings over Sγ ). Theorem [A.Mednykh, 1982] (the Hurwitz problem for smooth cov.). ( ) ∑ 1 ∑ n NSγ (n) = MSγ (m) µ d(2γ−2)m+2 n m|n md d| n m • MSγ (n):=#(n-index subgroups in the fundamental group of Sγ ). MSγ (n) is expressed in terms of irreducible characters of the symmetric group Sn [AM, 1982]. . . • Subsequently has been generalized to non-orientable surfaces and branched coverings (less transparently). A lot of results. Proof: initially heavy and rather artiﬁcial. Presently this is a particular case of a much more general and clear result! 9-b M¨ obius function: smooth coverings of surfaces Sγ : a closed orientable surface of genus γ. NSγ (n):=#(non-equivalent smooth n-sheeted coverings over Sγ ). Theorem [A.Mednykh, 1982] (the Hurwitz problem for smooth cov.). ( ) ∑ 1 ∑ n NSγ (n) = MSγ (m) µ d(2γ−2)m+2 n m|n md d| n m • MSγ (n):=#(n-index subgroups in the fundamental group of Sγ ). MSγ (n) is expressed in terms of irreducible characters of the symmetric group Sn [AM, 1982]. . . • Subsequently has been generalized to non-orientable surfaces and branched coverings (less transparently). A lot of results. • Proof: initially heavy and rather artiﬁcial. Presently this is a particular case of a much more general and clear result! 9-c Coverings vs subgroups Recall: For connected coverings of a manifold M with the fundamental group π1(M) • its pointed (rooted) n-fold coverings bijectively correspond to n-index subgroups of π1(M) • its n-fold coverings up to equivalence bijectively correspond to n-index subgroups of π1(M) up to conjugacy. 10 Non-conjugate subgroups vs epimorphisms onto Cm NG(n):=#(conjugacy classes of n-index subgroups of group G). Theorem [AM, 2006]. For any ﬁnitely generated group G, 1 ∑ ∑ NG(n) = |Epi(H, Cm)| n m|n H<G mk=n k H < G denotes summing over subgroups of index k, k |Epi(H, Cm)|:=#(epimorphisms of H onto cyclic Cm) Clariﬁes everything! Easily calculated for (S ), etc. Lemma [G.Jones, 1995]. If jHom(H; Cm)j:=#(homomorphisms), X n jEpi(H; Cn)j = d jHom(H; Cd)j 1 djn Corollary. For a free group Fr ; jEpi(Fr ; Cn)j = P djn n d dr : 11 Non-conjugate subgroups vs epimorphisms onto Cm NG(n):=#(conjugacy classes of n-index subgroups of group G). Theorem [AM, 2006]. For any ﬁnitely generated group G, 1 ∑ ∑ NG(n) = |Epi(H, Cm)| n m|n H<G mk=n k H < G denotes summing over subgroups of index k, k |Epi(H, Cm)|:=#(epimorphisms of H onto cyclic Cm) • Clariﬁes everything! Easily calculated for π1(S), etc. Lemma [G.Jones, 1995]. If |Hom(H, Cm)|:=#(homomorphisms), ∑ (n) |Hom(H, Cd)| |Epi(H, Cn)| = µ d d|n Corollary. For a free group Fr , |Epi(Fr , Cn)| = ∑ d|n ( ) µ nd dr . 11-a Jordan totient functions: deﬁnition ϕk (n):=#(k-tuples jointly coprime to n). Denoted often Jk (n). ϕk (n) := nk ∏ (1 − p−k ), k≥0 p|n prime • Deﬁned as a multiplicative function: ϕk (pa) := pk(a−1)(pk − 1), p prime, a ≥ 1 • In particular, ϕ1 = ϕ (Euler totient). In general, ϕ(n)| ϕk (n), k ≥ 1. Proposition. n k k (n) = d d djn X Easy. Is sometimes used as the deﬁnition of k(n). 12 Jordan totient functions: deﬁnition ϕk (n):=#(k-tuples jointly coprime to n). Denoted often Jk (n). ϕk (n) := nk ∏ (1 − p−k ), k≥0 p|n prime • Deﬁned as a multiplicative function: ϕk (pa) := pk(a−1)(pk − 1), p prime, a ≥ 1 • In particular, ϕ1 = ϕ (Euler totient). In general, ϕ(n)| ϕk (n), k ≥ 1. Proposition. ∑ (n) ϕk (n) = µ dk d d|n • Easy. Is sometimes used as the deﬁnition of ϕk (n). 12-a Jordan totient functions: free groups and coverings, revisited Recall for the conjugacy classes of subgroups of Fr : ∑ ( n ) 1∑ NFr (n) = MFr (m) µ d(r−1)m+1 n md m|n d| mn ( ) ∑ ( n ) (r−1)m+1 n . • As we just saw: µ md d = ϕ(r−1)m+1 m n d| m Therefore NFr X 1 (n ) = n mjn (r 1)m+1 n MFr (m) m – the very ﬁrst enumeration result with Jordan’s functions. Such a considerable rˆole of the Jordan function in this context has been realized (a simple observation) only recently [VL, 2003]. Rather popular presently. Similarly: non-equivalent smooth coverings NS (n), etc. 13 Jordan totient functions: free groups and coverings, revisited Recall for the conjugacy classes of subgroups of Fr : ∑ ( n ) 1∑ NFr (n) = MFr (m) µ d(r−1)m+1 n md m|n d| mn ( ) ∑ ( n ) (r−1)m+1 n . • As we just saw: µ md d = ϕ(r−1)m+1 m n d| m Therefore ( ) 1 ∑ n NFr (n) = MFr (m) ϕ(r−1)m+1 n m|n m – the very ﬁrst enumeration result with Jordan’s functions. • Such a considerable rˆ ole of the Jordan function in this context has been realized (a simple observation) only recently [VL, 2003]. Rather popular presently. Similarly: non-equivalent smooth coverings NS (n), etc. 13-a Jordan totient functions: free groups and coverings, revisited Recall for the conjugacy classes of subgroups of Fr : ∑ ( n ) 1∑ NFr (n) = MFr (m) µ d(r−1)m+1 n md m|n d| mn ( ) ∑ ( n ) (r−1)m+1 n . • As we just saw: µ md d = ϕ(r−1)m+1 m n d| m Therefore ( ) 1 ∑ n NFr (n) = MFr (m) ϕ(r−1)m+1 n m|n m – the very ﬁrst enumeration result with Jordan’s functions. • Such a considerable rˆ ole of the Jordan function in this context has been realized (a simple observation) only recently [VL, 2003]. Rather popular presently. • Similarly: non-equivalent smooth coverings NSγ (n), etc. 13-b Jordan totient functions: variations Modiﬁed “odd” Jordan totient function: ∏ k ϕodd k (n) := n (1 − p−k ), k≥0 p|n p odd prime • Deﬁned as a multiplicative function: a a ϕodd k (p ) := ϕk (p ), p odd; (For comparison: • ϕodd k (n) = ∑ d|n ( ) µ nd dk a ka ϕodd k (2 ) := 2 ϕk (2a) = 2k(a−1) (2k − 1).) n/d odd odd k (n) arises often in counting maps/coverings on non-orientable surfaces, or up to reﬂection [RN–AM &Co, 2008], or so-called circular maps [M.Deryagina, 2013]. Also even (n) . . . Less signiﬁcant. k 14 Jordan totient functions: variations Modiﬁed “odd” Jordan totient function: ∏ k ϕodd k (n) := n (1 − p−k ), k≥0 p|n p odd prime • Deﬁned as a multiplicative function: a a ϕodd k (p ) := ϕk (p ), p odd; (For comparison: • ϕodd k (n) = ∑ d|n ( ) µ nd dk a ka ϕodd k (2 ) := 2 ϕk (2a) = 2k(a−1) (2k − 1).) n/d odd • ϕodd k (n) arises often in counting maps/coverings on non-orientable surfaces, or up to reﬂection [RN–AM &Co, 2008], or so-called circular maps [M.Deryagina, 2013]. • Also ϕeven (n) . . . Less signiﬁcant. k 14-a Jordan functions: subgroups of F2 and dessins d’enfants ( ) ∑ 1 n M (m), n ≥ 1. r := 2. Free gr. F2. Recall: NF2 (n) = n ϕm+1 m F2 m|n NF2 (n)=#(transitive pairs of permutations up to joint conjugacy). =#(non-isomorphic dessins d’enfants (aka hypermaps)). • The initial numerical values: 1, 3, 7, 26, 97, 624, 4163,. . . (A057005 in the On-Line Encyclopedia of Integer Sequences). “7” for n edges (in a bipartite representation): =3 . [Pic.: L.Zapponi, Not. AMS, v.50, 2003] 15 Jordan functions: subgroups of F2 and dessins d’enfants ( ) ∑ 1 n M (m), n ≥ 1. r := 2. Free gr. F2. Recall: NF2 (n) = n ϕm+1 m F2 m|n NF2 (n)=#(transitive pairs of permutations up to joint conjugacy). =#(non-isomorphic dessins d’enfants (aka hypermaps)). • The initial numerical values: 1, 3, 7, 26, 97, 624, 4163,. . . (A057005 in the On-Line Encyclopedia of Integer Sequences). • “7” for n = 3 edges (in a bipartite representation): 3 2 3 1 2 1 Figure 1. The dessins with three edges. The cyclic ordering at each vertex is indicated geometrically. The last two are distinct because of different cyclic orders at the bottom vertex—(1, 3, 2) against (1, 2, 3) . [Pic.: L.Zapponi, Not. AMS, v.50, 2003] 15-a Dedekind totient function: cyclic regular dessins d’enfants 2 (n) . Equivalently: ψ(n). Related to Jordan functions: ψ(n) := ϕϕ(n) ψ(n) := n ∏ (1 + p−1) p|n prime • Deﬁned as a multiplicative function: ψ(pa) := pa−1(p + 1), p prime, a ≥ 1 A dessin [d’enfant] D is called cyclic regular if the group Aut(D) is cyclic and acts regularly on the edges. R(n):=#(non-isomorphic cyclic regular dessins with Theorem [R.Nedela & Co, 2014]. R(n) = n edges). (n) The very ﬁrst appearance of Dedekind’s psi in this context. 16 Dedekind totient function: cyclic regular dessins d’enfants 2 (n) . Equivalently: ψ(n). Related to Jordan functions: ψ(n) := ϕϕ(n) ψ(n) := n ∏ (1 + p−1) p|n prime • Deﬁned as a multiplicative function: ψ(pa) := pa−1(p + 1), p prime, a ≥ 1 • A dessin [d’enfant] D is called cyclic regular if the group Aut(D) is cyclic and acts regularly on the edges. R(n):=#(non-isomorphic cyclic regular dessins with n edges). Theorem [R.Nedela & Co, 2014]. R(n) = ψ(n) • The very ﬁrst appearance of Dedekind’s psi in this context. 16-a Sum/number of divisors: coverings of the torus/Klein bottle Two more multiplicative functions. Exclusive applications. Theorem. [AM, 1988]. For unrooted (connected) coverings of the (2-dim) torus T and the Klein bottle K: NT (n) = σ(n) NK (n) = d(n) if n is odd NK (n) = (5d(n/2) + σ(n/2))/2 if n ≡ 2 (mod 4) NK (n) = . . . if 4|n where σ(n) := d(n) := ∑ d|n ∑ d (the sum of divisors) 1 (the number of divisors) d|n Generalizations: dodd (n), . . . in the same context. 17 Sum/number of divisors: coverings of the torus/Klein bottle Two more multiplicative functions. Exclusive applications. Theorem. [AM, 1988]. For unrooted (connected) coverings of the (2-dim) torus T and the Klein bottle K: NT (n) = σ(n) NK (n) = d(n) if n is odd NK (n) = (5d(n/2) + σ(n/2))/2 if n ≡ 2 (mod 4) NK (n) = . . . if 4|n where σ(n) := d(n) := ∑ d|n ∑ d (the sum of divisors) 1 (the number of divisors) d|n Generalizations: dodd(n), . . . in the same context. 17-a II. A NEW MULTIVARIATE MULTIPLICATIVE FUNCTION Return to unrooted planar maps. Loosely speaking: 1( ′ + A0(n) + A0 (n) = 2n ∑ ) ′ ϕ(ℓ)Ab0(n/ℓ) ℓ≥2, ℓ|2n A+ 0 (n) := #(arbitrary non-isomorphic n-edge planar maps), A′0(n) := #(rooted n-edge planar maps), Ab′0(n/ℓ) := #(rooted planar quotient maps) with respect to rotations of order ℓ. What further for the torus and surfaces of greater genera? The answer expected long ago: a similar reduction to rooted maps, i.e. a summation formula in terms of #(rooted quotient maps) with some coeﬃcients. With respect to all possible ﬁnite automorphisms of the surface. However: Which terms?? Which coeﬃcients?? 18 II. A NEW MULTIVARIATE MULTIPLICATIVE FUNCTION Return to unrooted planar maps. Loosely speaking: 1( ′ + A0(n) + A0 (n) = 2n ∑ ) ′ ϕ(ℓ)Ab0(n/ℓ) ℓ≥2, ℓ|2n A+ 0 (n) := #(arbitrary non-isomorphic n-edge planar maps), A′0(n) := #(rooted n-edge planar maps), Ab′0(n/ℓ) := #(rooted planar quotient maps) with respect to rotations of order ℓ. • What further for the torus and surfaces of greater genera? The answer expected long ago: a similar reduction to rooted maps, i.e. a summation formula in terms of #(rooted quotient maps) with some coeﬃcients. With respect to all possible ﬁnite automorphisms of the surface. However: Which terms?? Which coeﬃcients?? 18-a II. A NEW MULTIVARIATE MULTIPLICATIVE FUNCTION Return to unrooted planar maps. Loosely speaking: 1( ′ + A0(n) + A0 (n) = 2n ∑ ) ′ ϕ(ℓ)Ab0(n/ℓ) ℓ≥2, ℓ|2n A+ 0 (n) := #(arbitrary non-isomorphic n-edge planar maps), A′0(n) := #(rooted n-edge planar maps), Ab′0(n/ℓ) := #(rooted planar quotient maps) with respect to rotations of order ℓ. • What further for the torus and surfaces of greater genera? The answer expected long ago: a similar reduction to rooted maps, i.e. a summation formula in terms of #(rooted quotient maps) with some coeﬃcients. With respect to all possible ﬁnite automorphisms of the surface. • However: Which terms?? Which coeﬃcients?? 18-b Non-planar maps: basic enumeration theorem A far-reaching generalization of the planar case! Sγ : a closed oriented surface of genus γ ≥ 0. A+ γ (n)=#(arbitrary unrooted maps with n edges on Sγ ). Theorem (A.Mednykh–R.Nedela, 2006). (Loosely) ∑ ∑ 1 ∑ + Aγ (n) = |Epio(π1(Ω), Cℓ)| #(rooted q.m.) . . . 2n ℓ|2n Ω∈Orb(Sγ /C ) ℓ Ω=Ω(g;m1 ,...,mr ) • Orb(Sγ /Cℓ): the set of all cyclic orbifolds: quotient spaces by orientation-preserving actions of the cyclic group Cℓ on Sγ . • Orbifold Ω = Ω(g; m1, . . . , mr ) ∈ Orb(Sγ /Cℓ): a closed surface with a distinguished ﬁnite set of branch points of signature (g; m1, . . . , mr ), where g:= its genus, mj := the orders of branch points. 19 Degression on rooted quotient maps #(rooted quotient maps) in the RHS. • Classes of generalized maps: with pendant semi-edges, etc. • Easily reduce to ordinary rooted maps on orientable surfaces: A′δ (n), δ ≤ γ. • Counted long ago with rather heavy formulae: T.Walsh, A.Giorgetti. A great progress presently: much more eﬃcient formulae, numerical results,. . . Outside of our topic. 20 Degression on rooted quotient maps #(rooted quotient maps) in the RHS. • Classes of generalized maps: with pendant semi-edges, etc. • Easily reduce to ordinary rooted maps on orientable surfaces: A′δ (n), δ ≤ γ. • Counted long ago with rather heavy formulae: T.Walsh, A.Giorgetti. • A great progress presently: much more eﬃcient formulae, numerical results,. . . Outside of our topic. 20-a Order preserving epimorphisms ∑ ∑ 1 ∑ + Aγ (n) = |Epio(π1(Ω), Cℓ)| #(rooted q.m.) 2n ℓ|2n Ω∈Orb(Sγ /C ) ℓ Ω=Ω(g;m1 ,...,mr ) Epio(π1(Ω), Cℓ) :={order preserving epimorphisms π1(Ω) → Cℓ}. The fundamental group: ⟨ π1(Ω) = x1, y1, . . . , xg , yg , z1, . . . , zr : g ∏ i=1 [xi, yi] r ∏ j=1 zj = 1, ⟩ mj zj = 1, j = 1, . . . , r • Order preserving epimorphism π1(Ω) → Cℓ: preserves the orders of the periodical generators zj , j = 1, . . . , r (aka: smooth epimorphism, or epimorphism with the torsion-free kernel). jEpi ( ( ); C`)j=?? o 1 21 Order preserving epimorphisms ∑ ∑ 1 ∑ + Aγ (n) = |Epio(π1(Ω), Cℓ)| #(rooted q.m.) 2n ℓ|2n Ω∈Orb(Sγ /C ) ℓ Ω=Ω(g;m1 ,...,mr ) Epio(π1(Ω), Cℓ) :={order preserving epimorphisms π1(Ω) → Cℓ}. The fundamental group: ⟨ π1(Ω) = x1, y1, . . . , xg , yg , z1, . . . , zr : g ∏ i=1 [xi, yi] r ∏ j=1 zj = 1, ⟩ mj zj = 1, j = 1, . . . , r • Order preserving epimorphism π1(Ω) → Cℓ: preserves the orders of the periodical generators zj , j = 1, . . . , r (aka: smooth epimorphism, or epimorphism with the torsion-free kernel). • |Epio(π1(Ω), Cℓ)|=?? 21-a Deﬁnition of the function E(m1, . . . , mr ) Theorem [AM–RN, 2006]. For Ω = Ω(g; m1, . . . , mr ) ∈ Orb(Sγ /Cℓ), |Epio(π1(Ω), Cℓ)| = m2g · ϕ2g (ℓ/m) · E(m1, . . . , mr ) Again, g , the Jordan totient function of order 2g. E (m ; : : : ; mr ) := the number of solutions of the system 9 x + + xr 0 (mod `) > > > = 2 1 (x ; `) = `=m ::: GCD(xr ; `) = `=mr where LCM(m ; : : : ; mr ) = m and mj`. E (m ; : : : ; mr ) does not depend on `. 1 GCD 1 1 > > > ; 1 1 Later I investigated this function and suggested to call it orbicyclic . It is a multivariate generalization of the Euler totient function. 22 Deﬁnition of the function E(m1, . . . , mr ) Theorem [AM–RN, 2006]. For Ω = Ω(g; m1, . . . , mr ) ∈ Orb(Sγ /Cℓ), |Epio(π1(Ω), Cℓ)| = m2g · ϕ2g (ℓ/m) · E(m1, . . . , mr ) • Again, ϕ2g , the Jordan totient function of order 2g. E m1; : : : ; mr := the number of solutions of the system ( ) x1 + + xr 0 (mod `) GCD(x1 ; `) = `=m1 ::: GCD(xr ; `) = `=mr where LCM(m1; : : : ; mr ) = m and mj`. E (m1; : : : ; mr ) does not depend on `. 9 > > > = > > > ; Later I investigated this function and suggested to call it orbicyclic . It is a multivariate generalization of the Euler totient function. 22-a Deﬁnition of the function E(m1, . . . , mr ) Theorem [AM–RN, 2006]. For Ω = Ω(g; m1, . . . , mr ) ∈ Orb(Sγ /Cℓ), |Epio(π1(Ω), Cℓ)| = m2g · ϕ2g (ℓ/m) · E(m1, . . . , mr ) • Again, ϕ2g , the Jordan totient function of order 2g. • E(m1, . . . , mr ) := the number of solutions of the system x1 + · · · + xr ≡ 0 (mod ℓ) GCD(x1 , ℓ) = ℓ/m1 ... GCD(xr , ℓ) = ℓ/mr where LCM(m1, . . . , mr ) = m and m|ℓ. E(m1, . . . , mr ) does not depend on ℓ. Later I investigated this function and suggested to call it orbicyclic . It is a multivariate generalization of the Euler totient function. 22-b Deﬁnition of the function E(m1, . . . , mr ) Theorem [AM–RN, 2006]. For Ω = Ω(g; m1, . . . , mr ) ∈ Orb(Sγ /Cℓ), |Epio(π1(Ω), Cℓ)| = m2g · ϕ2g (ℓ/m) · E(m1, . . . , mr ) • Again, ϕ2g , the Jordan totient function of order 2g. • E(m1, . . . , mr ) := the number of solutions of the system x1 + · · · + xr ≡ 0 (mod ℓ) GCD(x1 , ℓ) = ℓ/m1 ... GCD(xr , ℓ) = ℓ/mr where LCM(m1, . . . , mr ) = m and m|ℓ. E(m1, . . . , mr ) does not depend on ℓ. • Later I investigated this function and suggested to call it orbicyclic . It is a multivariate generalization of the Euler totient function. 22-c Orbicyclic function: initial formula and elementary properties Proposition [AM–RN]. For m := LCM(m1, . . . , mr ), m 1 ∑ cm (k) · · · cmr (k) E(m1, . . . , mr ) = m k=1 1 E(∅) := 1 (m := 1 for r = 0) where cn(k) is the famous Ramanujan (trigonometric) sum: ∑ cn(k) := d (mod n) GCD(d,n)=1 ( 2ikd exp n ) Inconvenient for calculations and study. E (m ; : : : ; mr ) is symmetric. mi = 1 play no role. (m ; : : : ; mr ) is called reduced if all mj > 1. E (m) = 0 for m > 1. E (m ; m ) = 0 for m 6= m . E (m; m) = (m) – just the coeﬃcients in our formula for 1 1 1 2 1 2 counting planar maps! 23 Orbicyclic function: initial formula and elementary properties Proposition [AM–RN]. For m := LCM(m1, . . . , mr ), m 1 ∑ cm (k) · · · cmr (k) E(m1, . . . , mr ) = m k=1 1 E(∅) := 1 (m := 1 for r = 0) where cn(k) is the famous Ramanujan (trigonometric) sum: cn(k) := ∑ d (mod n) GCD(d,n)=1 ( 2ikd exp n ) Inconvenient for calculations and study. • E(m1, . . . , mr ) is symmetric. • mi = 1 play no role. (m1, . . . , mr ) is called reduced if all mj > 1. • E(m) = 0 for m > 1. E(m1, m2) = 0 for m1 ̸= m2. • E(m, m) = ϕ(m) – just the coeﬃcients in our formula for counting planar maps! 23-a Properties of the Ramanujan sum H¨ older’s formula: ( ϕ(n) cn(k) = ( ϕ GCDn(k,n) n )µ GCD(k, n) ) • ϕ and µ again. Ramanujan’s identity: cn(k) = ∑ d|GCD(k,n) ( ) dµ k d Lemma. cn(k) is multiplicative by n and for any p prime and a ≥ 1, a−1 if (p − 1)p a−1 if cpa (k) = −p pa|k pa - k, pa−1|k 0 otherwise • cn(k) is alternating. Unlike E(m1, . . . , mr ). 24 Multivariate multiplicativity A multivariate function g = g(m1, . . . , mr ) is called multiplicative if g(1, . . . , 1) = 1 and g(m1, . . . , mr ) = g(m′1, . . . , m′r ) · g(m′′1, . . . , m′′r ) whenever mj = m′j m′′j , j = 1, . . . , r, and GCD(M ′, M ′′) = 1, where ∏ ∏ M ′ = j m′j and M ′′ = j m′′j . Theorem [VL, 2010]. The orbicyclic function E ( m 1 ; : : : ; mr ) is multiplicative. Thus: reduction to prime powers, i.e. E (m1; : : : ; mr ) is determined by its values when m=GCD(m1; : : : ; mr ) is a prime power. 25 Multivariate multiplicativity A multivariate function g = g(m1, . . . , mr ) is called multiplicative if g(1, . . . , 1) = 1 and g(m1, . . . , mr ) = g(m′1, . . . , m′r ) · g(m′′1, . . . , m′′r ) whenever mj = m′j m′′j , j = 1, . . . , r, and GCD(M ′, M ′′) = 1, where ∏ ∏ M ′ = j m′j and M ′′ = j m′′j . Theorem [VL, 2010]. The orbicyclic function E(m1, . . . , mr ) is multiplicative. • Thus: reduction to prime powers, i.e. E(m1, . . . , mr ) is determined by its values when m=GCD(m1, . . . , mr ) is a prime power. 25-a The primary case m = p a. Main eﬃcient formula Introduce three parameters: r = rp, s = sp, v = vp. For mj = paj , aj > 0, j = 1, . . . , r, without loss of generality, a1 = . . . = a s = a > as+1 · · · ≥ a r > 0. s := the multiplicity of the greatest exponent. v := ∑ (aj − 1) = j≥2 Theorem [VL, 2010]. hs(x) ∑ aj − r − a + 1 j≥1 E (pa1 ; : : : ; par ) = (p 1)r is the (“chromatic” ) polynomial h1(x) = 0 h2(x) = 1 h3(x) = x h4(x) = x2 h5(x) = (x s+1pv h s hs(x) := 2 3x + 3 2)(x 2x + 2) ( p) (x 1)s 1+( x 1)s : 2 26 The primary case m = p a. Main eﬃcient formula Introduce three parameters: r = rp, s = sp, v = vp. For mj = paj , aj > 0, j = 1, . . . , r, without loss of generality, a1 = . . . = a s = a > as+1 · · · ≥ a r > 0. s := the multiplicity of the greatest exponent. v := ∑ j≥2 (aj − 1) = ∑ aj − r − a + 1 j≥1 Theorem [VL, 2010]. E(pa1 , . . . , par ) = (p − 1)r−s+1pv hs(p) (x−1)s−1+(−1)s hs(x) is the (“chromatic” ) polynomial hs(x) := . x h1(x) = 0 h2(x) = 1 h3(x) = x − 2 h4(x) = x2 − 3x + 3 h5(x) = (x − 2)(x2 − 2x + 2) 26-a Further properties • E(pa1 , pa2 , . . . , par ) is non-negative and integer. • E(pa1 , pa2 , . . . , par ) vanishes iﬀ s = 1 or p = 2 and s is odd. • ϕ(m)|E(m1, . . . , mr ). fr (n) := E (n; : : : ; n) (the diagonal) is | {z } r the multiplicative function determined by fr (pa) = (p 1)p r ( 1)(a 1) hr (p); p prime; a 1: One more generalization of the Euler totient function: in particular f2 n n. ( )= ( ) () I called fr n (by certain historical reasons) the Rademacher–Brauer totient. 27 Further properties • E(pa1 , pa2 , . . . , par ) is non-negative and integer. • E(pa1 , pa2 , . . . , par ) vanishes iﬀ s = 1 or p = 2 and s is odd. • ϕ(m)|E(m1, . . . , mr ). • fr (n) := E(n, . . . , n) (the diagonal) is | {z } r the multiplicative function determined by fr (pa) = (p − 1)p(r−1)(a−1)hr (p), p prime, a ≥ 1. One more generalization of the Euler totient function: in particular f2(n) = ϕ(n). I called fr (n) (by certain historical reasons) the Rademacher–Brauer totient. 27-a Non-vanishing conditions for E(m1, . . . , mr ) Recall: |Epio(π1(Ω), Cℓ)| = m2g · ϕ2g (ℓ/m) · E(m1, . . . , mr ) (g; m ; : : : ; mr); where all mj 2; Orb(S =C`); ` 2; iﬀ its parameterssatisfy Corollary. 1. An orbifold exists and belongs to the Riemann–Hurwitz condition 1 2 2 = ` 2 2g r P j =1 1 1 mj 2g (`=m) and E (m1; : : : ; mr ) do not vanish. 2. 2g (`=m) = 0 iﬀ (e1) m - ` [by deﬁnition, any f (n) := 0 for a non-integer argument] or (e2) g = 0 and ` > m. 3. E (m1; : : : ; mr ) = 0 iﬀ (e3) sp = 1 for some odd prime pjm or (e4) 2jm and s2 is odd. [s2:= the multiplicity of the highest 2-power] and both (e3) & (e4) are equivalent to Harvey’s conditions (1966) on branching data of ﬁnite cyclic groups acting on Riemann surfaces. Unexpected enumerative reﬁnement. 28 Non-vanishing conditions for E(m1, . . . , mr ) Recall: |Epio(π1(Ω), Cℓ)| = m2g · ϕ2g (ℓ/m) · E(m1, . . . , mr ) Corollary. 1. An orbifold Ω(g; m1, . . . , mr ), where all mj ≥ 2, exists and belongs to Orb(Sγ /Cℓ), ℓ ≥ 2, iﬀ its parameters satisfy the Riemann–Hurwitz condition ( r ( )) ∑ 1 2 − 2γ = ℓ 2 − 2g − 1 − mj j=1 and both ϕ2g (ℓ/m) and E(m1, . . . , mr ) do not vanish. 2. ϕ2g (ℓ/m) = 0 iﬀ (e1) m - ℓ [by deﬁnition, any f (n) := 0 for a non-integer argument] or (e2) g = 0 and ℓ > m. ( =1 2j )=0 3. E m1; : : : ; mr iﬀ (e3) sp for some odd prime p m or (e4) m and s2 is odd. [s2:= the multiplicity of the highest 2-power] j (e3) & (e4) are equivalent to Harvey’s conditions (1966) on branching data of ﬁnite cyclic groups acting on Riemann surfaces. Unexpected enumerative reﬁnement. 28-a Non-vanishing conditions for E(m1, . . . , mr ) Recall: |Epio(π1(Ω), Cℓ)| = m2g · ϕ2g (ℓ/m) · E(m1, . . . , mr ) Corollary. 1. An orbifold Ω(g; m1, . . . , mr ), where all mj ≥ 2, exists and belongs to Orb(Sγ /Cℓ), ℓ ≥ 2, iﬀ its parameters satisfy the Riemann–Hurwitz condition ( r ( )) ∑ 1 2 − 2γ = ℓ 2 − 2g − 1 − mj j=1 and both ϕ2g (ℓ/m) and E(m1, . . . , mr ) do not vanish. 2. ϕ2g (ℓ/m) = 0 iﬀ (e1) m - ℓ [by deﬁnition, any f (n) := 0 for a non-integer argument] or (e2) g = 0 and ℓ > m. 3. E(m1, . . . , mr ) = 0 iﬀ (e3) sp = 1 for some odd prime p|m or (e4) 2|m and s2 is odd. [s2:= the multiplicity of the highest 2-power] • (e3) & (e4) are equivalent to Harvey’s conditions (1966) on branching data of ﬁnite cyclic groups acting on Riemann surfaces. Unexpected enumerative reﬁnement. 28-b Toth’s summation formula A compact number-theoretic representation: Theorem [L´ aszl´ o T´ oth, 2011]. (m ) ( mr ) d1 · · · dr 1 E(m1, . . . , mr ) = µ ···µ . LCM(d1 , . . . , dr ) d d r 1 d1 |m1,...,dr |mr ∑ Further generalizations . . . [L.T´ oth, 2014]: an explicit formula for the generalized average Es = Es(m1, . . . , mr ) := 1 m ∑ ms+1 k=1 kscm1 (k) · · · cmr (k) (so that E0 = E). 29 Concluding remark: what are (generalized) totients? • A lot of arithmetic functions (and their families) are called totients. Why? There is no generally accepted precise deﬁnition. Etymology: http://mathforum.org/kb/message.jspa?messageID=65511 J.Sylvester [1879] for Euler’s : from Latin “totiens” “that many” (similarly to “quotiens” “how many”). Totients are often enumerators. Tentative deﬁnition. An arithmetic function f (n) is called a totient iﬀ it is multiplicative and the values f p ; f p2 ; f p3 ,. . . () ( ) ( ) form a geometric progression for each prime p. Suggested (in an equivalent form) by R.Vaidyanathaswamy in the classical paper [The theory of multiplicative arithmetic functions, Trans. AMS, 1931]. All totient functionds are such! It makes sense to restrict this deﬁnition: non-completely multiplicative functions with non-negative integer values. . . 30 Concluding remark: what are (generalized) totients? • A lot of arithmetic functions (and their families) are called totients. Why? There is no generally accepted precise deﬁnition. • Etymology: http://mathforum.org/kb/message.jspa?messageID=65511 J.Sylvester [1879] for Euler’s ϕ: from Latin “totiens” ≈“that many” (similarly to “quotiens” ≈“how many”). Totients are often enumerators. Tentative deﬁnition. An arithmetic function f (n) is called a totient iﬀ it is multiplicative and the values f p ; f p2 ; f p3 ,. . . () ( ) ( ) form a geometric progression for each prime p. Suggested (in an equivalent form) by R.Vaidyanathaswamy in the classical paper [The theory of multiplicative arithmetic functions, Trans. AMS, 1931]. All totient functionds are such! It makes sense to restrict this deﬁnition: non-completely multiplicative functions with non-negative integer values. . . 30-a Concluding remark: what are (generalized) totients? • A lot of arithmetic functions (and their families) are called totients. Why? There is no generally accepted precise deﬁnition. • Etymology: http://mathforum.org/kb/message.jspa?messageID=65511 J.Sylvester [1879] for Euler’s ϕ: from Latin “totiens” ≈“that many” (similarly to “quotiens” ≈“how many”). Totients are often enumerators. • Tentative deﬁnition. An arithmetic function f (n) is called a totient iﬀ it is multiplicative and the values f (p), f (p2), f (p3),. . . form a geometric progression for each prime p. Suggested (in an equivalent form) by R.Vaidyanathaswamy in the classical paper [The theory of multiplicative arithmetic functions, Trans. AMS, 1931]. All totient functionds are such! It makes sense to restrict this deﬁnition: non-completely multiplicative functions with non-negative integer values. . . 30-b Concluding remark: what are (generalized) totients? • A lot of arithmetic functions (and their families) are called totients. Why? There is no generally accepted precise deﬁnition. • Etymology: http://mathforum.org/kb/message.jspa?messageID=65511 J.Sylvester [1879] for Euler’s ϕ: from Latin “totiens” ≈“that many” (similarly to “quotiens” ≈“how many”). Totients are often enumerators. • Tentative deﬁnition. An arithmetic function f (n) is called a totient iﬀ it is multiplicative and the values f (p), f (p2), f (p3),. . . form a geometric progression for each prime p. • Suggested (in an equivalent form) by R.Vaidyanathaswamy in the classical paper [The theory of multiplicative arithmetic functions, Trans. AMS, 1931]. All totient functionds are such! • It makes sense to restrict this deﬁnition: non-completely multiplicative functions with non-negative integer values . . . 30-c

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